Introduction
We consider either the following problem
{ − Δ u − γ ( − Δ ) s u = ∣ u ∣ r − 2 u in R n , u ≢ 0 in R n , \begin{align}
\left\{\begin{array}{rl}
-\Delta u - \gamma \, (-\Delta)^s u = |u|^{r-2}u & \textrm{ in } \mathbb{R}^n,\\
u\not\equiv0 & \textrm{ in } \mathbb{R}^n,
\end{array}\right. \tag*{(1.1)}
\end{align} { − Δ u − γ ( − Δ ) s u = ∣ u ∣ r − 2 u u ≡ 0 in R n , in R n , (1.1) or its version settled in Ω ⊂ R n \Omega \subset \mathbb{R}^n Ω ⊂ R n n ≥ 3 n \geq 3 n ≥ 3 ∂ Ω \partial \Omega ∂ Ω
{ − Δ u − γ ( − Δ ) s u = ∣ u ∣ r − 2 u in Ω , u ≢ 0 in Ω , u = 0 in R n ∖ Ω . \begin{align}
\left\{\begin{array}{rl}
-\Delta u - \gamma \, (-\Delta)^s u = |u|^{r-2}u & \textrm{ in } \Omega,\\
u\not\equiv0 & \textrm{ in } \Omega,\\
u= 0 & \textrm{ in } \mathbb{R}^{n} \setminus \Omega.
\end{array}\right. \tag*{(1.2)}
\end{align} ⎩ ⎨ ⎧ − Δ u − γ ( − Δ ) s u = ∣ u ∣ r − 2 u u ≡ 0 u = 0 in Ω , in Ω , in R n ∖ Ω. (1.2) Throughout the paper, we strengthen the regularity assumptions on ∂ Ω \partial\Omega ∂ Ω ( − Δ ) s (-\Delta)^s ( − Δ ) s s ∈ ( 0 , 1 ) s\in(0,1) s ∈ ( 0 , 1 ) u u u
( − Δ ) s u ( x ) : = ∫ R n 2 u ( x ) − u ( x + y ) − u ( x − y ) ∣ y ∣ n + 2 s d y . (-\Delta)^s u(x):=
\int_{\mathbb R^n}
\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\,dy. ( − Δ ) s u ( x ) := ∫ R n ∣ y ∣ n + 2 s 2 u ( x ) − u ( x + y ) − u ( x − y ) d y . In the whole-space problem, γ > 0 \gamma>0 γ > 0
0 < γ < C e m b ( s ) and 1 < r ≤ 2 ∗ : = 2 n n − 2 , 0<\gamma<C_{\rm emb}(s)\qquad\text{and}\qquad 1<r\le 2^*:=\frac{2n}{n-2}, 0 < γ < C emb ( s ) and 1 < r ≤ 2 ∗ := n − 2 2 n , where C e m b ( s ) C_{\rm emb}(s) C emb ( s )
C e m b ( s ) [ u ] s 2 ≤ ∥ ∇ u ∥ L 2 ( Ω ) 2 , u ∈ H 0 1 ( Ω ) , C_{\rm emb}(s)[u]_s^2\le \|\nabla u\|_{L^2(\Omega)}^2,
\qquad u\in H_0^1(\Omega), C emb ( s ) [ u ] s 2 ≤ ∥∇ u ∥ L 2 ( Ω ) 2 , u ∈ H 0 1 ( Ω ) , see, precisely, (2.9) . Concerning the constant C e m b ( s ) C_{\rm emb}(s) C emb ( s ) C e m b ( s ) C_{\rm emb}(s) C emb ( s ) λ \lambda λ
{ − Δ Φ = λ ( − Δ ) s Φ in Ω , Φ = 0 in R n ∖ Ω \begin{cases}
-\Delta\Phi=\lambda(-\Delta)^s\Phi & \text{in }\Omega,\\
\Phi=0 & \text{in }\mathbb R^n\setminus\Omega
\end{cases} { − ΔΦ = λ ( − Δ ) s Φ Φ = 0 in Ω , in R n ∖ Ω admits a nonzero weak solution (in the variational sense specified in § 2 ). Such values of λ \lambda λ C e m b ( s ) C_{\rm emb}(s) C emb ( s )
This terminology always refers to the above generalized problem; Dirichlet eigenvalues and eigenfunctions of − Δ -\Delta − Δ
As it is clear, both (1.2) and (1.1) can be interpreted as Lane-Emden-type equations for the operator
L γ , s : = − Δ − γ ( − Δ ) s , γ > 0. \begin{equation}
\mathcal L_{\gamma,s}:=-\Delta-\gamma(-\Delta)^s, \qquad \gamma >0. \tag*{(1.3)}
\end{equation} L γ , s := − Δ − γ ( − Δ ) s , γ > 0. (1.3) Before entering into the details of the results we will prove, let us mention that L γ , s \mathcal L_{\gamma,s} L γ , s
L μ : = ∫ [ 0 , 1 ] ( − Δ ) s d μ ( s ) , \begin{equation*}
L_{\mu} := \int_{[0,1]}(-\Delta)^s \, d\mu(s),
\end{equation*} L μ := ∫ [ 0 , 1 ] ( − Δ ) s d μ ( s ) , where μ = μ + − μ − \mu = \mu^+ - \mu^- μ = μ + − μ − [68 , 69 , 70 , 71 , 72 , 74 , 85 , 48 ] . We also mention [82 , 80 , 61 ] where the authors refer to L γ , s \mathcal L_{\gamma,s} L γ , s [63 , 60 ] in relation with the maximum principle) which is also experiencing an intense revival (see e.g., among the many,[62 , 57 , 86 , 49 , 45 , 46 , 65 , 77 , 78 , 52 ] ). All this interest can be easily understood in view of the possible applications to biology and other real–life situations, see e.g. [75 , 67 ] .
- Motivations. The classical Lane-Emden equation reads
− Δ u = u p , \begin{equation}
-\Delta u = u^p, \tag*{(1.4)}
\end{equation} − Δ u = u p , (1.4) and it has been the subject of several lines of research. One of the interesting phenomena related to this PDE is the existence of a critical exponent in the following sense: the numbers
\begin{equation*}
\inf \left\{p\in \mathbb{R}: \text{$\exists$ a positive solution to \eqref{eq:LANE-EMDEN} in $\mathbb{R}^n$}\right\},
\end{equation*}
and, considering a bounded, smooth and star-shaped set Ω \Omega Ω
\begin{equation*}
\sup \left\{p\in \mathbb{R}: \text{$\exists$ a positive solution $u$ to \eqref{eq:LANE-EMDEN} in $\Omega$ with $u = 0$ on $\partial\Omega$}\right\},
\end{equation*}
do coincide with 2 ∗ − 1 2^{\ast}-1 2 ∗ − 1 [79 , 59 ] . We briefly mention that a totally analogous phenomenon appears in the fractional counterpart of (1.4) , namely replacing − Δ -\Delta − Δ ( − Δ ) s (-\Delta)^s ( − Δ ) s 2 s ∗ − 1 = n + 2 s n − 2 s 2^{\ast}_s -1 = \tfrac{n+2s}{n-2s} 2 s ∗ − 1 = n − 2 s n + 2 s
A natural question related to the study of the operator of mixed order given by
L : = − Δ + ( − Δ ) s , s ∈ ( 0 , 1 ) , \mathcal{L}:= -\Delta + (-\Delta)^s, \qquad s \in (0,1), L := − Δ + ( − Δ ) s , s ∈ ( 0 , 1 ) , has been whether a similar notion of critical exponent does exist for L \mathcal{L} L [47 ] , where the authors showed that even if it exists, it cannot coincide with the one of − Δ -\Delta − Δ [50 ] . The starting point therein is the following simple observation: let Ω ⊆ R n \Omega \subseteq \mathbb{R}^{n} Ω ⊆ R n s ∈ ( 0 , 1 ) s \in (0,1) s ∈ ( 0 , 1 ) S n , s ( Ω ) > 0 \mathcal{S}_{n,s}(\Omega)>0 S n , s ( Ω ) > 0
S n , s ( Ω ) ∥ u ∥ L 2 ∗ ( R n ) 2 ≤ ∥ ∇ u ∥ L 2 ( R n ) 2 + [ u ] s 2 , \begin{equation}
\mathcal{S}_{n,s}(\Omega) \|u\|^{2}_{L^{2^{\ast}}(\mathbb{R}^{n})} \leq \|\nabla u\|^{2}_{L^{2}(\mathbb{R}^{n})} + [u]^2_{s}, \tag*{(1.5)}
\end{equation} S n , s ( Ω ) ∥ u ∥ L 2 ∗ ( R n ) 2 ≤ ∥∇ u ∥ L 2 ( R n ) 2 + [ u ] s 2 , (1.5) for every u ∈ C 0 ∞ ( Ω ) u \in C^{\infty}_{0}(\Omega) u ∈ C 0 ∞ ( Ω )
In principle, the constant S n , s ( Ω ) \mathcal{S}_{n,s}(\Omega) S n , s ( Ω ) n n n s s s Ω \Omega Ω [50 Theorem 1.1] ), indeed
S n , s ( Ω ) = S n \begin{equation*}
\mathcal{S}_{n,s}(\Omega) = S_{n}
\end{equation*} S n , s ( Ω ) = S n where S n S_n S n
Moreover, the energy functional
E ( u ) : = 1 2 ∫ R n ∣ ∇ u ∣ 2 d x + 1 2 [ u ] s 2 \begin{equation}
E(u) := \dfrac{1}{2} \int_{\mathbb{R}^n}|\nabla u|^2 \, dx + \dfrac{1}{2}[u]^{2}_s \tag*{(1.6)}
\end{equation} E ( u ) := 2 1 ∫ R n ∣∇ u ∣ 2 d x + 2 1 [ u ] s 2 (1.6) exhibits a lack of scaling invariance, namely
E ( u t ) = 1 2 ∫ R n ∣ ∇ u ∣ 2 d x + t 2 s − 2 2 [ u ] s 2 , \begin{equation}
E(u_{t}) = \dfrac{1}{2} \int_{\mathbb{R}^n}|\nabla u|^2 \, dx + \dfrac{t^{2s-2}}{2}[u]^{2}_s, \tag*{(1.7)}
\end{equation} E ( u t ) = 2 1 ∫ R n ∣∇ u ∣ 2 d x + 2 t 2 s − 2 [ u ] s 2 , (1.7) where u t ( x ) : = t ( n − 2 ) / 2 u ( t x ) u_t(x):=t^{(n-2)/2}u(tx) u t ( x ) := t ( n − 2 ) /2 u ( t x ) u u u R n \mathbb R^n R n ∥ ∇ u ∥ L 2 ( R n ) \|\nabla u\|_{L^2(\mathbb R^n)} ∥∇ u ∥ L 2 ( R n ) (1.5) is never achieved, see [50 Theorem 1.2] .
Since the first variation of the energy functional E E E (1.6) is given by the local-nonlocal operator L \mathcal{L} L [58 ] ) and with leading operator L \mathcal{L} L [50 , 54 , 55 ] .
Let us now come back to the topic of interest of the present paper. The first result is the following
Theorem 1.1. Let n ≥ 3 n\ge3 n ≥ 3 s ∈ ( 0 , 1 ) s\in(0,1) s ∈ ( 0 , 1 ) γ > 0 \gamma>0 γ > 0 u ∈ H 1 ( R n ) u\in H^1(\mathbb{R}^n) u ∈ H 1 ( R n ) − Δ u − γ ( − Δ ) s u ≥ 0 in R n
-\Delta u-\gamma(-\Delta)^s u\ge0
\qquad\text{in }\mathbb{R}^n
− Δ u − γ ( − Δ ) s u ≥ 0 in R n r > 1 r>1 r > 1 − Δ u − γ ( − Δ ) s u = u r − 1 in R n
-\Delta u-\gamma(-\Delta)^s u=u^{r-1}
\qquad\text{in }\mathbb{R}^n
− Δ u − γ ( − Δ ) s u = u r − 1 in R n H 1 ( R n ) H^1(\mathbb{R}^n) H 1 ( R n )
In this case the wrong sign of the fractional term yields a rather rigid Liouville property. The argument is based on a positive test function on which the operator L γ , s \mathcal L_{\gamma,s} L γ , s [84 ] and the references therein for an accurate description of the method and the several applications found by the authors) and it has also been recently applied in a local–nonlocal parabolic setting in [53 ] to detect the corresponding Fujita exponent.
As a final comment, let us also compare our Theorem 1.1 with the recent results appeared in [81 ] (for the linear operator − Δ + ( − Δ ) s -\Delta + (-\Delta)^s − Δ + ( − Δ ) s [64 ] (for the sum of possibly fractional p p p n n − 2 s \tfrac{n}{n-2s} n − 2 s n [81 ] .
The second part of the paper is devoted to the study of (1.2) settled in a bounded set Ω \Omega Ω § 2 , and in particular to (2.6) , the exhaustive discussion on the function space X 1 , 2 ( Ω ) \mathcal{X}^{1,2}(\Omega) X 1 , 2 ( Ω )
X 1 , 2 ( Ω ) = { u ∈ H 1 ( R n ) : u ∣ Ω ∈ H 0 1 ( Ω ) and u ≡ 0 a.e. in R n ∖ Ω } . \mathcal{X}^{1,2}(\Omega) = \big\{u\in H^1(\mathbb{R}^n):\,\text{$u|_\Omega\in H_0^1(\Omega)$ and
$u\equiv 0$ a.e.\,in $\mathbb{R}^n\setminus\Omega$}\big\}. X 1 , 2 ( Ω ) = { u ∈ H 1 ( R n ) : u ∣ Ω ∈ H 0 1 ( Ω ) and u ≡ 0 a.e. in R n ∖ Ω } . A first result in the Sobolev critical case (i.e. for r = 2 ∗ r=2^{\ast} r = 2 ∗ [51 ] and we believe it is worth briefly recalling it. Let us first define the functional
Q γ ( u ) : = ∫ Ω ∣ ∇ u ∣ 2 d x − γ [ u ] s 2 , u ∈ X 1 , 2 ( Ω ) , \begin{equation}
Q_{\gamma}(u) := \int_{\Omega}|\nabla u|^{2} \, dx - \gamma \, [u]^{2}_{s}, \quad u \in \mathcal{X}^{1,2}(\Omega), \tag*{(1.8)}
\end{equation} Q γ ( u ) := ∫ Ω ∣∇ u ∣ 2 d x − γ [ u ] s 2 , u ∈ X 1 , 2 ( Ω ) , (1.8) and let us consider the following constrained minimization problem
S ( γ ) : = inf { Q γ ( u ) : u ∈ X 1 , 2 ( Ω ) , ∥ u ∥ L 2 ∗ ( Ω ) = 1 } . \begin{equation}
S(\gamma) := \inf \big\{ Q_{\gamma}(u) : u \in \mathcal{X}^{1,2}(\Omega), \|u\|_{L^{2^{\ast}}(\Omega)}=1 \big\}. \tag*{(1.9)}
\end{equation} S ( γ ) := inf { Q γ ( u ) : u ∈ X 1 , 2 ( Ω ) , ∥ u ∥ L 2 ∗ ( Ω ) = 1 } . (1.9) The main result obtained in [51 ] reads as follows:
Theorem 1.2 ([51 ] - Theorem 1.1). Let n ≥ 3 n\geq 3 n ≥ 3 Ω ⊂ R n \Omega \subset \mathbb{R}^{n} Ω ⊂ R n
If n ≥ 5 n\geq 5 n ≥ 5 (1.9) is achieved for every γ ∈ ( 0 , C e m b ( s ) ) \gamma\in (0,C_{\rm emb}(s)) γ ∈ ( 0 , C emb ( s ))
If n = 3 , 4 n = 3,4 n = 3 , 4 γ ∗ ∈ [ 0 , C e m b ( s ) ) \gamma^*\in [0,C_{\rm emb}(s)) γ ∗ ∈ [ 0 , C emb ( s )) (1.9) is achieved for every γ ∈ ( γ ∗ , C e m b ( s ) ) \gamma\in(\gamma^*,C_{\rm emb}(s)) γ ∈ ( γ ∗ , C emb ( s ))
In both cases, if u 0 ∈ X 1 , 2 ( Ω ) u_0\in \mathcal{X}^{1,2}(\Omega) u 0 ∈ X 1 , 2 ( Ω ) (1.9) , then u ~ = S ( γ ) 1 / ( 2 ∗ − 2 ) u 0
\widetilde{u} = S(\gamma)^{1/(2^*-2)}u_0
u = S ( γ ) 1/ ( 2 ∗ − 2 ) u 0 (1.2) with r = 2 ∗ r=2^{\ast} r = 2 ∗
As it is clear, the sign of the solution has not been investigated, and it can neither be a simple consequence of a maximum principle nor be based on a trivial test function argument (i.e. testing the equation by u + u^+ u + [74 ] to prove, for small fractional orders, the sign-definiteness of generalized eigenfunctions associated with the least eigenvalue; see § 2 for more details. We stress that this approach is kind of transversal and naturally appears in other problems, see e.g. [83 ] , where the authors study a shape optimization problem associated with a functional which exhibits competitive terms like in our case.
As the argument in [74 ] suggests, there is actually no reason to restrict our interest only to the Sobolev critical case r = 2 ∗ r=2^{\ast} r = 2 ∗ 0 < γ < C e m b ( s ) 0<\gamma<C_{\rm emb}(s) 0 < γ < C emb ( s ) 1 < r ≤ 2 ∗ 1<r\le2^* 1 < r ≤ 2 ∗
S r ( γ ) : = inf { Q γ ( u ) : u ∈ X 1 , 2 ( Ω ) , ∥ u ∥ L r ( Ω ) = 1 } . \begin{equation}
S_r(\gamma):=\inf\left\{Q_\gamma(u):u\in \mathcal{X}^{1,2}(\Omega),\
\left\lVert u\right\rVert_{L^r(\Omega)}=1\right\}. \tag*{(1.11)}
\end{equation} S r ( γ ) := inf { Q γ ( u ) : u ∈ X 1 , 2 ( Ω ) , ∥ u ∥ L r ( Ω ) = 1 } . (1.11) It is pretty easy to see that the infimum in (1.11) is achieved in the Sobolev subcritical case r ∈ ( 1 , 2 ∗ ) r \in (1,2^{\ast}) r ∈ ( 1 , 2 ∗ ) 4.1 for the standard details).
Once this is established, we have the following
Theorem 1.3. Let n ≥ 3 n\ge3 n ≥ 3 Ω ⊂ R n \Omega\subset\mathbb{R}^n Ω ⊂ R n s ∈ ( 0 , 1 ) s\in(0,1) s ∈ ( 0 , 1 ) 1 < r ≤ 2 ∗ 1<r\le2^* 1 < r ≤ 2 ∗ γ \gamma γ S r ( γ ) S_r(\gamma) S r ( γ ) u γ , r u_{\gamma,r} u γ , r r < 2 ∗ r<2^* r < 2 ∗ ( 0 , C e m b ( s ) ) (0,C_{\rm emb}(s)) ( 0 , C emb ( s )) r = 2 ∗ r=2^* r = 2 ∗ [51 Theorem 1.1] . Then the following assertions hold.
S r ( γ ) ⟶ 0 as γ → C e m b ( s ) − .
S_r(\gamma)\longrightarrow0
\qquad\text{as }\gamma\to C_{\rm emb}(s)^-.
S r ( γ ) ⟶ 0 as γ → C emb ( s ) − .
If γ k → C e m b ( s ) − \gamma_k\to C_{\rm emb}(s)^- γ k → C emb ( s ) − u γ k , r u_{\gamma_k,r} u γ k , r u ∞ ∈ X 1 , 2 ( Ω ) u_\infty\in \mathcal{X}^{1,2}(\Omega) u ∞ ∈ X 1 , 2 ( Ω ) u γ k , r ⟶ u ∞ strongly in X 1 , 2 ( Ω ) ,
u_{\gamma_k,r}\longrightarrow u_\infty
\qquad\text{strongly in }\mathcal{X}^{1,2}(\Omega),
u γ k , r ⟶ u ∞ strongly in X 1 , 2 ( Ω ) , ∥ u ∞ ∥ L r ( Ω ) = 1 , ∥ ∇ u ∞ ∥ L 2 ( Ω ) 2 = C e m b ( s ) [ u ∞ ] s 2 .
\left\lVert u_\infty\right\rVert_{L^r(\Omega)}=1,
\qquad
\left\lVert \nabla u_\infty\right\rVert_{L^2(\Omega)}^2
=C_{\rm emb}(s)[u_\infty]_s^2.
∥ u ∞ ∥ L r ( Ω ) = 1 , ∥ ∇ u ∞ ∥ L 2 ( Ω ) 2 = C emb ( s ) [ u ∞ ] s 2 . u ∞ u_\infty u ∞ L r L^r L r C e m b ( s ) C_{\rm emb}(s) C emb ( s ) C e m b ( s ) C_{\rm emb}(s) C emb ( s ) Φ s \Phi_s Φ s L r L^r L r u ∞ = Φ s or u ∞ = − Φ s .
u_\infty=\Phi_s\qquad\text{or}\qquad u_\infty=-\Phi_s.
u ∞ = Φ s or u ∞ = − Φ s .
For every R ∈ [ 1 , + ∞ ) R\in[1,+\infty) R ∈ [ 1 , + ∞ ) γ R < C e m b ( s ) \gamma_R<C_{\rm emb}(s) γ R < C emb ( s ) M R > 0 M_R>0 M R > 0 ∥ u γ , r ∥ L R ( Ω ) ≤ M R
\left\lVert u_{\gamma,r}\right\rVert_{L^R(\Omega)}\le M_R
∥ u γ , r ∥ L R ( Ω ) ≤ M R γ ∈ ( γ R , C e m b ( s ) ) \gamma\in(\gamma_R,C_{\rm emb}(s)) γ ∈ ( γ R , C emb ( s )) u γ , r u_{\gamma,r} u γ , r
We notice that the first part i) of the above Theorem 1.3 has been already noticed for r = 2 ∗ r=2^{\ast} r = 2 ∗ [51 ] . Regarding assertion ii), the normalized minimizers converge, along subsequences, to generalized eigenfunctions associated with C e m b ( s ) C_{\rm emb}(s) C emb ( s ) (1.12) . Under the additional connectedness and boundary regularity assumptions of Lemma 2.4 , the corresponding L r L^r L r Φ s \Phi_s Φ s s s s
Theorem 1.4. Assume in addition that 0 < s < 1 / 2 0<s<1/2 0 < s < 1/2 Ω \Omega Ω C 1 , 1 C^{1,1} C 1 , 1 q ∈ [ 2 , + ∞ ) q\in[2,+\infty) q ∈ [ 2 , + ∞ ) γ q < C e m b ( s ) \gamma_q<C_{\rm emb}(s) γ q < C emb ( s ) C q > 0 C_q>0 C q > 0 ∥ u γ , r ∥ W 2 , q ( Ω ) ≤ C q
\left\lVert u_{\gamma,r}\right\rVert_{W^{2,q}(\Omega)}\le C_q
∥ u γ , r ∥ W 2 , q ( Ω ) ≤ C q γ ∈ ( γ q , C e m b ( s ) ) \gamma\in(\gamma_q,C_{\rm emb}(s)) γ ∈ ( γ q , C emb ( s )) u γ , r u_{\gamma,r} u γ , r q > n q>n q > n α = 1 − n / q \alpha=1-n/q α = 1 − n / q ∥ u γ , r ∥ C 1 , α ( Ω ‾ ) ≤ C q
\left\lVert u_{\gamma,r}\right\rVert_{C^{1,\alpha}(\overline\Omega)}\le C_q
∥ u γ , r ∥ C 1 , α ( Ω ) ≤ C q γ ∈ ( γ q , C e m b ( s ) ) \gamma\in(\gamma_q,C_{\rm emb}(s)) γ ∈ ( γ q , C emb ( s )) u γ , r u_{\gamma,r} u γ , r
Moreover, if γ k → C e m b ( s ) − \gamma_k\to C_{\rm emb}(s)^- γ k → C emb ( s ) − u γ k , r → u ∞ u_{\gamma_k,r}\to u_\infty u γ k , r → u ∞ X 1 , 2 ( Ω ) \mathcal{X}^{1,2}(\Omega) X 1 , 2 ( Ω ) 1.3 , then u γ k , r ⟶ u ∞ strongly in W 2 , q ( Ω )
u_{\gamma_k,r}\longrightarrow u_\infty
\qquad\text{strongly in }W^{2,q}(\Omega)
u γ k , r ⟶ u ∞ strongly in W 2 , q ( Ω ) q ∈ [ 2 , + ∞ ) q\in[2,+\infty) q ∈ [ 2 , + ∞ ) q > n q>n q > n u γ k , r ⟶ u ∞ strongly in C 1 ( Ω ‾ ) .
u_{\gamma_k,r}\longrightarrow u_\infty
\qquad\text{strongly in }C^1(\overline\Omega).
u γ k , r ⟶ u ∞ strongly in C 1 ( Ω ) . C e m b ( s ) C_{\rm emb}(s) C emb ( s ) u ∞ = ± Φ s u_\infty=\pm\Phi_s u ∞ = ± Φ s (1.12) .
We stress that the C 1 , 1 C^{1,1} C 1 , 1 1.4 is used to apply the global boundary regularity theory from [74 ] . Once the strong C 1 C^1 C 1 s s s
Theorem 1.6. Assume that Ω \Omega Ω C 1 , 1 C^{1,1} C 1 , 1 1 < r ≤ 2 ∗ 1<r\le2^* 1 < r ≤ 2 ∗ s 1 ∈ ( 0 , 1 / 2 ) s_1\in(0,1/2) s 1 ∈ ( 0 , 1/2 ) s ∈ ( 0 , s 1 ) s\in(0,s_1) s ∈ ( 0 , s 1 ) γ ‾ r , s < C e m b ( s )
\overline\gamma_{r,s}<C_{\rm emb}(s)
γ r , s < C emb ( s ) γ ∈ ( γ ‾ r , s , C e m b ( s ) ) \gamma\in(\overline\gamma_{r,s},C_{\rm emb}(s)) γ ∈ ( γ r , s , C emb ( s )) u γ , r u_{\gamma,r} u γ , r Ω \Omega Ω − 1 -1 − 1 u γ , r > 0 in Ω .
u_{\gamma,r}>0\qquad\text{in }\Omega.
u γ , r > 0 in Ω.
The connectedness assumption is needed in order to apply the small-s s s 2.4 . Indeed, on disconnected domains the spectral picture for the generalized eigenvalue problem introduced above may be substantially different. In particular, the result in [74 Theorem 1.4] shows that generalized eigenfunctions associated with C e m b ( s ) C_{\rm emb}(s) C emb ( s ) Φ s \Phi_s Φ s
Once Theorem 1.6 is established, from the attainment result in Proposition 4.1 we immediately deduce the following
Corollary 1.7. Assume the hypotheses of Theorem 1.6 , and let 1 < r ≤ 2 ∗ 1<r\le2^* 1 < r ≤ 2 ∗ r ≠ 2 r\neq2 r = 2 s > 0 s>0 s > 0 γ < C e m b ( s ) \gamma<C_{\rm emb}(s) γ < C emb ( s ) C e m b ( s ) C_{\rm emb}(s) C emb ( s ) { − Δ v − γ ( − Δ ) s v = ∣ v ∣ r − 2 v in Ω , v = 0 in R n ∖ Ω
\begin{cases}
-\Delta v - \gamma(-\Delta)^s v = |v|^{r-2}v&\text{in }\Omega,\\
v=0&\text{in }\mathbb{R}^n\setminus\Omega
\end{cases}
{ − Δ v − γ ( − Δ ) s v = ∣ v ∣ r − 2 v v = 0 in Ω , in R n ∖ Ω u γ , r u_{\gamma,r} u γ , r v γ : = S r ( γ ) 1 / ( r − 2 ) u γ , r
v_\gamma:=S_{r}(\gamma)^{1/(r-2)}u_{\gamma,r}
v γ := S r ( γ ) 1/ ( r − 2 ) u γ , r (1.13) .
The above Corollary 1.7 holds for every r ∈ ( 1 , 2 ∗ ] , r ≠ 2 r\in(1,2^*],\,r\neq 2 r ∈ ( 1 , 2 ∗ ] , r = 2 s s s L γ , s \mathcal L_{\gamma,s} L γ , s
− Δ + ( − Δ ) s . -\Delta+(-\Delta)^s. − Δ + ( − Δ ) s . However, if one wonders about the existence of a critical exponent for L γ , s \mathcal{L}_{\gamma,s} L γ , s
Theorem 1.8. Let Ω ⊂ R n \Omega \subset \mathbb{R}^n Ω ⊂ R n C 2 C^2 C 2 ∂ Ω \partial \Omega ∂ Ω ν \nu ν ∂ Ω \partial \Omega ∂ Ω s ∈ ( 0 , 1 / 2 ) s \in (0,1/2) s ∈ ( 0 , 1/2 ) p > n + 2 n − 2 p > \tfrac{n+2}{n-2} p > n − 2 n + 2
Then, there are no nontrivial solutions u ∈ C 2 ( Ω ‾ ) ∩ C ( R n ) u\in C^2(\overline\Omega)\cap C(\mathbb R^n) u ∈ C 2 ( Ω ) ∩ C ( R n ) { − Δ u − γ ( − Δ ) s u = ∣ u ∣ p − 1 u in Ω , u = 0 in R n ∖ Ω ,
\begin{cases}
-\Delta u - \gamma\, (-\Delta)^s u =|u|^{p-1}u &\text{in }\Omega,\\
u=0&\text{in }\mathbb{R}^n\setminus\Omega,
\end{cases}
{ − Δ u − γ ( − Δ ) s u = ∣ u ∣ p − 1 u u = 0 in Ω , in R n ∖ Ω , 0 < γ < C e m b ( n − 2 2 − n p + 1 n − 2 s 2 − n p + 1 ) .
0 < \gamma < C_{\rm emb}\, \left(\dfrac{\tfrac{n-2}{2}-\tfrac{n}{p+1}}{\tfrac{n-2s}{2}-\tfrac{n}{p+1}} \right).
0 < γ < C emb ( 2 n − 2 s − p + 1 n 2 n − 2 − p + 1 n ) .
Notice that the upper bound
C e m b ( n − 2 2 − n p + 1 n − 2 s 2 − n p + 1 ) , \begin{equation*}
C_{\rm emb}\, \left(\dfrac{\tfrac{n-2}{2}-\tfrac{n}{p+1}}{\tfrac{n-2s}{2}-\tfrac{n}{p+1}} \right),
\end{equation*} C emb ( 2 n − 2 s − p + 1 n 2 n − 2 − p + 1 n ) , can be close to C e m b C_{\rm emb} C emb s s s n n n § 6 for further comments on Theorem 1.8 , especially regarding the assumptions on s s s
- Plan of the paper. The paper is organized as follows. In § 2 we collect the functional setting and the preliminary results, including the spectral information from [74 ] that is crucial for the positivity argument.
In § 3 we prove the whole-space Liouville property stated in Theorem 1.1 by constructing a suitable rescaling of the fractional Sobolev bubble.
§ 4 is devoted to the variational problem in bounded domains: we prove attainment in the subcritical case, convergence of the normalized minimizers to the eigenspace associated with C e m b ( s ) C_{\rm emb}(s) C emb ( s ) γ → C e m b ( s ) − \gamma\to C_{\rm emb}(s)^- γ → C emb ( s ) − 1.3 .
In § 5 we establish the global W 2 , q W^{2,q} W 2 , q C 1 C^1 C 1 1.4 .
Finally, § 6 contains the positivity argument leading to Theorem 1.6 , and concludes with the Pohozaev-based supercritical nonexistence result of Theorem 1.8 .
- Acknowledgements. The Authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). S.B. and E.V. are partially supported by the Indam-GNAMPA project Esistenza, regolarità e studio delle proprietà qualitative per problemi nonlineari.
The Authors thank B. Ruffini for the fruitful discussions on the topics of [83 ] .
- AI disclosure. The authors used GPT-6 to assist with the detection of typographical errors, consistency checks, and the verification of bibliographic references. All mathematical arguments and proofs were developed, written, and verified by the authors, who take full responsibility for the entire content of the manuscript.
- Conflict of interest. The authors state no conflict of interest.
- Data availability statement. There are no data associated with this research.
Preliminary results
Throughout the paper, we tacitly exploit all the notation listed below; we thus refer the Reader to this list for any non-standard notation encountered.
Ω ⊂ R n \Omega\subset\mathbb R^n Ω ⊂ R n ∂ Ω \partial\Omega ∂ Ω
Given any set E E E ∣ E ∣ |E| ∣ E ∣ n n n
Given any set E E E E c E^c E c R n \mathbb{R}^n R n E c : = R n ∖ E E^c:=\mathbb{R}^n\setminus E E c := R n ∖ E
Given x 0 ∈ R n x_0 \in \mathbb{R}^n x 0 ∈ R n r > 0 r>0 r > 0 B r ( x 0 ) B_{r}(x_0) B r ( x 0 ) x 0 x_0 x 0 r r r x 0 = 0 x_0=0 x 0 = 0 B r B_r B r
Given any two vectors v , w ∈ R n v,w \in \mathbb{R}^{n} v , w ∈ R n v ⋅ w v\cdot w v ⋅ w
Given any p ∈ [ 1 , n ) p \in [1,n) p ∈ [ 1 , n ) p ∗ p^{*} p ∗ R n \mathbb{R}^n R n
p ∗ : = n p n − p . p^{*} := \frac{np}{n-p}. p ∗ := n − p n p . Given any r ∈ ( 1 , + ∞ ) r \in (1,+\infty) r ∈ ( 1 , + ∞ ) r ′ r' r ′ r r r
r ′ : = r r − 1 . r' := \frac{r}{r-1}. r ′ := r − 1 r . Given a reflexive Banach space V V V V ′ V' V ′ ⟨ ⋅ , ⋅ ⟩ \langle \cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩
Given a measurable set E E E χ E \chi_{E} χ E
Given any function v v v v v v
v + : = max { v , 0 } and v − : = − min { v , 0 } . \begin{equation}
v^{+}:= \max\{v,0\} \quad \textrm{ and } \quad v^{-}:= -\min\{v,0\}. \tag*{(2.1)}
\end{equation} v + := max { v , 0 } and v − := − min { v , 0 } . (2.1) Functional setting. We now set the adequate functional setting to study problem (1.2) . We refer to e.g. [50 Section 2] for more details.
Let s ∈ ( 0 , 1 ) s\in (0,1) s ∈ ( 0 , 1 ) u : R n → R u:\mathbb{R}^n\to\mathbb{R} u : R n → R
[ u ] s : = ( ∬ R 2 n ∣ u ( x ) − u ( y ) ∣ 2 ∣ x − y ∣ n + 2 s d x d y ) 1 / 2 , \begin{equation}
[u]_s := \left(\iint_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\right)^{1/2}, \tag*{(2.2)}
\end{equation} [ u ] s := ( ∬ R 2 n ∣ x − y ∣ n + 2 s ∣ u ( x ) − u ( y ) ∣ 2 d x d y ) 1/2 , (2.2) and we refer to [ u ] s [u]_s [ u ] s u u u s s s
E s ( u , v ) : = ∬ R 2 n ( u ( x ) − u ( y ) ) ( v ( x ) − v ( y ) ) ∣ x − y ∣ n + 2 s d x d y , \begin{equation}
\mathcal E_s(u,v):=\iint_{\mathbb{R}^{2n}}
\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy, \tag*{(2.3)}
\end{equation} E s ( u , v ) := ∬ R 2 n ∣ x − y ∣ n + 2 s ( u ( x ) − u ( y )) ( v ( x ) − v ( y )) d x d y , (2.3) so that [ u ] s 2 = E s ( u , u ) [u]_s^2= \mathcal E_s(u,u) [ u ] s 2 = E s ( u , u )
Let ∅ ≠ Ω ⊆ R n \varnothing\neq \Omega\subseteq\mathbb{R}^n ∅ = Ω ⊆ R n n ≥ 3 n\geq 3 n ≥ 3 X 1 , 2 ( Ω ) \mathcal{X}^{1,2}(\Omega) X 1 , 2 ( Ω )
X 1 , 2 ( Ω ) : = C 0 ∞ ( Ω ) ‾ ρ ( ⋅ ) \begin{equation*}
\mathcal{X}^{1,2}(\Omega) := \overline{C_0^\infty(\Omega)}^{\,\,\rho(\cdot)}
\end{equation*} X 1 , 2 ( Ω ) := C 0 ∞ ( Ω ) ρ ( ⋅ ) i.e. as the completion of C 0 ∞ ( Ω ) C_0^\infty(\Omega) C 0 ∞ ( Ω )
ρ ( u ) : = ( ∥ ∇ u ∥ L 2 ( R n ) 2 + [ u ] s 2 ) 1 / 2 . \begin{equation}
\rho(u) := \left(\|\nabla u\|^2_{L^2(\mathbb{R}^n)}+[u]^2_s\right)^{1/2}. \tag*{(2.4)}
\end{equation} ρ ( u ) := ( ∥∇ u ∥ L 2 ( R n ) 2 + [ u ] s 2 ) 1/2 . (2.4) We stress that the norm ρ ( ⋅ ) \rho(\cdot) ρ ( ⋅ )
⟨ u , v ⟩ ρ : = ∫ R n ∇ u ⋅ ∇ v d x + ∬ R 2 n ( u ( x ) − u ( y ) ) ( v ( x ) − v ( y ) ) ∣ x − y ∣ n + 2 s d x d y , \langle u,v\rangle_{\rho} := \int_{\mathbb{R}^n}\nabla u\cdot\nabla v\,dx
+ \iint_{\mathbb{R}^{2n}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy, ⟨ u , v ⟩ ρ := ∫ R n ∇ u ⋅ ∇ v d x + ∬ R 2 n ∣ x − y ∣ n + 2 s ( u ( x ) − u ( y )) ( v ( x ) − v ( y )) d x d y , and therefore X 1 , 2 ( Ω ) \mathcal{X}^{1,2}(\Omega) X 1 , 2 ( Ω )
It is known, see e.g. [66 ] , that the Sobolev space H 1 ( R n ) H^{1}(\mathbb{R}^n) H 1 ( R n ) H s ( R n ) H^{s}(\mathbb{R}^n) H s ( R n ) s ∈ ( 0 , 1 ) s\in (0,1) s ∈ ( 0 , 1 ) C > 0 C>0 C > 0 s s s n n n
C [ u ] s 2 ≤ ∥ u ∥ H 1 ( R n ) 2 , \begin{equation}
C \, [u]^{2}_{s}\leq \|u\|^2_{H^{1}(\mathbb{R}^n)}, \tag*{(2.5)}
\end{equation} C [ u ] s 2 ≤ ∥ u ∥ H 1 ( R n ) 2 , (2.5) for every u ∈ H 1 ( R n ) u \in H^{1}(\mathbb{R}^n) u ∈ H 1 ( R n )
For our purposes, we will also be interested in the case Ω \Omega Ω (2.5) , and combining it with the classical Poincaré inequality, we find that ρ ( ⋅ ) \rho(\cdot) ρ ( ⋅ ) H 1 H^1 H 1 R n \mathbb{R}^n R n C 0 ∞ ( Ω ) C^\infty_0(\Omega) C 0 ∞ ( Ω )
X 1 , 2 ( Ω ) = C 0 ∞ ( Ω ) ‾ ∥ ⋅ ∥ H 1 ( R n ) = { u ∈ H 1 ( R n ) : u ∣ Ω ∈ H 0 1 ( Ω ) and u ≡ 0 a.e. in R n ∖ Ω } . \begin{equation}
\begin{aligned}
\mathcal{X}^{1,2}(\Omega) & = \overline{C_0^\infty(\Omega)}^{\,\,\|\cdot\|_{H^1(\mathbb{R}^n)}} \\
& = \big\{u\in H^1(\mathbb{R}^n):\,\text{$u|_\Omega\in H_0^1(\Omega)$ and
$u\equiv 0$ a.e.\,in $\mathbb{R}^n\setminus\Omega$}\big\}.
\end{aligned} \tag*{(2.6)}
\end{equation} X 1 , 2 ( Ω ) = C 0 ∞ ( Ω ) ∥ ⋅ ∥ H 1 ( R n ) = { u ∈ H 1 ( R n ) : u ∣ Ω ∈ H 0 1 ( Ω ) and u ≡ 0 a.e. in R n ∖ Ω } . (2.6) We recall the following standard facts.
We now give the precise definition of weak solution to (1.2) .
Definition 2.2. We say that a function u : R n → R u:\mathbb{R}^n\to\mathbb{R} u : R n → R (1.2) if it satisfies the following properties:
u ∈ X 1 , 2 ( Ω ) u\in\mathcal{X}^{1,2}(\Omega) u ∈ X 1 , 2 ( Ω ) u ≢ 0 u\not\equiv 0 u ≡ 0 Ω \Omega Ω
for every test function v ∈ X 1 , 2 ( Ω ) v\in \mathcal{X}^{1,2}(\Omega) v ∈ X 1 , 2 ( Ω )
∫ Ω ∇ u ⋅ ∇ v d x − γ ∬ R 2 n ( u ( x ) − u ( y ) ) ( v ( x ) − v ( y ) ) ∣ x − y ∣ n + 2 s d x d y = ∫ Ω ∣ u ∣ r − 2 u v d x . \begin{align*}
\int_{\Omega}\nabla u\cdot\nabla v\,dx
-\gamma \, \iint_{\mathbb{R}^{2n}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy = \int_\Omega |u|^{r-2}uv\,dx.
\end{align*} ∫ Ω ∇ u ⋅ ∇ v d x − γ ∬ R 2 n ∣ x − y ∣ n + 2 s ( u ( x ) − u ( y )) ( v ( x ) − v ( y )) d x d y = ∫ Ω ∣ u ∣ r − 2 uv d x . A generalized eigenvalue problem. Starting from (2.5) and applying the classical Poincaré inequality, we can easily infer the existence of a positive constant C > 0 C > 0 C > 0 Ω \Omega Ω
C [ u ] s 2 ≤ ∫ Ω ∣ ∇ u ∣ 2 d x = ∥ ∇ u ∥ L 2 ( Ω ) 2 , \begin{equation}
C[{u}]^2_s\leq \int_\Omega|\nabla u|^2\,dx = \|\nabla u\|^2_{L^2(\Omega)}, \tag*{(2.8)}
\end{equation} C [ u ] s 2 ≤ ∫ Ω ∣∇ u ∣ 2 d x = ∥∇ u ∥ L 2 ( Ω ) 2 , (2.8) for every u ∈ H 0 1 ( Ω ) u\in H_0^1(\Omega) u ∈ H 0 1 ( Ω )
C e m b ( s ) : = inf { ∥ ∇ u ∥ L 2 ( Ω ) 2 : u ∈ H 0 1 ( Ω ) , [ u ] s 2 = 1 } . \begin{equation}
C_{\rm emb}(s) := \inf \big\{ \|\nabla u\|^{2}_{L^2(\Omega)} : u\in H_0^1(\Omega),\,[u]_{s}^2 =1\big\}. \tag*{(2.9)}
\end{equation} C emb ( s ) := inf { ∥∇ u ∥ L 2 ( Ω ) 2 : u ∈ H 0 1 ( Ω ) , [ u ] s 2 = 1 } . (2.9) As observed in [51 Remark 2.1(iii)] , the infimum in (2.9) is achieved. When no confusion arises, we will simply write C e m b C_{\rm emb} C emb s s s (2.9) reads
∫ Ω ∇ Φ ⋅ ∇ φ d x = C e m b E s ( Φ , φ ) for every φ ∈ X 1 , 2 ( Ω ) , \begin{equation}
\int_\Omega \nabla\Phi\cdot\nabla\varphi\,dx
=C_{\rm emb}\, \mathcal E_s(\Phi,\varphi)
\qquad\text{for every }\varphi\in \mathcal{X}^{1,2}(\Omega), \tag*{(2.10)}
\end{equation} ∫ Ω ∇Φ ⋅ ∇ φ d x = C emb E s ( Φ , φ ) for every φ ∈ X 1 , 2 ( Ω ) , (2.10) for every minimizer Φ \Phi Φ (2.9) . As usual, (2.10) can be seen as the weak formulation (in X 1 , 2 ( Ω ) \mathcal{X}^{1,2}(\Omega) X 1 , 2 ( Ω )
− Δ Φ = C e m b ( − Δ ) s Φ in Ω , Φ = 0 in R n ∖ Ω . -\Delta\Phi=C_{\rm emb}(-\Delta)^s\Phi\quad\text{in }\Omega,
\qquad
\Phi=0\quad\text{in }\mathbb{R}^n\setminus\Omega. − ΔΦ = C emb ( − Δ ) s Φ in Ω , Φ = 0 in R n ∖ Ω. The latter is a particular instance of a problem recently studied in [74 ] and referred to therein as a nonlocal eigenvalue problem. To justify the terminology anticipated in the Introduction, let λ \lambda λ
Φ ∈ X 1 , 2 ( Ω ) ∖ { 0 } \Phi\in \mathcal{X}^{1,2}(\Omega)\setminus\{0\} Φ ∈ X 1 , 2 ( Ω ) ∖ { 0 } be an associated generalized eigenfunction.
Testing its weak equation with Φ \Phi Φ
λ = ∥ ∇ Φ ∥ L 2 ( Ω ) 2 [ Φ ] s 2 ≥ C e m b . \lambda
=\frac{\|\nabla\Phi\|_{L^2(\Omega)}^2}{[\Phi]_s^2}
\ge C_{\rm emb}. λ = [ Φ ] s 2 ∥∇Φ ∥ L 2 ( Ω ) 2 ≥ C emb . Since the infimum in (2.9) is attained and its minimizers satisfy (2.10) , C e m b C_{\rm emb} C emb Φ ∈ X 1 , 2 ( Ω ) \Phi\in \mathcal{X}^{1,2}(\Omega) Φ ∈ X 1 , 2 ( Ω )
∥ ∇ Φ ∥ L 2 ( Ω ) 2 = C e m b [ Φ ] s 2 . \|\nabla\Phi\|_{L^2(\Omega)}^2=C_{\rm emb}[\Phi]_s^2. ∥∇Φ ∥ L 2 ( Ω ) 2 = C emb [ Φ ] s 2 . In particular, the minimizers of (2.9) are precisely the elements of this eigenspace satisfying [ Φ ] s = 1 [\Phi]_s=1 [ Φ ] s = 1
The next lemma collects some results obtained in [74 ] . To be more precise, it is a direct consequence of the perturbative spectral results in [74 Theorems 1.2 and 1.3 and the proof of Theorem 1.3] , together with the global regularity estimate in [74 Theorem 1.8] .
Lemma 2.4. Let Ω \Omega Ω C 1 , 1 C^{1,1} C 1 , 1 s 0 ∈ ( 0 , 1 / 2 ) s_0\in(0,1/2) s 0 ∈ ( 0 , 1/2 ) s ∈ ( 0 , s 0 ) s\in(0,s_0) s ∈ ( 0 , s 0 ) C e m b C_{\rm emb} C emb − Δ Φ = λ ( − Δ ) s Φ in Ω , Φ = 0 in R n ∖ Ω ,
-\Delta\Phi=\lambda(-\Delta)^s\Phi\quad\text{in }\Omega,
\qquad
\Phi=0\quad\text{in }\mathbb R^n\setminus\Omega,
− ΔΦ = λ ( − Δ ) s Φ in Ω , Φ = 0 in R n ∖ Ω , C e m b C_{\rm emb} C emb Ω \Omega Ω
Moreover, for every fixed r ∈ ( 1 , 2 ∗ ] r\in(1,2^*] r ∈ ( 1 , 2 ∗ ] φ 1 \varphi_1 φ 1 − Δ -\Delta − Δ ∥ φ 1 ∥ L r ( Ω ) = 1 \|\varphi_1\|_{L^r(\Omega)}=1 ∥ φ 1 ∥ L r ( Ω ) = 1 Φ s \Phi_s Φ s C e m b C_{\rm emb} C emb ∥ Φ s ∥ L r ( Ω ) = 1 \|\Phi_s\|_{L^r(\Omega)}=1 ∥ Φ s ∥ L r ( Ω ) = 1 Φ s → φ 1 strongly in C 1 ( Ω ‾ ) as s → 0 + .
\Phi_s\to \varphi_1
\qquad\text{strongly in }C^1(\overline\Omega)
\quad\text{as }s\to0^+.
Φ s → φ 1 strongly in C 1 ( Ω ) as s → 0 + .
A regularity estimate. We end this section proving the following elementary lower-order estimate, valid on the whole space. Despite this, it will be useful both for (1.2) and (1.1) .
Lemma 2.7. There exists a constant C = C ( n , s ) > 0 C=C(n,s)>0 C = C ( n , s ) > 0 v ∈ H 1 ( R n ) v\in H^1(\mathbb{R}^n) v ∈ H 1 ( R n ) ρ > 0 \rho>0 ρ > 0 [ v ] s 2 ≤ C ρ 2 − 2 s ∥ ∇ v ∥ 2 2 + C ρ − 2 s ∥ v ∥ 2 2 .
[v]_s^2\le
C\rho^{2-2s}\left\lVert \nabla v\right\rVert_2^2
+C\rho^{-2s}\left\lVert v\right\rVert_2^2.
[ v ] s 2 ≤ C ρ 2 − 2 s ∥ ∇ v ∥ 2 2 + C ρ − 2 s ∥ v ∥ 2 2 . ε > 0 \varepsilon>0 ε > 0 C ε > 0 C_\varepsilon>0 C ε > 0 [ v ] s 2 ≤ ε ∥ ∇ v ∥ 2 2 + C ε ∥ v ∥ 2 2 for every v ∈ H 1 ( R n ) .
[v]_s^2\le\varepsilon\left\lVert \nabla v\right\rVert_2^2
+C_\varepsilon\left\lVert v\right\rVert_2^2
\qquad\text{for every }v\in H^1(\mathbb{R}^n).
[ v ] s 2 ≤ ε ∥ ∇ v ∥ 2 2 + C ε ∥ v ∥ 2 2 for every v ∈ H 1 ( R n ) .
Proof. Writing h = y − x h=y-x h = y − x
[ v ] s 2 = ∫ R n ∥ v ( ⋅ + h ) − v ∥ 2 2 ∣ h ∣ n + 2 s d h . [v]_s^2=
\int_{\mathbb{R}^n}\frac{\left\lVert v(\cdot+h)-v\right\rVert_2^2}{|h|^{n+2s}}\,dh. [ v ] s 2 = ∫ R n ∣ h ∣ n + 2 s ∥ v ( ⋅ + h ) − v ∥ 2 2 d h . For ∣ h ∣ ≤ ρ |h|\le\rho ∣ h ∣ ≤ ρ
∥ v ( ⋅ + h ) − v ∥ 2 ≤ ∣ h ∣ ∥ ∇ v ∥ 2 , \left\lVert v(\cdot+h)-v\right\rVert_2\le |h|\left\lVert \nabla v\right\rVert_2, ∥ v ( ⋅ + h ) − v ∥ 2 ≤ ∣ h ∣ ∥ ∇ v ∥ 2 , whereas, for every h ∈ R n h\in\mathbb{R}^n h ∈ R n
∥ v ( ⋅ + h ) − v ∥ 2 ≤ 2 ∥ v ∥ 2 . \left\lVert v(\cdot+h)-v\right\rVert_2\le2\left\lVert v\right\rVert_2. ∥ v ( ⋅ + h ) − v ∥ 2 ≤ 2 ∥ v ∥ 2 . Splitting the integral into ∣ h ∣ ≤ ρ |h|\le\rho ∣ h ∣ ≤ ρ ∣ h ∣ > ρ |h|>\rho ∣ h ∣ > ρ (2.13) . Formula (2.14) follows by choosing ρ \rho ρ
Proof of Theorem 1.1
In this section we prove Theorem 1.1 . We first fix the notion of weak supersolution used below and then construct a positive function on which L γ , s \mathcal L_{\gamma,s} L γ , s
Throughout the section, n ≥ 3 n\ge3 n ≥ 3 s ∈ ( 0 , 1 ) s\in(0,1) s ∈ ( 0 , 1 ) γ > 0 \gamma>0 γ > 0
Definition 3.1. We say that u ∈ H 1 ( R n ) u\in H^1(\mathbb{R}^n) u ∈ H 1 ( R n )
L γ , s u = 0 in R n \begin{equation}
\mathcal L_{\gamma,s}u = 0
\qquad\text{in }\mathbb{R}^n \tag*{(3.1)}
\end{equation} L γ , s u = 0 in R n (3.1) if u ≥ 0 u\ge0 u ≥ 0 R n \mathbb{R}^n R n
∫ R n ∇ u ⋅ ∇ φ d x − γ E s ( u , φ ) ≥ 0 ∀ φ ∈ C 0 ∞ ( R n ) , φ ≥ 0. \begin{equation*}
\int_{\mathbb{R}^n}\nabla u\cdot\nabla\varphi\,dx
-\gamma \, \mathcal E_s(u,\varphi)\ge0\qquad\forall\,\,\varphi\in C_0^\infty(\mathbb{R}^n),\,\varphi\geq 0.
\end{equation*} ∫ R n ∇ u ⋅ ∇ φ d x − γ E s ( u , φ ) ≥ 0 ∀ φ ∈ C 0 ∞ ( R n ) , φ ≥ 0. We now set
β : = n − 2 s 2 \begin{equation}
\beta:=\frac{n-2s}{2} \tag*{(3.2)}
\end{equation} β := 2 n − 2 s (3.2) and define
ψ ( x ) : = 1 ( 1 + ∣ x ∣ 2 ) β = 1 ( 1 + ∣ x ∣ 2 ) ( n − 2 s ) / 2 . \begin{equation}
\psi(x):=\frac{1}{(1+|x|^2)^\beta}
=\frac{1}{(1+|x|^2)^{(n-2s)/2}}. \tag*{(3.3)}
\end{equation} ψ ( x ) := ( 1 + ∣ x ∣ 2 ) β 1 = ( 1 + ∣ x ∣ 2 ) ( n − 2 s ) /2 1 . (3.3) The latter is a scalar multiple of what is usually called fractional Sobolev bubble. Therefore, with the choice of β \beta β (3.2) , we have both a known sign and decay of ( − Δ ) s ψ (-\Delta)^{s}\psi ( − Δ ) s ψ
Lemma 3.2. There exists a constant a n , s > 0 a_{n,s}>0 a n , s > 0 ( − Δ ) s ψ ( x ) = a n , s ( 1 + ∣ x ∣ 2 ) − ( n + 2 s ) / 2 for every x ∈ R n .
(-\Delta)^s\psi(x)
=a_{n,s}(1+|x|^2)^{-(n+2s)/2}
\qquad\text{for every }x\in\mathbb{R}^n.
( − Δ ) s ψ ( x ) = a n , s ( 1 + ∣ x ∣ 2 ) − ( n + 2 s ) /2 for every x ∈ R n . ( − Δ ) s ψ > 0 (-\Delta)^s\psi>0 ( − Δ ) s ψ > 0 R n \mathbb{R}^n R n C 0 = C 0 ( n , s ) > 0 C_0=C_0(n,s)>0 C 0 = C 0 ( n , s ) > 0 − Δ ψ ≤ C 0 ( − Δ ) s ψ in R n .
-\Delta\psi\le C_0(-\Delta)^s\psi
\qquad\text{in }\mathbb{R}^n.
− Δ ψ ≤ C 0 ( − Δ ) s ψ in R n .
Proof. The explicit formula (3.4) follows from [76 formulas (6.5)–(6.6)] . We briefly spell out the specialization, also in order to keep track of the sign. In the notation of [76 ] , consider
f β ~ ( x ) : = − ( 1 + ∣ x ∣ 2 ) − β ~ / 2 , β ~ : = n − 2 s , f_{\widetilde\beta}(x)
:=-(1+|x|^2)^{-\widetilde\beta/2},
\qquad
\widetilde\beta:=n-2s, f β ( x ) := − ( 1 + ∣ x ∣ 2 ) − β /2 , β := n − 2 s , and take α : = 2 s \alpha:=2s α := 2 s [76 ] gives
− ( − Δ ) α / 2 f β ~ ( x ) = C ( α , β ~ , n ) 2 F 1 ( a , b ; c ; − ∣ x ∣ 2 ) , -(-\Delta)^{\alpha/2}f_{\widetilde\beta}(x)
=
C(\alpha,\widetilde\beta,n)
{}_2F_1\left(a,b;c;-|x|^2\right), − ( − Δ ) α /2 f β ( x ) = C ( α , β , n ) 2 F 1 ( a , b ; c ; − ∣ x ∣ 2 ) , where C ( α , β ~ , n ) > 0 C(\alpha,\widetilde\beta,n)>0 C ( α , β , n ) > 0
a = n + α 2 = n + 2 s 2 , b = α + β ~ 2 = n 2 , c = n 2 . a=\frac{n+\alpha}{2}=\frac{n+2s}{2},
\qquad
b=\frac{\alpha+\widetilde\beta}{2}=\frac n2,
\qquad
c=\frac n2. a = 2 n + α = 2 n + 2 s , b = 2 α + β = 2 n , c = 2 n . Since b = c b=c b = c
2 F 1 ( a , c ; c ; z ) = ( 1 − z ) − a {}_2F_1(a,c;c;z)=(1-z)^{-a} 2 F 1 ( a , c ; c ; z ) = ( 1 − z ) − a yields
2 F 1 ( n + 2 s 2 , n 2 ; n 2 ; − ∣ x ∣ 2 ) = ( 1 + ∣ x ∣ 2 ) − ( n + 2 s ) / 2 . {}_2F_1\left(
\frac{n+2s}{2},\frac n2;\frac n2;-|x|^2
\right)
=
(1+|x|^2)^{-(n+2s)/2}. 2 F 1 ( 2 n + 2 s , 2 n ; 2 n ; − ∣ x ∣ 2 ) = ( 1 + ∣ x ∣ 2 ) − ( n + 2 s ) /2 . Recalling that f β ~ = − ψ f_{\widetilde\beta}=-\psi f β = − ψ
( − Δ ) s ψ ( x ) = a n , s ( 1 + ∣ x ∣ 2 ) − ( n + 2 s ) / 2 , (-\Delta)^s\psi(x)
=
a_{n,s}(1+|x|^2)^{-(n+2s)/2}, ( − Δ ) s ψ ( x ) = a n , s ( 1 + ∣ x ∣ 2 ) − ( n + 2 s ) /2 , for a suitable constant a n , s > 0 a_{n,s}>0 a n , s > 0 a n , s a_{n,s} a n , s
We now compute the local part. More generally, if ψ β ( x ) : = ( 1 + ∣ x ∣ 2 ) − β \psi_\beta(x):=(1+|x|^2)^{-\beta} ψ β ( x ) := ( 1 + ∣ x ∣ 2 ) − β
Δ ψ β ( x ) = 2 β ( 1 + ∣ x ∣ 2 ) β + 2 [ ( 2 β − n + 2 ) ∣ x ∣ 2 − n ] . \begin{equation*}
\Delta\psi_\beta(x)
=\frac{2\beta}{(1+|x|^2)^{\beta+2}}
\left[(2\beta-n+2)|x|^2-n\right].
\end{equation*} Δ ψ β ( x ) = ( 1 + ∣ x ∣ 2 ) β + 2 2 β [ ( 2 β − n + 2 ) ∣ x ∣ 2 − n ] . For β = ( n − 2 s ) / 2 \beta=(n-2s)/2 β = ( n − 2 s ) /2
− Δ ψ ( x ) = ( n − 2 s ) n − 2 ( 1 − s ) ∣ x ∣ 2 ( 1 + ∣ x ∣ 2 ) ( n + 4 − 2 s ) / 2 . \begin{equation}
-\Delta\psi(x)
=(n-2s)
\frac{n-2(1-s)|x|^2}
{(1+|x|^2)^{(n+4-2s)/2}}. \tag*{(3.6)}
\end{equation} − Δ ψ ( x ) = ( n − 2 s ) ( 1 + ∣ x ∣ 2 ) ( n + 4 − 2 s ) /2 n − 2 ( 1 − s ) ∣ x ∣ 2 . (3.6) Combining (3.4) and (3.6) , we can rewrite the latter identity as
− Δ ψ ( x ) = n − 2 s a n , s n − 2 ( 1 − s ) ∣ x ∣ 2 ( 1 + ∣ x ∣ 2 ) 2 ( 1 − s ) ( − Δ ) s ψ ( x ) . \begin{equation*}
-\Delta\psi(x)
=\frac{n-2s}{a_{n,s}}
\frac{n-2(1-s)|x|^2}{(1+|x|^2)^{2(1-s)}}
(-\Delta)^s\psi(x).
\end{equation*} − Δ ψ ( x ) = a n , s n − 2 s ( 1 + ∣ x ∣ 2 ) 2 ( 1 − s ) n − 2 ( 1 − s ) ∣ x ∣ 2 ( − Δ ) s ψ ( x ) . Since
n − 2 ( 1 − s ) ∣ x ∣ 2 ( 1 + ∣ x ∣ 2 ) 2 ( 1 − s ) ≤ n for every x ∈ R n , \frac{n-2(1-s)|x|^2}{(1+|x|^2)^{2(1-s)}}\le n
\qquad\text{for every }x\in\mathbb{R}^n, ( 1 + ∣ x ∣ 2 ) 2 ( 1 − s ) n − 2 ( 1 − s ) ∣ x ∣ 2 ≤ n for every x ∈ R n , we obtain (3.5) , for instance with
C 0 : = n ( n − 2 s ) a n , s . \begin{equation*}
C_0:=\frac{n(n-2s)}{a_{n,s}}.
\end{equation*} C 0 := a n , s n ( n − 2 s ) . This ends the proof. ◻
We now consider a suitable rescaling of the function ψ \psi ψ R > 0 R>0 R > 0
ψ R ( x ) : = ψ ( x R ) . \begin{equation*}
\psi_R(x):=\psi\left(\frac{x}{R}\right).
\end{equation*} ψ R ( x ) := ψ ( R x ) . As already mentioned, − Δ -\Delta − Δ ( − Δ ) s (-\Delta)^s ( − Δ ) s
− Δ ψ R ( x ) = R − 2 ( − Δ ψ ) ( x R ) , ( − Δ ) s ψ R ( x ) = R − 2 s ( − Δ ) s ψ ( x R ) . \begin{equation}
-\Delta\psi_R(x)
=R^{-2}(-\Delta\psi)\left(\frac{x}{R}\right),
\qquad
(-\Delta)^s\psi_R(x)
=R^{-2s}(-\Delta)^s\psi\left(\frac{x}{R}\right). \tag*{(3.7)}
\end{equation} − Δ ψ R ( x ) = R − 2 ( − Δ ψ ) ( R x ) , ( − Δ ) s ψ R ( x ) = R − 2 s ( − Δ ) s ψ ( R x ) . (3.7) Consequently, Lemma 3.2 gives
L γ , s ψ R ( x ) = R − 2 ( − Δ ψ ) ( x R ) − γ R − 2 s ( − Δ ) s ψ ( x R ) ≤ R − 2 s ( C 0 R 2 s − 2 − γ ) ( − Δ ) s ψ ( x R ) . \begin{align}
\mathcal L_{\gamma,s}\psi_R(x)
&=R^{-2}(-\Delta\psi)\left(\frac{x}{R}\right)
-\gamma R^{-2s}(-\Delta)^s\psi\left(\frac{x}{R}\right)\notag\\
&\le R^{-2s}\left(C_0R^{2s-2}-\gamma\right)
(-\Delta)^s\psi\left(\frac{x}{R}\right). \tag*{(3.8)}
\end{align} L γ , s ψ R ( x ) = R − 2 ( − Δ ψ ) ( R x ) − γ R − 2 s ( − Δ ) s ψ ( R x ) ≤ R − 2 s ( C 0 R 2 s − 2 − γ ) ( − Δ ) s ψ ( R x ) . (3.8) Since s < 1 s<1 s < 1 R 2 s − 2 → 0 R^{2s-2}\to0 R 2 s − 2 → 0 R → + ∞ R\to+\infty R → + ∞ R 0 = R 0 ( n , s , γ ) > 0 R_0=R_0(n,s,\gamma)>0 R 0 = R 0 ( n , s , γ ) > 0
C 0 R 2 s − 2 ≤ γ 2 for every R ≥ R 0 . \begin{equation}
C_0R^{2s-2}\le\frac\gamma2
\qquad\text{for every }R\ge R_0. \tag*{(3.9)}
\end{equation} C 0 R 2 s − 2 ≤ 2 γ for every R ≥ R 0 . (3.9) It follows from (3.4) and (3.8) that, for every R ≥ R 0 R\ge R_0 R ≥ R 0
L γ , s ψ R ( x ) ≤ − γ 2 R − 2 s ( − Δ ) s ψ ( x R ) < 0 for every x ∈ R n . \begin{equation}
\mathcal L_{\gamma,s}\psi_R(x)
\le-\frac\gamma2 R^{-2s}
(-\Delta)^s\psi\left(\frac{x}{R}\right)<0
\qquad\text{for every }x\in\mathbb{R}^n. \tag*{(3.10)}
\end{equation} L γ , s ψ R ( x ) ≤ − 2 γ R − 2 s ( − Δ ) s ψ ( R x ) < 0 for every x ∈ R n . (3.10) The function ψ R \psi_R ψ R (3.3) we have
ψ R ( x ) ≍ ∣ x ∣ − ( n − 2 s ) as ∣ x ∣ → + ∞ , \begin{equation}
\psi_R(x)\asymp |x|^{-(n-2s)}
\qquad\text{as }|x|\to+\infty, \tag*{(3.11)}
\end{equation} ψ R ( x ) ≍ ∣ x ∣ − ( n − 2 s ) as ∣ x ∣ → + ∞ , (3.11) and therefore
ψ R ∈ L 2 ( R n ) ⟺ n > 4 s . \begin{equation*}
\psi_R\in L^2(\mathbb{R}^n)
\quad\Longleftrightarrow\quad n>4s.
\end{equation*} ψ R ∈ L 2 ( R n ) ⟺ n > 4 s . No restriction of this type is needed, however. The relevant observation is that ψ R \psi_R ψ R
Lemma 3.3. For every R > 0 R>0 R > 0 ∥ ∇ ψ R ∥ L 2 ( R n ) , [ ψ R ] s < + ∞ .
\|\nabla \psi_R\|_{L^2(\mathbb{R}^n)},\,[\psi_R]_s<+\infty.
∥∇ ψ R ∥ L 2 ( R n ) , [ ψ R ] s < + ∞. φ k ∈ C 0 ∞ ( R n ) \varphi_k\in C_0^\infty(\mathbb{R}^n) φ k ∈ C 0 ∞ ( R n ) φ k ≥ 0 \varphi_k\ge0 φ k ≥ 0 ∥ ∇ ( φ k − ψ R ) ∥ 2 → 0 , [ φ k − ψ R ] s → 0 as k → + ∞ .
\left\lVert \nabla(\varphi_k-\psi_R)\right\rVert_2\to 0,
\qquad
[\varphi_k-\psi_R]_s\to 0
\qquad\text{as }k\to+\infty.
∥ ∇ ( φ k − ψ R ) ∥ 2 → 0 , [ φ k − ψ R ] s → 0 as k → + ∞.
Proof. The fact that [ ψ R ] s < + ∞ [\psi_R]_s<+\infty [ ψ R ] s < + ∞ (3.3) . As for the local energy, (3.3) gives, for ∣ x ∣ |x| ∣ x ∣
∣ ∇ ψ R ( x ) ∣ ≤ C R ∣ x ∣ − ( n − 2 s + 1 ) . \begin{equation*}
|\nabla\psi_R(x)|\le C_R|x|^{-(n-2s+1)}.
\end{equation*} ∣∇ ψ R ( x ) ∣ ≤ C R ∣ x ∣ − ( n − 2 s + 1 ) . Thus ∇ ψ R ∈ L 2 ( R n ) \nabla\psi_R\in L^2(\mathbb{R}^n) ∇ ψ R ∈ L 2 ( R n )
2 ( n − 2 s + 1 ) > n ⟺ n + 2 > 4 s , 2(n-2s+1)>n
\quad\Longleftrightarrow\quad n+2>4s, 2 ( n − 2 s + 1 ) > n ⟺ n + 2 > 4 s , and the latter inequality follows from n ≥ 3 n\ge3 n ≥ 3 s < 1 s<1 s < 1
Let now η ∈ C 0 ∞ ( R n ) \eta\in C_0^\infty(\mathbb{R}^n) η ∈ C 0 ∞ ( R n )
0 ≤ η ≤ 1 , η ≡ 1 in B 1 , η ≡ 0 in R n ∖ B 2 , \begin{equation*}
0\le\eta\le1,
\qquad
\eta\equiv1\ \text{in }B_1,
\qquad
\eta\equiv0\ \text{in }\mathbb{R}^n\setminus B_2,
\end{equation*} 0 ≤ η ≤ 1 , η ≡ 1 in B 1 , η ≡ 0 in R n ∖ B 2 , and define
η k ( x ) : = η ( x k ) , φ k ( x ) : = η k ( x ) ψ R ( x ) . \begin{equation*}
\eta_k(x):=\eta\left(\frac{x}{k}\right),
\qquad
\varphi_k(x):=\eta_k(x)\psi_R(x).
\end{equation*} η k ( x ) := η ( k x ) , φ k ( x ) := η k ( x ) ψ R ( x ) . Clearly, φ k ∈ C 0 ∞ ( R n ) \varphi_k\in C_0^\infty(\mathbb{R}^n) φ k ∈ C 0 ∞ ( R n ) φ k ≥ 0 \varphi_k\ge0 φ k ≥ 0
We first consider the local energy. Since
∇ ( φ k − ψ R ) = ( η k − 1 ) ∇ ψ R + ψ R ∇ η k , \nabla(\varphi_k-\psi_R)
=(\eta_k-1)\nabla\psi_R+\psi_R\nabla\eta_k, ∇ ( φ k − ψ R ) = ( η k − 1 ) ∇ ψ R + ψ R ∇ η k , the first term converges to zero in L 2 ( R n ) L^2(\mathbb{R}^n) L 2 ( R n ) ∣ ∇ η k ∣ ≤ C / k |\nabla\eta_k|\le C/k ∣∇ η k ∣ ≤ C / k (3.11) , we have
∥ ψ R ∇ η k ∥ 2 2 ≤ C k 2 ∫ { k ≤ ∣ x ∣ ≤ 2 k } ψ R ( x ) 2 d x ≤ C R k − n + 4 s − 2 → 0 , \begin{align*}
\left\lVert \psi_R\nabla\eta_k\right\rVert_2^2
&\le\frac{C}{k^2}
\int_{\{k\le|x|\le2k\}}\psi_R(x)^2\,dx\le C_R k^{-n+4s-2}\to 0,
\end{align*} ∥ ψ R ∇ η k ∥ 2 2 ≤ k 2 C ∫ { k ≤ ∣ x ∣ ≤ 2 k } ψ R ( x ) 2 d x ≤ C R k − n + 4 s − 2 → 0 , since − n + 4 s − 2 < 0 -n+4s-2<0 − n + 4 s − 2 < 0 n ≥ 3 n\ge3 n ≥ 3 s < 1 s<1 s < 1 (3.12) .
We next consider the fractional seminorm. Set
g k : = ( 1 − η k ) ψ R = ψ R − φ k . \begin{equation*}
g_k:=(1-\eta_k)\psi_R=\psi_R-\varphi_k.
\end{equation*} g k := ( 1 − η k ) ψ R = ψ R − φ k . For every x , y ∈ R n x,y\in\mathbb{R}^n x , y ∈ R n
g k ( x ) − g k ( y ) = ( 1 − η k ( x ) ) ( ψ R ( x ) − ψ R ( y ) ) + ψ R ( y ) ( η k ( y ) − η k ( x ) ) . g_k(x)-g_k(y)
=(1-\eta_k(x))(\psi_R(x)-\psi_R(y))
+\psi_R(y)(\eta_k(y)-\eta_k(x)). g k ( x ) − g k ( y ) = ( 1 − η k ( x )) ( ψ R ( x ) − ψ R ( y )) + ψ R ( y ) ( η k ( y ) − η k ( x )) . Hence
[ g k ] s 2 ≤ 2 I k + 2 J k , \begin{equation}
[g_k]_s^2\le2I_k+2J_k, \tag*{(3.13)}
\end{equation} [ g k ] s 2 ≤ 2 I k + 2 J k , (3.13) where
I k : = ∬ R 2 n ( 1 − η k ( x ) ) 2 ∣ ψ R ( x ) − ψ R ( y ) ∣ 2 ∣ x − y ∣ n + 2 s d x d y , J k : = ∫ R n ψ R ( y ) 2 Θ k ( y ) d y , \begin{align*}
I_k&:=\iint_{\mathbb{R}^{2n}}
\frac{(1-\eta_k(x))^2|\psi_R(x)-\psi_R(y)|^2}
{|x-y|^{n+2s}}\,dx\,dy,\\
J_k&:=\int_{\mathbb{R}^n}\psi_R(y)^2\Theta_k(y)\,dy,
\end{align*} I k J k := ∬ R 2 n ∣ x − y ∣ n + 2 s ( 1 − η k ( x ) ) 2 ∣ ψ R ( x ) − ψ R ( y ) ∣ 2 d x d y , := ∫ R n ψ R ( y ) 2 Θ k ( y ) d y , and
Θ k ( y ) : = ∫ R n ∣ η k ( x ) − η k ( y ) ∣ 2 ∣ x − y ∣ n + 2 s d x . \begin{equation}
\Theta_k(y):=\int_{\mathbb{R}^n}
\frac{|\eta_k(x)-\eta_k(y)|^2}{|x-y|^{n+2s}}\,dx. \tag*{(3.14)}
\end{equation} Θ k ( y ) := ∫ R n ∣ x − y ∣ n + 2 s ∣ η k ( x ) − η k ( y ) ∣ 2 d x . (3.14) Since [ ψ R ] s < + ∞ [\psi_R]_s<+\infty [ ψ R ] s < + ∞ 0 ≤ ( 1 − η k ) 2 ≤ 1 0\le(1-\eta_k)^2\le1 0 ≤ ( 1 − η k ) 2 ≤ 1 1 − η k ( x ) → 0 1-\eta_k(x)\to0 1 − η k ( x ) → 0 x x x
I k → 0. \begin{equation}
I_k\to 0. \tag*{(3.15)}
\end{equation} I k → 0. (3.15) To estimate J k J_k J k y y y
J k = ∫ { ∣ y ∣ ≤ 3 k } ψ R ( y ) 2 Θ k ( y ) d y + ∫ { ∣ y ∣ > 3 k } ψ R ( y ) 2 Θ k ( y ) d y = J 1 , k + J 2 , k , \begin{split}
J_k
&=\int_{\{|y|\le3k\}}\psi_R(y)^2\Theta_k(y)\,dy
+\int_{\{|y|>3k\}}\psi_R(y)^2\Theta_k(y)\,dy=J_{1,k}+J_{2,k},
\end{split} J k = ∫ { ∣ y ∣ ≤ 3 k } ψ R ( y ) 2 Θ k ( y ) d y + ∫ { ∣ y ∣ > 3 k } ψ R ( y ) 2 Θ k ( y ) d y = J 1 , k + J 2 , k , and we study the two terms separately.
- Study of J 1 , k J_{1,k} J 1 , k 0 ≤ η k ≤ 1 0\leq \eta_k\leq 1 0 ≤ η k ≤ 1 ∣ ∇ η k ∣ ≤ C / k |\nabla\eta_k|\leq C/k ∣∇ η k ∣ ≤ C / k
∣ η k ( x ) − η k ( y ) ∣ ≤ C min { 1 , ∣ x − y ∣ k } . |\eta_k(x)-\eta_k(y)|
\le C\min\left\{1,\frac{|x-y|}{k}\right\}. ∣ η k ( x ) − η k ( y ) ∣ ≤ C min { 1 , k ∣ x − y ∣ } . Thus, we obtain the uniform estimate
Consequently,
J 1 , k = ∫ { ∣ y ∣ ≤ 3 k } ψ R ( y ) 2 Θ k ( y ) d y ≤ C k − 2 s ∫ { ∣ y ∣ ≤ 3 k } ψ R ( y ) 2 d y . \begin{equation}
J_{1,k} = \int_{\{|y|\le3k\}}\psi_R(y)^2\Theta_k(y)\,dy
\le Ck^{-2s}\int_{\{|y|\le3k\}}\psi_R(y)^2\,dy. \tag*{(3.17)}
\end{equation} J 1 , k = ∫ { ∣ y ∣ ≤ 3 k } ψ R ( y ) 2 Θ k ( y ) d y ≤ C k − 2 s ∫ { ∣ y ∣ ≤ 3 k } ψ R ( y ) 2 d y . (3.17) By (3.11) ,
∫ { ∣ y ∣ ≤ 3 k } ψ R ( y ) 2 d y ≤ C R { 1 , n > 4 s , 1 + log k , n = 4 s , 1 + k 4 s − n , n < 4 s . \begin{equation*}
\int_{\{|y|\le3k\}}\psi_R(y)^2\,dy
\le C_R
\begin{cases}
1,&n>4s,\\
1+\log k,&n=4s,\\
1+k^{4s-n},&n<4s.
\end{cases}
\end{equation*} ∫ { ∣ y ∣ ≤ 3 k } ψ R ( y ) 2 d y ≤ C R ⎩ ⎨ ⎧ 1 , 1 + log k , 1 + k 4 s − n , n > 4 s , n = 4 s , n < 4 s . Since n > 2 s n>2s n > 2 s J 1 , k → 0 J_{1,k}\to 0 J 1 , k → 0
- Study of J 2 , k J_{2,k} J 2 , k { ∣ y ∣ > 3 k } \{|y|>3k\} { ∣ y ∣ > 3 k } η k ( y ) = 0 \eta_k(y)=0 η k ( y ) = 0 η k ( x ) ≠ 0 \eta_k(x)\ne0 η k ( x ) = 0 ∣ x ∣ < 2 k |x|<2k ∣ x ∣ < 2 k ∣ x − y ∣ ≥ ∣ y ∣ / 3 |x-y|\ge |y|/3 ∣ x − y ∣ ≥ ∣ y ∣/3 (3.14) is nonzero, and we obtain the following stronger version of (3.16) :
Θ k ( y ) = ∫ { ∣ x ∣ < 2 k } η k ( x ) 2 ∣ x − y ∣ n + 2 s d x ≤ C ∣ y ∣ − n − 2 s ∫ { ∣ x ∣ < 2 k } η k ( x ) 2 d x ≤ C k n ∣ y ∣ − n − 2 s for all ∣ y ∣ > 3 k . \begin{equation*}
\begin{split}
\Theta_k(y)& =
\int_{\{|x|<2k\}}\frac{\eta_k(x)^2}{|x-y|^{n+2s}}\,dx
\leq
C|y|^{-n-2s}\int_{\{|x|<2k\}}\eta_k(x)^2\,dx \\
& \leq Ck^n|y|^{-n-2s}
\qquad\text{for all }|y|>3k.
\end{split}
\end{equation*} Θ k ( y ) = ∫ { ∣ x ∣ < 2 k } ∣ x − y ∣ n + 2 s η k ( x ) 2 d x ≤ C ∣ y ∣ − n − 2 s ∫ { ∣ x ∣ < 2 k } η k ( x ) 2 d x ≤ C k n ∣ y ∣ − n − 2 s for all ∣ y ∣ > 3 k . Using (3.11) , we conclude that
J 2 , k = ∫ { ∣ y ∣ > 3 k } ψ R ( y ) 2 Θ k ( y ) d y ≤ C R k n ∫ 3 k + ∞ r − 2 n + 2 s − 1 d r ≤ C R k − n + 2 s → 0 , \begin{align}
J_{2,k} = \int_{\{|y|>3k\}}\psi_R(y)^2\Theta_k(y)\,dy
&\le C_Rk^n\int_{3k}^{+\infty}r^{-2n+2s-1}\,dr\le C_Rk^{-n+2s}\to 0, \tag*{(3.18)}
\end{align} J 2 , k = ∫ { ∣ y ∣ > 3 k } ψ R ( y ) 2 Θ k ( y ) d y ≤ C R k n ∫ 3 k + ∞ r − 2 n + 2 s − 1 d r ≤ C R k − n + 2 s → 0 , (3.18) again because n > 2 s n>2s n > 2 s (3.17) –(3.18) , we obtain J k → 0 J_k\to0 J k → 0
Together with (3.13) and (3.15) , this gives [ g k ] s → 0 [g_k]_s\to0 [ g k ] s → 0 (3.12) , and the proof is complete. ◻
We are now in a position to prove the Liouville property stated in Theorem 1.1 . We split it, proving first the following
Theorem 3.4. Let n ≥ 3 n\ge3 n ≥ 3 s ∈ ( 0 , 1 ) s\in(0,1) s ∈ ( 0 , 1 ) γ > 0 \gamma>0 γ > 0 u ∈ H 1 ( R n ) u\in H^1(\mathbb{R}^n) u ∈ H 1 ( R n ) (3.1) , then u ≡ 0.
u\equiv0.
u ≡ 0.
Proof. Assume by contradiction that u ≢ 0 u\not\equiv0 u ≡ 0 R ≥ R 0 R\ge R_0 R ≥ R 0 R 0 R_0 R 0 (3.9) , so that (3.10) holds. Let ( φ k ) k (\varphi_k)_k ( φ k ) k 3.3 . Since u u u
∫ R n ∇ u ⋅ ∇ φ k d x − γ E s ( u , φ k ) ≥ 0 for every k ∈ N . \begin{equation}
\int_{\mathbb{R}^n}\nabla u\cdot\nabla\varphi_k\,dx
-\gamma\mathcal E_s(u,\varphi_k)\ge0
\qquad\text{for every }k\in\mathbb{N}. \tag*{(3.19)}
\end{equation} ∫ R n ∇ u ⋅ ∇ φ k d x − γ E s ( u , φ k ) ≥ 0 for every k ∈ N . (3.19) The convergences in (3.12) , together with the Cauchy–Schwarz inequality, allow us to pass to the limit in (3.19) , obtaining
∫ R n ∇ u ⋅ ∇ ψ R d x − γ E s ( u , ψ R ) ≥ 0. \begin{equation}
\int_{\mathbb{R}^n}\nabla u\cdot\nabla\psi_R\,dx
-\gamma\mathcal E_s(u,\psi_R)\ge0. \tag*{(3.20)}
\end{equation} ∫ R n ∇ u ⋅ ∇ ψ R d x − γ E s ( u , ψ R ) ≥ 0. (3.20) We next move the operator onto ψ R \psi_R ψ R (3.6) , (3.4) and the scaling relations (3.7) , we have
− Δ ψ R ∈ L 2 ( R n ) , ( − Δ ) s ψ R ∈ L 2 ( R n ) . \begin{equation}
-\Delta\psi_R\in L^2(\mathbb{R}^n),
\qquad
(-\Delta)^s\psi_R\in L^2(\mathbb{R}^n). \tag*{(3.21)}
\end{equation} − Δ ψ R ∈ L 2 ( R n ) , ( − Δ ) s ψ R ∈ L 2 ( R n ) . (3.21) Indeed, at infinity these functions decay respectively as ∣ x ∣ − n − 2 + 2 s |x|^{-n-2+2s} ∣ x ∣ − n − 2 + 2 s ∣ x ∣ − n − 2 s |x|^{-n-2s} ∣ x ∣ − n − 2 s
Take u j ∈ C 0 ∞ ( R n ) u_j\in C_0^\infty(\mathbb{R}^n) u j ∈ C 0 ∞ ( R n )
u j → u strongly in H 1 ( R n ) . \begin{equation}
u_j\to u
\qquad\text{strongly in }H^1(\mathbb{R}^n). \tag*{(3.22)}
\end{equation} u j → u strongly in H 1 ( R n ) . (3.22) Since u j u_j u j
∫ R n ∇ u j ⋅ ∇ ψ R d x = ∫ R n u j ( − Δ ψ R ) d x . \int_{\mathbb{R}^n}\nabla u_j\cdot\nabla\psi_R\,dx
=\int_{\mathbb{R}^n}u_j(-\Delta\psi_R)\,dx. ∫ R n ∇ u j ⋅ ∇ ψ R d x = ∫ R n u j ( − Δ ψ R ) d x . Using (3.21) and (3.22) , we may pass to the limit and obtain
∫ R n ∇ u ⋅ ∇ ψ R d x = ∫ R n u ( − Δ ψ R ) d x . \begin{equation}
\int_{\mathbb{R}^n}\nabla u\cdot\nabla\psi_R\,dx
=\int_{\mathbb{R}^n}u(-\Delta\psi_R)\,dx. \tag*{(3.23)}
\end{equation} ∫ R n ∇ u ⋅ ∇ ψ R d x = ∫ R n u ( − Δ ψ R ) d x . (3.23) The same argument applies to the fractional part. Indeed, (3.22) and Lemma 2.7 imply u j → u u_j\to u u j → u H s ( R n ) H^s(\mathbb{R}^n) H s ( R n )
∣ E s ( u j − u , ψ R ) ∣ ≤ [ u j − u ] s [ ψ R ] s ⟶ 0. \begin{equation*}
|\mathcal E_s(u_j-u,\psi_R)|
\le [u_j-u]_s[\psi_R]_s\longrightarrow0.
\end{equation*} ∣ E s ( u j − u , ψ R ) ∣ ≤ [ u j − u ] s [ ψ R ] s ⟶ 0. On the other hand, by the fractional integration-by-parts formula, with the normalization associated with (2.3) ,
E s ( u j , ψ R ) = ∫ R n u j ( − Δ ) s ψ R d x . \mathcal E_s(u_j,\psi_R)
=\int_{\mathbb{R}^n}u_j(-\Delta)^s\psi_R\,dx. E s ( u j , ψ R ) = ∫ R n u j ( − Δ ) s ψ R d x . Using again (3.21) and passing to the limit gives
E s ( u , ψ R ) = ∫ R n u ( − Δ ) s ψ R d x . \begin{equation}
\mathcal E_s(u,\psi_R)
=\int_{\mathbb{R}^n}u(-\Delta)^s\psi_R\,dx. \tag*{(3.24)}
\end{equation} E s ( u , ψ R ) = ∫ R n u ( − Δ ) s ψ R d x . (3.24) Combining (3.23) and (3.24) , we therefore find
∫ R n ∇ u ⋅ ∇ ψ R d x − γ E s ( u , ψ R ) = ∫ R n u [ − Δ ψ R − γ ( − Δ ) s ψ R ] d x = ∫ R n u L γ , s ψ R d x . \begin{align}
\int_{\mathbb{R}^n}\nabla u\cdot\nabla\psi_R\,dx
-\gamma\mathcal E_s(u,\psi_R)
&=\int_{\mathbb{R}^n}u
\left[-\Delta\psi_R-\gamma(-\Delta)^s\psi_R\right]dx\notag\\
&=\int_{\mathbb{R}^n}u\,\mathcal L_{\gamma,s}\psi_R\,dx. \tag*{(3.25)}
\end{align} ∫ R n ∇ u ⋅ ∇ ψ R d x − γ E s ( u , ψ R ) = ∫ R n u [ − Δ ψ R − γ ( − Δ ) s ψ R ] d x = ∫ R n u L γ , s ψ R d x . (3.25) The last integral is finite by (3.21) and u ∈ L 2 ( R n ) u\in L^2(\mathbb{R}^n) u ∈ L 2 ( R n )
Finally, u ≥ 0 u\ge0 u ≥ 0 u ≢ 0 u\not\equiv0 u ≡ 0 L γ , s ψ R < 0 \mathcal L_{\gamma,s}\psi_R<0 L γ , s ψ R < 0 (3.10) . Hence
∫ R n u L γ , s ψ R d x < 0 , \begin{equation*}
\int_{\mathbb{R}^n}u\,\mathcal L_{\gamma,s}\psi_R\,dx<0,
\end{equation*} ∫ R n u L γ , s ψ R d x < 0 , which contradicts (3.20) and (3.25) . Thus u ≡ 0 u\equiv0 u ≡ 0
The preceding theorem immediately gives the desired nonexistence result for power nonlinearities, without any restriction on the exponent.
Corollary 3.5. Let r > 1 r>1 r > 1 − Δ u − γ ( − Δ ) s u = u r − 1 in R n
-\Delta u-\gamma(-\Delta)^s u=u^{r-1}
\qquad\text{in }\mathbb{R}^n
− Δ u − γ ( − Δ ) s u = u r − 1 in R n u ∈ H 1 ( R n ) u\in H^1(\mathbb{R}^n) u ∈ H 1 ( R n ) r r r u r − 1 ∈ L l o c 1 ( R n ) u^{r-1}\in L^1_{\rm loc}(\mathbb{R}^n) u r − 1 ∈ L loc 1 ( R n ) ∫ R n ∇ u ⋅ ∇ φ d x − γ E s ( u , φ ) = ∫ R n u r − 1 φ d x
\int_{\mathbb{R}^n}\nabla u\cdot\nabla\varphi\,dx
-\gamma\mathcal E_s(u,\varphi)
=\int_{\mathbb{R}^n}u^{r-1}\varphi\,dx
∫ R n ∇ u ⋅ ∇ φ d x − γ E s ( u , φ ) = ∫ R n u r − 1 φ d x φ ∈ C 0 ∞ ( R n ) \varphi\in C_0^\infty(\mathbb{R}^n) φ ∈ C 0 ∞ ( R n )
Proof. If u ≥ 0 u\ge0 u ≥ 0 (3.26) , then for every nonnegative φ ∈ C 0 ∞ ( R n ) \varphi\in C_0^\infty(\mathbb{R}^n) φ ∈ C 0 ∞ ( R n )
∫ R n ∇ u ⋅ ∇ φ d x − γ E s ( u , φ ) = ∫ R n u r − 1 φ d x ≥ 0. \int_{\mathbb{R}^n}\nabla u\cdot\nabla\varphi\,dx
-\gamma\mathcal E_s(u,\varphi)
=\int_{\mathbb{R}^n}u^{r-1}\varphi\,dx\ge0. ∫ R n ∇ u ⋅ ∇ φ d x − γ E s ( u , φ ) = ∫ R n u r − 1 φ d x ≥ 0. Thus u u u (3.1) , and Theorem 3.4 gives u ≡ 0 u\equiv0 u ≡ 0
Proof of Theorem 1.3
In this section we prove Theorem 1.3 . We first establish the attainment of the constant S r ( γ ) S_r(\gamma) S r ( γ ) γ → C e m b − \gamma\to C_{\rm emb}^- γ → C emb − C e m b C_{\rm emb} C emb L R L^R L R
Recall that
Q γ ( u ) : = ∫ Ω ∣ ∇ u ∣ 2 d x − γ [ u ] s 2 , u ∈ X 1 , 2 ( Ω ) , \begin{equation*}
Q_{\gamma}(u) := \int_{\Omega}|\nabla u|^{2} \, dx - \gamma \, [u]^{2}_{s}, \quad u \in \mathcal{X}^{1,2}(\Omega),
\end{equation*} Q γ ( u ) := ∫ Ω ∣∇ u ∣ 2 d x − γ [ u ] s 2 , u ∈ X 1 , 2 ( Ω ) , and notice that, by the definition of C e m b C_{\rm emb} C emb (2.9) ,
Q γ ( u ) ≥ ( 1 − γ C e m b ) ∥ ∇ u ∥ L 2 ( Ω ) 2 . \begin{equation}
Q_\gamma(u)\ge
\left(1-\frac\gamma{C_{\rm emb}}\right)
\left\lVert \nabla u\right\rVert_{L^2(\Omega)}^2. \tag*{(4.1)}
\end{equation} Q γ ( u ) ≥ ( 1 − C emb γ ) ∥ ∇ u ∥ L 2 ( Ω ) 2 . (4.1) Proposition 4.1. Let 1 < r ≤ 2 ∗ 1<r\le2^* 1 < r ≤ 2 ∗ 0 < γ < C e m b 0<\gamma<C_{\rm emb} 0 < γ < C emb
If r < 2 ∗ r<2^* r < 2 ∗ (1.11) is achieved for every γ ∈ ( 0 , C e m b ) \gamma\in(0,C_{\rm emb}) γ ∈ ( 0 , C emb )
If r = 2 ∗ r=2^* r = 2 ∗ [51 Theorem 1.1] : if n ≥ 5 n\ge5 n ≥ 5 γ ∈ ( 0 , C e m b ) \gamma\in(0,C_{\rm emb}) γ ∈ ( 0 , C emb ) n = 3 , 4 n=3,4 n = 3 , 4 γ ∗ ∈ [ 0 , C e m b ) \gamma^*\in[0,C_{\rm emb}) γ ∗ ∈ [ 0 , C emb ) γ ∈ ( γ ∗ , C e m b ) \gamma\in(\gamma^*,C_{\rm emb}) γ ∈ ( γ ∗ , C emb )
In either case, every normalized minimizer u γ , r u_{\gamma,r} u γ , r ∫ Ω ∇ u γ , r ⋅ ∇ φ d x − γ E s ( u γ , r , φ ) = S r ( γ ) ∫ Ω ∣ u γ , r ∣ r − 2 u γ , r φ d x
\int_\Omega\nabla u_{\gamma,r}\cdot\nabla\varphi\,dx
-\gamma\mathcal E_s(u_{\gamma,r},\varphi)
=S_r(\gamma)\int_\Omega
|u_{\gamma,r}|^{r-2}u_{\gamma,r}\varphi\,dx
∫ Ω ∇ u γ , r ⋅ ∇ φ d x − γ E s ( u γ , r , φ ) = S r ( γ ) ∫ Ω ∣ u γ , r ∣ r − 2 u γ , r φ d x φ ∈ X 1 , 2 ( Ω ) \varphi\in \mathcal{X}^{1,2}(\Omega) φ ∈ X 1 , 2 ( Ω )
Proof. In view of Theorem 1.2 , only the case r < 2 ∗ r<2^* r < 2 ∗ ( u j ) (u_j) ( u j ) ∥ u j ∥ r = 1 \left\lVert u_j\right\rVert_r=1 ∥ u j ∥ r = 1 (4.1) , ( u j ) (u_j) ( u j ) X 1 , 2 ( Ω ) \mathcal{X}^{1,2}(\Omega) X 1 , 2 ( Ω )
u j ⇀ u in X 1 , 2 ( Ω ) , u j → u in L r ( Ω ) ∩ H s ( R n ) . u_j\rightharpoonup u\quad\text{in }\mathcal{X}^{1,2}(\Omega),
\qquad
u_j\to u\quad\text{in }L^r(\Omega)\cap H^s(\mathbb{R}^n). u j ⇀ u in X 1 , 2 ( Ω ) , u j → u in L r ( Ω ) ∩ H s ( R n ) . Therefore ∥ u ∥ r = 1 \left\lVert u\right\rVert_r=1 ∥ u ∥ r = 1 H s ( R n ) H^s(\mathbb{R}^n) H s ( R n )
Q γ ( u ) ≤ lim inf j → ∞ Q γ ( u j ) = S r ( γ ) . Q_\gamma(u)\le\liminf_{j\to\infty}Q_\gamma(u_j)=S_r(\gamma). Q γ ( u ) ≤ j → ∞ lim inf Q γ ( u j ) = S r ( γ ) . Thus, setting u γ , r : = u u_{\gamma,r}:=u u γ , r := u (1.11) . The Euler equation follows from the Lagrange multiplier rule. Testing it with u γ , r u_{\gamma,r} u γ , r ∥ u γ , r ∥ r = 1 \left\lVert u_{\gamma,r}\right\rVert_r=1 ∥ u γ , r ∥ r = 1 S r ( γ ) S_r(\gamma) S r ( γ )
For the rest of the paper, whenever r = 2 ∗ r=2^* r = 2 ∗ γ \gamma γ (1.11) is achieved, according to Proposition 4.1 . This is no restriction when studying the limit
γ → C e m b − . \gamma\to C_{\rm emb}^-. γ → C emb − . Let Φ r \Phi_r Φ r C e m b C_{\rm emb} C emb
∥ Φ r ∥ L r ( Ω ) = 1. \left\lVert \Phi_r\right\rVert_{L^r(\Omega)}=1. ∥ Φ r ∥ L r ( Ω ) = 1. In particular,
∥ ∇ Φ r ∥ L 2 ( Ω ) 2 = C e m b [ Φ r ] s 2 . \left\lVert \nabla\Phi_r\right\rVert_{L^2(\Omega)}^2
=C_{\rm emb}[\Phi_r]_s^2. ∥ ∇ Φ r ∥ L 2 ( Ω ) 2 = C emb [ Φ r ] s 2 . Proposition 4.2. For every 1 < r ≤ 2 ∗ 1<r\le2^* 1 < r ≤ 2 ∗ S r ( γ ) ⟶ 0 as γ → C e m b − .
S_r(\gamma)\longrightarrow0
\qquad\text{as }\gamma\to C_{\rm emb}^-.
S r ( γ ) ⟶ 0 as γ → C emb − . 0 < S r ( γ ) ≤ ( C e m b − γ ) [ Φ r ] s 2 .
0<S_r(\gamma)\le(C_{\rm emb}-\gamma)[\Phi_r]_s^2.
0 < S r ( γ ) ≤ ( C emb − γ ) [ Φ r ] s 2 . u γ , r u_{\gamma,r} u γ , r [ u γ , r ] s 2 ≤ [ Φ r ] s 2
[u_{\gamma,r}]_s^2\le[\Phi_r]_s^2
[ u γ , r ] s 2 ≤ [ Φ r ] s 2 ∥ ∇ u γ , r ∥ L 2 ( Ω ) 2 ≤ C e m b [ Φ r ] s 2 .
\left\lVert \nabla u_{\gamma,r}\right\rVert_{L^2(\Omega)}^2
\le C_{\rm emb}[\Phi_r]_s^2.
∥ ∇ u γ , r ∥ L 2 ( Ω ) 2 ≤ C emb [ Φ r ] s 2 .
Proof. Strict positivity of S r ( γ ) S_r(\gamma) S r ( γ ) (4.1) and the Sobolev–Poincaré inequality. Since ∥ ∇ Φ r ∥ 2 2 = C e m b [ Φ r ] s 2 \left\lVert \nabla\Phi_r\right\rVert_2^2=C_{\rm emb}[\Phi_r]_s^2 ∥ ∇ Φ r ∥ 2 2 = C emb [ Φ r ] s 2 S r ( γ ) S_r(\gamma) S r ( γ )
S r ( γ ) ≤ Q γ ( Φ r ) = ( C e m b − γ ) [ Φ r ] s 2 , S_r(\gamma)\le Q_\gamma(\Phi_r)
=(C_{\rm emb}-\gamma)[\Phi_r]_s^2, S r ( γ ) ≤ Q γ ( Φ r ) = ( C emb − γ ) [ Φ r ] s 2 , which proves (4.4) and (4.3) .
Set
Q C e m b ( u ) : = ∥ ∇ u ∥ 2 2 − C e m b [ u ] s 2 ≥ 0. Q_{C_{\rm emb}}(u):=\left\lVert \nabla u\right\rVert_2^2-C_{\rm emb}[u]_s^2\ge0. Q C emb ( u ) := ∥ ∇ u ∥ 2 2 − C emb [ u ] s 2 ≥ 0. Since
Q γ ( u ) = Q C e m b ( u ) + ( C e m b − γ ) [ u ] s 2 , Q_\gamma(u)=Q_{C_{\rm emb}}(u)+(C_{\rm emb}-\gamma)[u]_s^2, Q γ ( u ) = Q C emb ( u ) + ( C emb − γ ) [ u ] s 2 , we obtain
( C e m b − γ ) [ u γ , r ] s 2 ≤ Q γ ( u γ , r ) = S r ( γ ) ≤ ( C e m b − γ ) [ Φ r ] s 2 , (C_{\rm emb}-\gamma)[u_{\gamma,r}]_s^2
\le Q_\gamma(u_{\gamma,r})=S_r(\gamma)
\le(C_{\rm emb}-\gamma)[\Phi_r]_s^2, ( C emb − γ ) [ u γ , r ] s 2 ≤ Q γ ( u γ , r ) = S r ( γ ) ≤ ( C emb − γ ) [ Φ r ] s 2 , which gives (4.5) . Finally,
∥ ∇ u γ , r ∥ 2 2 = S r ( γ ) + γ [ u γ , r ] s 2 ≤ ( C e m b − γ ) [ Φ r ] s 2 + γ [ Φ r ] s 2 = C e m b [ Φ r ] s 2 . \begin{split}
\left\lVert \nabla u_{\gamma,r}\right\rVert_2^2
&=S_r(\gamma)+\gamma[u_{\gamma,r}]_s^2\\
&\le(C_{\rm emb}-\gamma)[\Phi_r]_s^2+\gamma[\Phi_r]_s^2
=C_{\rm emb}[\Phi_r]_s^2.
\end{split} ∥ ∇ u γ , r ∥ 2 2 = S r ( γ ) + γ [ u γ , r ] s 2 ≤ ( C emb − γ ) [ Φ r ] s 2 + γ [ Φ r ] s 2 = C emb [ Φ r ] s 2 . This closes the proof. ◻
Theorem 4.3. Let ( γ k ) k ⊂ ( 0 , C e m b ) (\gamma_k)_k\subset(0,C_{\rm emb}) ( γ k ) k ⊂ ( 0 , C emb ) γ k → C e m b − \gamma_k\to C_{\rm emb}^- γ k → C emb − u γ k , r u_{\gamma_k,r} u γ k , r 0 ≢ u ∞ ∈ X 1 , 2 ( Ω ) 0\not\equiv u_\infty\in \mathcal{X}^{1,2}(\Omega) 0 ≡ u ∞ ∈ X 1 , 2 ( Ω ) u γ k , r → u ∞ strongly in X 1 , 2 ( Ω ) ,
u_{\gamma_k,r}\to u_\infty\qquad\text{strongly in }\mathcal{X}^{1,2}(\Omega),
u γ k , r → u ∞ strongly in X 1 , 2 ( Ω ) , ∥ u ∞ ∥ L r ( Ω ) = 1 , ∥ ∇ u ∞ ∥ 2 2 = C e m b [ u ∞ ] s 2 .
\left\lVert u_\infty\right\rVert_{L^r(\Omega)}=1,
\qquad
\left\lVert \nabla u_\infty\right\rVert_2^2=C_{\rm emb}[u_\infty]_s^2.
∥ u ∞ ∥ L r ( Ω ) = 1 , ∥ ∇ u ∞ ∥ 2 2 = C emb [ u ∞ ] s 2 . u ∞ u_\infty u ∞ L r L^r L r C e m b C_{\rm emb} C emb C e m b C_{\rm emb} C emb Φ s \Phi_s Φ s L r L^r L r u ∞ = Φ s or u ∞ = − Φ s .
u_\infty=\Phi_s\qquad\text{or}\qquad u_\infty=-\Phi_s.
u ∞ = Φ s or u ∞ = − Φ s .
Proof. By Proposition 4.2 , ( u γ k , r ) k (u_{\gamma_k,r})_k ( u γ k , r ) k X 1 , 2 ( Ω ) \mathcal{X}^{1,2}(\Omega) X 1 , 2 ( Ω )
u γ k , r ⇀ u ∞ in X 1 , 2 ( Ω ) , u γ k , r → u ∞ in H s ( R n ) . u_{\gamma_k,r}\rightharpoonup u_\infty\quad\text{in }\mathcal{X}^{1,2}(\Omega),
\qquad
u_{\gamma_k,r}\to u_\infty\quad\text{in }H^s(\mathbb{R}^n). u γ k , r ⇀ u ∞ in X 1 , 2 ( Ω ) , u γ k , r → u ∞ in H s ( R n ) . In particular,
[ u γ k , r ] s 2 → [ u ∞ ] s 2 . \begin{equation}
[u_{\gamma_k,r}]_s^2\to[u_\infty]_s^2. \tag*{(4.7)}
\end{equation} [ u γ k , r ] s 2 → [ u ∞ ] s 2 . (4.7) Moreover,
Q C e m b ( u γ k , r ) = Q γ k ( u γ k , r ) − ( C e m b − γ k ) [ u γ k , r ] s 2 = S r ( γ k ) − ( C e m b − γ k ) [ u γ k , r ] s 2 ⟶ 0 , \begin{equation}
\begin{split}
Q_{C_{\rm emb}}(u_{\gamma_k,r})
&=Q_{\gamma_k}(u_{\gamma_k,r})-(C_{\rm emb}-\gamma_k)[u_{\gamma_k,r}]_s^2\\
&=S_r(\gamma_k)-(C_{\rm emb}-\gamma_k)[u_{\gamma_k,r}]_s^2\longrightarrow0,
\end{split} \tag*{(4.8)}
\end{equation} Q C emb ( u γ k , r ) = Q γ k ( u γ k , r ) − ( C emb − γ k ) [ u γ k , r ] s 2 = S r ( γ k ) − ( C emb − γ k ) [ u γ k , r ] s 2 ⟶ 0 , (4.8) where we used (4.3) and (4.5) . By weak lower semicontinuity and (4.7) ,
Q C e m b ( u ∞ ) ≤ lim inf k → ∞ Q C e m b ( u γ k , r ) = 0. Q_{C_{\rm emb}}(u_\infty)
\le\liminf_{k\to\infty}Q_{C_{\rm emb}}(u_{\gamma_k,r})=0. Q C emb ( u ∞ ) ≤ k → ∞ lim inf Q C emb ( u γ k , r ) = 0. Since Q C e m b ≥ 0 Q_{C_{\rm emb}}\ge0 Q C emb ≥ 0
Q C e m b ( u ∞ ) = 0. \begin{equation}
Q_{C_{\rm emb}}(u_\infty)=0. \tag*{(4.9)}
\end{equation} Q C emb ( u ∞ ) = 0. (4.9) Combining (4.8) , (4.7) and (4.9) , we obtain
∥ ∇ u γ k , r ∥ 2 2 = Q C e m b ( u γ k , r ) + C e m b [ u γ k , r ] s 2 ⟶ C e m b [ u ∞ ] s 2 = ∥ ∇ u ∞ ∥ 2 2 . \left\lVert \nabla u_{\gamma_k,r}\right\rVert_2^2
=Q_{C_{\rm emb}}(u_{\gamma_k,r})+C_{\rm emb}[u_{\gamma_k,r}]_s^2
\longrightarrow C_{\rm emb}[u_\infty]_s^2
=\left\lVert \nabla u_\infty\right\rVert_2^2. ∥ ∇ u γ k , r ∥ 2 2 = Q C emb ( u γ k , r ) + C emb [ u γ k , r ] s 2 ⟶ C emb [ u ∞ ] s 2 = ∥ ∇ u ∞ ∥ 2 2 . Weak convergence together with convergence of the norms gives strong convergence in H 0 1 ( Ω ) H^1_0(\Omega) H 0 1 ( Ω ) (4.7) , in X 1 , 2 ( Ω ) \mathcal{X}^{1,2}(\Omega) X 1 , 2 ( Ω )
u γ k , r → u ∞ u_{\gamma_k,r}\to u_\infty u γ k , r → u ∞ strongly in L r ( Ω ) L^r(\Omega) L r ( Ω ) r = 2 ∗ r=2^* r = 2 ∗ ∥ u ∞ ∥ r = 1 \|u_\infty\|_r=1 ∥ u ∞ ∥ r = 1 (4.9) identifies u ∞ u_\infty u ∞ C e m b C_{\rm emb} C emb C e m b C_{\rm emb} C emb
This closes the proof. ◻
We now prove estimates which are uniform as γ → C e m b − \gamma\to C_{\rm emb}^- γ → C emb − [87 , 88 ] ; here, the main difficulty is the unfavorable sign of the fractional term in (4.2) .
For β > 1 \beta>1 β > 1 T ≥ 1 T\ge1 T ≥ 1
G β , T ( t ) : = t min { ∣ t ∣ , T } β − 1 , H β , T ( t ) : = t min { ∣ t ∣ , T } 2 β − 2 . \begin{equation}
G_{\beta,T}(t):=t\min\{|t|,T\}^{\beta-1},
\qquad
H_{\beta,T}(t):=t\min\{|t|,T\}^{2\beta-2}. \tag*{(4.10)}
\end{equation} G β , T ( t ) := t min { ∣ t ∣ , T } β − 1 , H β , T ( t ) := t min { ∣ t ∣ , T } 2 β − 2 . (4.10) Lemma 4.4. For every β > 1 \beta>1 β > 1 T ≥ 1 T\ge1 T ≥ 1 (4.10) satisfy H β , T ′ ( t ) ≥ c β ∣ G β , T ′ ( t ) ∣ 2 for a.e. t ∈ R , c β : = 2 β − 1 β 2 ,
H_{\beta,T}'(t)\ge c_\beta|G_{\beta,T}'(t)|^2
\quad\text{for a.e. }t\in\mathbb{R},
\qquad
c_\beta:=\frac{2\beta-1}{\beta^2},
H β , T ′ ( t ) ≥ c β ∣ G β , T ′ ( t ) ∣ 2 for a.e. t ∈ R , c β := β 2 2 β − 1 , ( a − b ) ( H β , T ( a ) − H β , T ( b ) ) ≤ 2 ∣ G β , T ( a ) − G β , T ( b ) ∣ 2 for every a , b ∈ R ,
(a-b)\bigl(H_{\beta,T}(a)-H_{\beta,T}(b)\bigr)
\le2|G_{\beta,T}(a)-G_{\beta,T}(b)|^2
\qquad\text{for every }a,b\in\mathbb{R},
( a − b ) ( H β , T ( a ) − H β , T ( b ) ) ≤ 2∣ G β , T ( a ) − G β , T ( b ) ∣ 2 for every a , b ∈ R , t H β , T ( t ) = G β , T ( t ) 2 for every t ∈ R .
tH_{\beta,T}(t)=G_{\beta,T}(t)^2
\qquad\text{for every }t\in\mathbb{R}.
t H β , T ( t ) = G β , T ( t ) 2 for every t ∈ R .
Proof. Set h ( t ) : = min { ∣ t ∣ , T } β − 1 h(t):=\min\{|t|,T\}^{\beta-1} h ( t ) := min { ∣ t ∣ , T } β − 1 G ( t ) = t h ( t ) G(t)=th(t) G ( t ) = t h ( t ) H ( t ) = t h ( t ) 2 H(t)=th(t)^2 H ( t ) = t h ( t ) 2 ∣ t ∣ < T |t|<T ∣ t ∣ < T
H ′ ( t ) = ( 2 β − 1 ) ∣ t ∣ 2 β − 2 , ∣ G ′ ( t ) ∣ 2 = β 2 ∣ t ∣ 2 β − 2 , H'(t)=(2\beta-1)|t|^{2\beta-2},
\qquad
|G'(t)|^2=\beta^2|t|^{2\beta-2}, H ′ ( t ) = ( 2 β − 1 ) ∣ t ∣ 2 β − 2 , ∣ G ′ ( t ) ∣ 2 = β 2 ∣ t ∣ 2 β − 2 , while, if ∣ t ∣ > T |t|>T ∣ t ∣ > T
H ′ ( t ) = T 2 β − 2 , ∣ G ′ ( t ) ∣ 2 = T 2 β − 2 . H'(t)=T^{2\beta-2},
\qquad
|G'(t)|^2=T^{2\beta-2}. H ′ ( t ) = T 2 β − 2 , ∣ G ′ ( t ) ∣ 2 = T 2 β − 2 . Since c β ≤ 1 c_\beta\le1 c β ≤ 1 (4.11) ; (4.13) is immediate.
For (4.12) , a direct computation gives
( a − b ) ( H ( a ) − H ( b ) ) = ( G ( a ) − G ( b ) ) 2 − a b ( h ( a ) − h ( b ) ) 2 . \begin{equation}
(a-b)(H(a)-H(b))
=(G(a)-G(b))^2-ab\,(h(a)-h(b))^2. \tag*{(4.14)}
\end{equation} ( a − b ) ( H ( a ) − H ( b )) = ( G ( a ) − G ( b ) ) 2 − ab ( h ( a ) − h ( b ) ) 2 . (4.14) If a b ≥ 0 ab\ge0 ab ≥ 0 a b < 0 ab<0 ab < 0 ∣ a ∣ ≥ ∣ b ∣ |a|\ge|b| ∣ a ∣ ≥ ∣ b ∣ h h h ∣ t ∣ |t| ∣ t ∣
∣ a b ∣ ( h ( a ) − h ( b ) ) 2 ≤ ∣ a ∣ ∣ b ∣ h ( a ) 2 ≤ ∣ a ∣ 2 h ( a ) 2 ≤ ∣ G ( a ) − G ( b ) ∣ 2 , |ab|\,(h(a)-h(b))^2
\le |a||b|h(a)^2
\le |a|^2h(a)^2
\le |G(a)-G(b)|^2, ∣ ab ∣ ( h ( a ) − h ( b ) ) 2 ≤ ∣ a ∣∣ b ∣ h ( a ) 2 ≤ ∣ a ∣ 2 h ( a ) 2 ≤ ∣ G ( a ) − G ( b ) ∣ 2 , because G ( a ) G(a) G ( a ) G ( b ) G(b) G ( b ) (4.14) then yields (4.12) . ◻
Proposition 4.5. Let R ∈ [ 1 , + ∞ ) R\in[1,+\infty) R ∈ [ 1 , + ∞ ) γ R < C e m b \gamma_R<C_{\rm emb} γ R < C emb M R > 0 M_R>0 M R > 0 ∥ u γ , r ∥ L R ( Ω ) ≤ M R
\left\lVert u_{\gamma,r}\right\rVert_{L^R(\Omega)}\le M_R
∥ u γ , r ∥ L R ( Ω ) ≤ M R γ ∈ ( γ R , C e m b ) \gamma\in(\gamma_R,C_{\rm emb}) γ ∈ ( γ R , C emb ) u γ , r u_{\gamma,r} u γ , r
Proof. Fix β > 1 \beta>1 β > 1 T ≥ 1 T\ge1 T ≥ 1
G γ , T : = G β , T ( u γ , r ) , H γ , T : = H β , T ( u γ , r ) . G_{\gamma,T}:=G_{\beta,T}(u_{\gamma,r}),\qquad
H_{\gamma,T}:=H_{\beta,T}(u_{\gamma,r}). G γ , T := G β , T ( u γ , r ) , H γ , T := H β , T ( u γ , r ) . Since H β , T H_{\beta,T} H β , T H γ , T ∈ X 1 , 2 ( Ω ) H_{\gamma,T}\in \mathcal{X}^{1,2}(\Omega) H γ , T ∈ X 1 , 2 ( Ω ) (4.2) . By (4.11) ,
∫ Ω ∇ u γ , r ⋅ ∇ H γ , T d x ≥ c β ∥ ∇ G γ , T ∥ 2 2 . \begin{equation}
\int_\Omega\nabla u_{\gamma,r}\cdot\nabla H_{\gamma,T}\,dx
\ge c_\beta\left\lVert \nabla G_{\gamma,T}\right\rVert_2^2. \tag*{(4.16)}
\end{equation} ∫ Ω ∇ u γ , r ⋅ ∇ H γ , T d x ≥ c β ∥ ∇ G γ , T ∥ 2 2 . (4.16) On the other hand, (4.12) gives
E s ( u γ , r , H γ , T ) ≤ 2 [ G γ , T ] s 2 . \begin{equation}
\mathcal E_s(u_{\gamma,r},H_{\gamma,T})\le2[G_{\gamma,T}]_s^2. \tag*{(4.17)}
\end{equation} E s ( u γ , r , H γ , T ) ≤ 2 [ G γ , T ] s 2 . (4.17) By Remark 2.8 , choosing ε = c β / ( 4 C e m b ) \varepsilon=c_\beta/(4C_{\rm emb}) ε = c β / ( 4 C emb ) γ < C e m b \gamma<C_{\rm emb} γ < C emb (4.16) –(4.17) that
∫ Ω ∇ u γ , r ⋅ ∇ H γ , T d x − γ E s ( u γ , r , H γ , T ) ≥ c β 2 ∥ ∇ G γ , T ∥ 2 2 − C β ∥ G γ , T ∥ 2 2 , \begin{equation}
\int_\Omega\nabla u_{\gamma,r}\cdot\nabla H_{\gamma,T}\,dx-\gamma\mathcal E_s(u_{\gamma,r},H_{\gamma,T})
\ge\frac{c_\beta}{2}\left\lVert \nabla G_{\gamma,T}\right\rVert_2^2-C_\beta\left\lVert G_{\gamma,T}\right\rVert_2^2, \tag*{(4.18)}
\end{equation} ∫ Ω ∇ u γ , r ⋅ ∇ H γ , T d x − γ E s ( u γ , r , H γ , T ) ≥ 2 c β ∥ ∇ G γ , T ∥ 2 2 − C β ∥ G γ , T ∥ 2 2 , (4.18) where C β > 0 C_\beta>0 C β > 0 T T T γ \gamma γ u γ , r u_{\gamma,r} u γ , r (4.13) , the right-hand side of (4.2) becomes
S r ( γ ) ∫ Ω ∣ u γ , r ∣ r − 2 G γ , T 2 d x . \begin{equation*}
S_r(\gamma)\int_\Omega|u_{\gamma,r}|^{r-2}G_{\gamma,T}^2\,dx.
\end{equation*} S r ( γ ) ∫ Ω ∣ u γ , r ∣ r − 2 G γ , T 2 d x . We distinguish two cases. If 2 ≤ r ≤ 2 ∗ 2\le r\le2^* 2 ≤ r ≤ 2 ∗ ∥ u γ , r ∥ r = 1 \left\lVert u_{\gamma,r}\right\rVert_r=1 ∥ u γ , r ∥ r = 1
∫ Ω ∣ u γ , r ∣ r − 2 G γ , T 2 d x ≤ ∥ G γ , T ∥ r 2 ≤ C ∥ G γ , T ∥ 2 ∗ 2 . \int_\Omega|u_{\gamma,r}|^{r-2}G_{\gamma,T}^2\,dx\le\left\lVert G_{\gamma,T}\right\rVert_r^2
\le C\left\lVert G_{\gamma,T}\right\rVert_{2^*}^2. ∫ Ω ∣ u γ , r ∣ r − 2 G γ , T 2 d x ≤ ∥ G γ , T ∥ r 2 ≤ C ∥ G γ , T ∥ 2 ∗ 2 . Using (2.7) in (4.18) and recalling that S r ( γ ) → 0 S_r(\gamma)\to0 S r ( γ ) → 0 γ \gamma γ C e m b C_{\rm emb} C emb
∥ G γ , T ∥ 2 ∗ 2 ≤ K β ( 1 + ∥ G γ , T ∥ 2 2 ) , \begin{equation}
\left\lVert G_{\gamma,T}\right\rVert_{2^*}^2\le K_\beta\bigl(1+\left\lVert G_{\gamma,T}\right\rVert_2^2\bigr), \tag*{(4.19)}
\end{equation} ∥ G γ , T ∥ 2 ∗ 2 ≤ K β ( 1 + ∥ G γ , T ∥ 2 2 ) , (4.19) with K β K_\beta K β T T T γ \gamma γ u γ , r u_{\gamma,r} u γ , r
If 1 < r < 2 1<r<2 1 < r < 2
∣ u γ , r ∣ r − 2 G γ , T 2 ≤ 1 + G γ , T 2 . \begin{equation*}
|u_{\gamma,r}|^{r-2}G_{\gamma,T}^2\le1+G_{\gamma,T}^2.
\end{equation*} ∣ u γ , r ∣ r − 2 G γ , T 2 ≤ 1 + G γ , T 2 . Indeed, on { ∣ u γ , r ∣ ≤ 1 } \{|u_{\gamma,r}|\le1\} { ∣ u γ , r ∣ ≤ 1 } 1 1 1 { ∣ u γ , r ∣ > 1 } \{|u_{\gamma,r}|>1\} { ∣ u γ , r ∣ > 1 } ∣ u γ , r ∣ r ≤ ∣ u γ , r ∣ 2 |u_{\gamma,r}|^r\le|u_{\gamma,r}|^2 ∣ u γ , r ∣ r ≤ ∣ u γ , r ∣ 2 S r ( γ ) S_r(\gamma) S r ( γ ) γ \gamma γ C e m b C_{\rm emb} C emb (4.19) follows again from (4.18) and (2.7) .
We now remove the L 2 L^2 L 2 (4.19) without any iteration. Let P : = 2 ∗ P:=2^* P := 2 ∗ M ≥ 1 M\ge1 M ≥ 1
E M : = { x ∈ Ω : ∣ u γ , r ( x ) ∣ > M } . E_M:=\{x\in\Omega:|u_{\gamma,r}(x)|>M\}. E M := { x ∈ Ω : ∣ u γ , r ( x ) ∣ > M } . Since ∥ u γ , r ∥ r = 1 \left\lVert u_{\gamma,r}\right\rVert_r=1 ∥ u γ , r ∥ r = 1
∣ E M ∣ ≤ M − r . \begin{equation*}
|E_M|\le M^{-r}.
\end{equation*} ∣ E M ∣ ≤ M − r . On Ω ∖ E M \Omega\setminus E_M Ω ∖ E M ∣ G γ , T ∣ ≤ ∣ u γ , r ∣ β ≤ M β |G_{\gamma,T}|\le|u_{\gamma,r}|^\beta\le M^\beta ∣ G γ , T ∣ ≤ ∣ u γ , r ∣ β ≤ M β E M E_M E M
∫ E M ∣ G γ , T ∣ 2 d x ≤ ∣ E M ∣ 1 − 2 / P ∥ G γ , T ∥ P 2 ≤ M − 2 r / n ∥ G γ , T ∥ P 2 . \int_{E_M}|G_{\gamma,T}|^2\,dx
\le |E_M|^{1-2/P}\left\lVert G_{\gamma,T}\right\rVert_P^2
\le M^{-2r/n}\left\lVert G_{\gamma,T}\right\rVert_P^2. ∫ E M ∣ G γ , T ∣ 2 d x ≤ ∣ E M ∣ 1 − 2/ P ∥ G γ , T ∥ P 2 ≤ M − 2 r / n ∥ G γ , T ∥ P 2 . Consequently,
∥ G γ , T ∥ 2 2 ≤ ∣ Ω ∣ M 2 β + M − 2 r / n ∥ G γ , T ∥ P 2 . \begin{equation}
\left\lVert G_{\gamma,T}\right\rVert_2^2\le|\Omega|M^{2\beta}
+M^{-2r/n}\left\lVert G_{\gamma,T}\right\rVert_P^2. \tag*{(4.20)}
\end{equation} ∥ G γ , T ∥ 2 2 ≤ ∣Ω∣ M 2 β + M − 2 r / n ∥ G γ , T ∥ P 2 . (4.20) Inserting (4.20) into (4.19) , and choosing M M M K β M − 2 r / n ≤ 1 / 2 K_\beta M^{-2r/n}\le1/2 K β M − 2 r / n ≤ 1/2
∥ G γ , T ∥ P 2 ≤ A β , \begin{equation*}
\left\lVert G_{\gamma,T}\right\rVert_P^2\le A_\beta,
\end{equation*} ∥ G γ , T ∥ P 2 ≤ A β , where A β A_\beta A β T T T γ \gamma γ u γ , r u_{\gamma,r} u γ , r T → + ∞ T\to+\infty T → + ∞
∥ u γ , r ∥ L β P ( Ω ) 2 β ≤ A β . \begin{equation}
\left\lVert u_{\gamma,r}\right\rVert_{L^{\beta P}(\Omega)}^{2\beta}\le A_\beta. \tag*{(4.21)}
\end{equation} ∥ u γ , r ∥ L β P ( Ω ) 2 β ≤ A β . (4.21) Given R < + ∞ R<+\infty R < + ∞ β > 1 \beta>1 β > 1 R ≤ β P R\le\beta P R ≤ β P Ω \Omega Ω (4.15) follows from (4.21) . ◻
Corollary 4.6. For every fixed q ∈ [ 1 , + ∞ ) q\in[1,+\infty) q ∈ [ 1 , + ∞ ) S r ( γ ) ∣ u γ , r ∣ r − 2 u γ , r ⟶ 0 strongly in L q ( Ω )
S_r(\gamma)|u_{\gamma,r}|^{r-2}u_{\gamma,r}
\longrightarrow0
\qquad\text{strongly in }L^q(\Omega)
S r ( γ ) ∣ u γ , r ∣ r − 2 u γ , r ⟶ 0 strongly in L q ( Ω ) γ → C e m b − \gamma\to C_{\rm emb}^- γ → C emb − u γ , r u_{\gamma,r} u γ , r
Proof. Set R : = max { 1 , q ( r − 1 ) } R:=\max\{1,q(r-1)\} R := max { 1 , q ( r − 1 )} 4.5 and Hölder’s inequality give
∥ S r ( γ ) ∣ u γ , r ∣ r − 2 u γ , r ∥ q ≤ S r ( γ ) ∣ Ω ∣ 1 / q − ( r − 1 ) / R ∥ u γ , r ∥ R r − 1 ≤ C R S r ( γ ) , \left\lVert S_r(\gamma)|u_{\gamma,r}|^{r-2}u_{\gamma,r}\right\rVert_q
\le S_r(\gamma)|\Omega|^{1/q-(r-1)/R}\left\lVert u_{\gamma,r}\right\rVert_R^{r-1}
\le C_RS_r(\gamma), S r ( γ ) ∣ u γ , r ∣ r − 2 u γ , r q ≤ S r ( γ ) ∣Ω ∣ 1/ q − ( r − 1 ) / R ∥ u γ , r ∥ R r − 1 ≤ C R S r ( γ ) , and the conclusion follows from (4.3) . ◻
Proof of Theorem 1.3 . This is now a simple combination of Proposition 4.2 , Theorem 4.3 and Proposition 4.5 . This closes the proof. ◻
Proof of Theorem 1.4
In this section we prove Theorem 1.4. Throughout the section, we assume that
and that Ω \Omega Ω C 1 , 1 C^{1,1} C 1 , 1
We shall use directly the global regularity estimate in [74 Theorem 1.8] . There is only one normalization issue to keep track of. In [74 ] , the fractional Laplacian contains the standard positive factor c n , s c_{n,s} c n , s (2.2) corresponds to c n , s = 1 c_{n,s}=1 c n , s = 1
− Δ v − γ ( − Δ ) s v = g -\Delta v-\gamma(-\Delta)^s v=g − Δ v − γ ( − Δ ) s v = g corresponds in the notation of [74 ] to the signed measure
μ γ = − γ c n , s δ s . \begin{equation*}
\mu_\gamma=-\frac{\gamma}{c_{n,s}}\delta_s.
\end{equation*} μ γ = − c n , s γ δ s . Since s s s 0 ≤ γ ≤ C e m b 0\le\gamma\le C_{\rm emb} 0 ≤ γ ≤ C emb q ∈ [ 2 , + ∞ ) q\in[2,+\infty) q ∈ [ 2 , + ∞ ) [74 Theorem 1.8, formula (1.26)] yields
∥ v ∥ W 2 , q ( Ω ) ≤ C ( ∥ g ∥ L q ( Ω ) + ∥ v ∥ L 2 ( Ω ) ) , \begin{equation}
\left\lVert v\right\rVert_{W^{2,q}(\Omega)}
\le C\bigl(\left\lVert g\right\rVert_{L^q(\Omega)}+\left\lVert v\right\rVert_{L^2(\Omega)}\bigr), \tag*{(5.1)}
\end{equation} ∥ v ∥ W 2 , q ( Ω ) ≤ C ( ∥ g ∥ L q ( Ω ) + ∥ v ∥ L 2 ( Ω ) ) , (5.1) where C C C γ ∈ [ 0 , C e m b ] \gamma\in[0,C_{\rm emb}] γ ∈ [ 0 , C emb ]
Proof. Apply (5.1) to (4.2) , with
g = S r ( γ ) ∣ u γ , r ∣ r − 2 u γ , r . g=S_r(\gamma)|u_{\gamma,r}|^{r-2}u_{\gamma,r}. g = S r ( γ ) ∣ u γ , r ∣ r − 2 u γ , r . The L q L^q L q g g g 4.6 , while the L 2 L^2 L 2 u γ , r u_{\gamma,r} u γ , r 4.2 . This proves (5.2) . The second assertion follows from the Sobolev–Morrey embedding. ◻
Proposition 5.2. Let γ k → C e m b − \gamma_k\to C_{\rm emb}^{-} γ k → C emb − u γ k , r u_{\gamma_k,r} u γ k , r u ∞ u_\infty u ∞ 4.3 , so that u γ k , r ⟶ u ∞ strongly in X 1 , 2 ( Ω ) . u_{\gamma_k,r}\longrightarrow u_\infty
\qquad\text{strongly in }\mathcal{X}^{1,2}(\Omega). u γ k , r ⟶ u ∞ strongly in X 1 , 2 ( Ω ) . q ∈ [ 2 , + ∞ ) q\in[2,+\infty) q ∈ [ 2 , + ∞ ) u γ k , r → u ∞ strongly in W 2 , q ( Ω ) .
u_{\gamma_k,r}\to u_\infty\qquad\text{strongly in }W^{2,q}(\Omega).
u γ k , r → u ∞ strongly in W 2 , q ( Ω ) . q > n q>n q > n u γ k , r → u ∞ strongly in C 1 ( Ω ‾ ) .
u_{\gamma_k,r}\to u_\infty\qquad\text{strongly in }C^1(\overline\Omega).
u γ k , r → u ∞ strongly in C 1 ( Ω ) . C e m b C_{\rm emb} C emb σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 } u γ k , r → σ Φ s strongly in W 2 , q ( Ω ) and C 1 ( Ω ‾ ) ,
u_{\gamma_k,r}\to\sigma\Phi_s
\qquad\text{strongly in }W^{2,q}(\Omega)\text{ and }C^1(\overline\Omega),
u γ k , r → σ Φ s strongly in W 2 , q ( Ω ) and C 1 ( Ω ) , Φ s \Phi_s Φ s L r L^r L r C e m b C_{\rm emb} C emb
Proof. The limit u ∞ u_\infty u ∞
− Δ u ∞ − C e m b ( − Δ ) s u ∞ = 0. -\Delta u_\infty-C_{\rm emb}(-\Delta)^s u_\infty=0. − Δ u ∞ − C emb ( − Δ ) s u ∞ = 0. Applying (5.1) at γ = C e m b \gamma=C_{\rm emb} γ = C emb u ∞ ∈ W 2 , q ( Ω ) u_\infty\in W^{2,q}(\Omega) u ∞ ∈ W 2 , q ( Ω )
( − Δ ) s u ∞ = − Δ u ∞ C e m b ∈ L q ( Ω ) . \begin{equation}
(-\Delta)^s u_\infty
=\frac{-\Delta u_\infty}{C_{\rm emb}}\in L^q(\Omega). \tag*{(5.5)}
\end{equation} ( − Δ ) s u ∞ = C emb − Δ u ∞ ∈ L q ( Ω ) . (5.5) Set
f γ k : = S r ( γ k ) ∣ u γ k , r ∣ r − 2 u γ k , r , w k : = u γ k , r − u ∞ . f_{\gamma_k}:=S_r(\gamma_k)|u_{\gamma_k,r}|^{r-2}u_{\gamma_k,r},
\qquad
w_k:=u_{\gamma_k,r}-u_\infty. f γ k := S r ( γ k ) ∣ u γ k , r ∣ r − 2 u γ k , r , w k := u γ k , r − u ∞ . Then
− Δ w k − γ k ( − Δ ) s w k = f γ k − ( C e m b − γ k ) ( − Δ ) s u ∞ . \begin{equation*}
-\Delta w_k-\gamma_k(-\Delta)^s w_k
=f_{\gamma_k}-(C_{\rm emb}-\gamma_k)(-\Delta)^s u_\infty.
\end{equation*} − Δ w k − γ k ( − Δ ) s w k = f γ k − ( C emb − γ k ) ( − Δ ) s u ∞ . Using (5.1) once more, together with Corollary 4.6 , (5.5) , and the strong X 1 , 2 \mathcal{X}^{1,2} X 1 , 2 4.3 , we obtain
∥ u γ k , r − u ∞ ∥ W 2 , q ( Ω ) ≤ C ( ∥ f γ k ∥ q + ( C e m b − γ k ) ∥ ( − Δ ) s u ∞ ∥ q + ∥ u γ k , r − u ∞ ∥ 2 ) ⟶ 0. \begin{split}
\left\lVert u_{\gamma_k,r}-u_\infty\right\rVert_{W^{2,q}(\Omega)}
\le C\bigl(&\left\lVert f_{\gamma_k}\right\rVert_q
+(C_{\rm emb}-\gamma_k)\left\lVert (-\Delta)^s u_\infty\right\rVert_q\\
&+\left\lVert u_{\gamma_k,r}-u_\infty\right\rVert_2\bigr)\longrightarrow0.
\end{split} ∥ u γ k , r − u ∞ ∥ W 2 , q ( Ω ) ≤ C ( ∥ f γ k ∥ q + ( C emb − γ k ) ∥ ( − Δ ) s u ∞ ∥ q + ∥ u γ k , r − u ∞ ∥ 2 ) ⟶ 0. This proves (5.3) ; (5.4) follows for q > n q>n q > n (4.6) . ◻
Proof of Theorem 1.4 . Combining Proposition 5.1 and Proposition 5.2 we get the desired conclusion. This closes the proof. ◻
Proof of Theorems 1.6 and 1.8
In this last section, we complete the analysis of the bounded-domain problem. We first prove Theorem 1.6 by combining the convergence obtained in § 4 and § 5 with the small-s s s 2.4 and the Hopf lemma. We then turn to the Sobolev supercritical regime and prove Theorem 1.8 by means of a Pohozaev identity.
We begin with the positivity of the normalized minimizers. For this part of the section, we assume that Ω \Omega Ω C 1 , 1 C^{1,1} C 1 , 1 1 < r ≤ 2 ∗ 1<r\le2^* 1 < r ≤ 2 ∗
Let φ 1 \varphi_1 φ 1 − Δ -\Delta − Δ ∥ φ 1 ∥ L r ( Ω ) = 1 \|\varphi_1\|_{L^r(\Omega)}=1 ∥ φ 1 ∥ L r ( Ω ) = 1
φ 1 > 0 in Ω , ∂ ν i n φ 1 > 0 on ∂ Ω , \begin{equation*}
\varphi_1>0\quad\text{in }\Omega,
\qquad
\partial_{\nu_{\rm in}}\varphi_1>0\quad\text{on }\partial\Omega,
\end{equation*} φ 1 > 0 in Ω , ∂ ν in φ 1 > 0 on ∂ Ω , where ν i n \nu_{\rm in} ν in
Lemma 6.1. Let ϕ ∈ C 1 ( Ω ‾ ) \phi\in C^1(\overline\Omega) ϕ ∈ C 1 ( Ω ) ϕ = 0 on ∂ Ω , ϕ > 0 in Ω , inf ∂ Ω ∂ ν i n ϕ > 0. \phi=0\quad\text{on }\partial\Omega,
\qquad
\phi>0\quad\text{in }\Omega,
\qquad
\inf_{\partial\Omega}\partial_{\nu_{\rm in}}\phi>0. ϕ = 0 on ∂ Ω , ϕ > 0 in Ω , inf ∂ Ω ∂ ν in ϕ > 0. η > 0 \eta>0 η > 0 v ∈ C 1 ( Ω ‾ ) v\in C^1(\overline\Omega) v ∈ C 1 ( Ω ) v = 0 v=0 v = 0 ∂ Ω \partial\Omega ∂ Ω ∥ v − ϕ ∥ C 1 ( Ω ‾ ) < η \left\lVert v-\phi\right\rVert_{C^1(\overline\Omega)}<\eta ∥ v − ϕ ∥ C 1 ( Ω ) < η Ω \Omega Ω
Proof. Set
a : = min ∂ Ω ∂ ν i n ϕ > 0. a:=\min_{\partial\Omega}\partial_{\nu_{\rm in}}\phi>0. a := ∂ Ω min ∂ ν in ϕ > 0. By continuity, there exists δ > 0 \delta>0 δ > 0 ϕ \phi ϕ 3 a / 4 3a/4 3 a /4
Ω δ : = { x ∈ Ω : dist ( x , ∂ Ω ) < δ } . \Omega_\delta:=\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)<\delta\}. Ω δ := { x ∈ Ω : dist ( x , ∂ Ω ) < δ } . If ∥ v − ϕ ∥ C 1 < a / 4 \left\lVert v-\phi\right\rVert_{C^1}<a/4 ∥ v − ϕ ∥ C 1 < a /4 v v v a / 2 a/2 a /2 v = 0 v=0 v = 0 ∂ Ω \partial\Omega ∂ Ω
v ( x ) ≥ a 2 dist ( x , ∂ Ω ) > 0 for x ∈ Ω δ . v(x)\ge\frac a2\operatorname{dist}(x,\partial\Omega)>0
\qquad\text{for }x\in\Omega_\delta. v ( x ) ≥ 2 a dist ( x , ∂ Ω ) > 0 for x ∈ Ω δ . On the compact set K δ : = { x ∈ Ω : dist ( x , ∂ Ω ) ≥ δ } K_\delta:=\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)\ge\delta\} K δ := { x ∈ Ω : dist ( x , ∂ Ω ) ≥ δ } m δ : = min K δ ϕ > 0 m_\delta:=\min_{K_\delta}\phi>0 m δ := min K δ ϕ > 0 ∥ v − ϕ ∥ C 0 < m δ / 2 \left\lVert v-\phi\right\rVert_{C^0}<m_\delta/2 ∥ v − ϕ ∥ C 0 < m δ /2 v > 0 v>0 v > 0 K δ K_\delta K δ
Thanks to the above lemma, we can now provide the
Proof of Theorem 1.6 . Let η > 0 \eta>0 η > 0 6.1 with ϕ = φ 1 \phi=\varphi_1 ϕ = φ 1 2.4 , reducing s 0 s_0 s 0 s 1 ∈ ( 0 , s 0 ) s_1\in(0,s_0) s 1 ∈ ( 0 , s 0 )
∥ Φ s − φ 1 ∥ C 1 ( Ω ‾ ) < η 2 for every s ∈ ( 0 , s 1 ) , \begin{equation*}
\left\lVert \Phi_s-\varphi_1\right\rVert_{C^1(\overline\Omega)}<\frac\eta2
\qquad\text{for every }s\in(0,s_1),
\end{equation*} ∥ Φ s − φ 1 ∥ C 1 ( Ω ) < 2 η for every s ∈ ( 0 , s 1 ) , where Φ s \Phi_s Φ s L r L^r L r C e m b C_{\rm emb} C emb s ∈ ( 0 , s 1 ) s\in(0,s_1) s ∈ ( 0 , s 1 ) γ k → C e m b − \gamma_k\to C_{\rm emb}^{-} γ k → C emb − u γ k , r u_{\gamma_k,r} u γ k , r Ω \Omega Ω
By Theorem 4.3 and the simplicity of C e m b C_{\rm emb} C emb σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 }
u γ k , r → σ Φ s strongly in X 1 , 2 ( Ω ) . u_{\gamma_k,r}\to\sigma\Phi_s\qquad\text{strongly in }\mathcal{X}^{1,2}(\Omega). u γ k , r → σ Φ s strongly in X 1 , 2 ( Ω ) . Proposition 5.2 , with any q > n q>n q > n
u γ k , r → σ Φ s strongly in C 1 ( Ω ‾ ) . \begin{equation*}
u_{\gamma_k,r}\to\sigma\Phi_s
\qquad\text{strongly in }C^1(\overline\Omega).
\end{equation*} u γ k , r → σ Φ s strongly in C 1 ( Ω ) . Therefore, for k k k
∥ σ u γ k , r − φ 1 ∥ C 1 ( Ω ‾ ) ≤ ∥ σ u γ k , r − Φ s ∥ C 1 ( Ω ‾ ) + ∥ Φ s − φ 1 ∥ C 1 ( Ω ‾ ) < η . \left\lVert \sigma u_{\gamma_k,r}-\varphi_1\right\rVert_{C^1(\overline\Omega)}
\le\left\lVert \sigma u_{\gamma_k,r}-\Phi_s\right\rVert_{C^1(\overline\Omega)}
+\left\lVert \Phi_s-\varphi_1\right\rVert_{C^1(\overline\Omega)}<\eta. ∥ σ u γ k , r − φ 1 ∥ C 1 ( Ω ) ≤ ∥ σ u γ k , r − Φ s ∥ C 1 ( Ω ) + ∥ Φ s − φ 1 ∥ C 1 ( Ω ) < η . Lemma 6.1 gives σ u γ k , r > 0 \sigma u_{\gamma_k,r}>0 σ u γ k , r > 0 Ω \Omega Ω u γ k , r u_{\gamma_k,r} u γ k , r γ ‾ r , s \overline\gamma_{r,s} γ r , s
We now turn to establish Theorem 1.8 . The proof is based on a Pohozaev-type identity adapted to the wrong-sign operator. We first recall the identity in the form needed below.
Proposition 6.2. Let Ω ⊂ R n \Omega \subset \mathbb{R}^n Ω ⊂ R n C 2 C^2 C 2 ∂ Ω \partial \Omega ∂ Ω ν \nu ν ∂ Ω \partial \Omega ∂ Ω f : R → R f:\mathbb{R}\to \mathbb{R} f : R → R s ∈ ( 0 , 1 / 2 ) s \in (0,1/2) s ∈ ( 0 , 1/2 ) u ∈ C 2 ( Ω ‾ ) ∩ C ( R n ) u \in C^{2}(\overline{\Omega})\cap C(\mathbb{R}^n) u ∈ C 2 ( Ω ) ∩ C ( R n ) { − Δ u − γ ( − Δ ) s u = f ( u ) in Ω , u = 0 in R n ∖ Ω .
\begin{cases}
-\Delta u - \gamma(-\Delta)^s u =f(u)&\text{in }\Omega,\\
u=0&\text{in }\mathbb{R}^n\setminus\Omega.
\end{cases}
{ − Δ u − γ ( − Δ ) s u = f ( u ) u = 0 in Ω , in R n ∖ Ω. n − 2 2 ∫ Ω ∣ ∇ u ∣ 2 − γ n − 2 s 2 [ u ] s 2 − n ∫ Ω F ( u ) = − 1 2 ∫ ∂ Ω ( x ⋅ ν ) ( ∂ u ∂ ν ) 2 d σ ,
\dfrac{n-2}{2} \int_{\Omega}|\nabla u|^2 - \gamma \, \dfrac{n-2s}{2} \, [u]^2_s - n \int_{\Omega}F(u) =-\dfrac{1}{2}\int_{\partial\Omega}(x\cdot \nu) \, \left(\dfrac{\partial u}{\partial \nu}\right)^2\,d\sigma,
2 n − 2 ∫ Ω ∣∇ u ∣ 2 − γ 2 n − 2 s [ u ] s 2 − n ∫ Ω F ( u ) = − 2 1 ∫ ∂ Ω ( x ⋅ ν ) ( ∂ ν ∂ u ) 2 d σ , F ( t ) : = ∫ 0 t f ( τ ) d τ F(t):=\int_{0}^{t}f(\tau)\, d\tau F ( t ) := ∫ 0 t f ( τ ) d τ
The identity is the same as the one proved in [56 Theorem 1.1] . We only need to keep track of two minor points. First, our fractional Laplacian differs from the normalization used in [56 ] by a positive multiplicative constant, which is absorbed into γ \gamma γ [56 ] states the equation with a nonnegative coefficient in front of the fractional term, the proof of the Pohozaev identity treats the local and nonlocal contributions separately, and the sign of this coefficient does not enter the computation once the required regularity is available. In the present formulation this regularity is immediate: since u ∈ C 2 ( Ω ‾ ) u\in C^2(\overline\Omega) u ∈ C 2 ( Ω ) u = 0 u=0 u = 0 R n ∖ Ω \mathbb R^n\setminus\Omega R n ∖ Ω s < 1 / 2 s<1/2 s < 1/2 [56 ] applies with a negative fractional coefficient and yields the identity above.
Proof of Theorem 1.8 . Up to a translation of coordinates, we may assume that Ω \Omega Ω x ⋅ ν ( x ) ≥ 0 x\cdot\nu(x)\ge0 x ⋅ ν ( x ) ≥ 0 x ∈ ∂ Ω x\in\partial\Omega x ∈ ∂ Ω
Let us start noticing that, due to the assumption p > n + 2 n − 2 p>\tfrac{n+2}{n-2} p > n − 2 n + 2
n − 2 2 − n p + 1 > 0 , \begin{equation}
\dfrac{n-2}{2}-\dfrac{n}{p+1}>0, \tag*{(6.3)}
\end{equation} 2 n − 2 − p + 1 n > 0 , (6.3) and
n − 2 s 2 − n p + 1 > 0. \begin{equation}
\dfrac{n-2s}{2}-\dfrac{n}{p+1}>0. \tag*{(6.4)}
\end{equation} 2 n − 2 s − p + 1 n > 0. (6.4) Assume now that there exists a non-trivial solution to (1.14) and let us test (1.14) with u u u
∫ Ω ∣ ∇ u ∣ 2 − γ [ u ] s 2 = ∫ Ω ∣ u ∣ p + 1 . \begin{equation}
\int_{\Omega}|\nabla u|^2 -\gamma \, [u]^{2}_{s} = \int_{\Omega}|u|^{p+1}. \tag*{(6.5)}
\end{equation} ∫ Ω ∣∇ u ∣ 2 − γ [ u ] s 2 = ∫ Ω ∣ u ∣ p + 1 . (6.5) Plugging the latter into (6.2) , we get
n − 2 2 ∫ Ω ∣ ∇ u ∣ 2 − γ n − 2 s 2 [ u ] s 2 − n p + 1 ∫ Ω ∣ ∇ u ∣ 2 + γ n p + 1 [ u ] s 2 = − 1 2 ∫ ∂ Ω ( x ⋅ ν ) ( ∂ u ∂ ν ) 2 d σ , \begin{equation*}
\begin{aligned}
\dfrac{n-2}{2} \int_{\Omega}|\nabla u|^2 &- \gamma \, \dfrac{n-2s}{2} \, [u]^2_s - \dfrac{n}{p+1}\int_{\Omega}|\nabla u|^2 + \dfrac{\gamma \, n}{p+1}[u]^{2}_{s} \\
& =-\dfrac{1}{2}\int_{\partial\Omega}(x\cdot \nu) \, \left(\dfrac{\partial u}{\partial \nu}\right)^2\,d\sigma,
\end{aligned}
\end{equation*} 2 n − 2 ∫ Ω ∣∇ u ∣ 2 − γ 2 n − 2 s [ u ] s 2 − p + 1 n ∫ Ω ∣∇ u ∣ 2 + p + 1 γ n [ u ] s 2 = − 2 1 ∫ ∂ Ω ( x ⋅ ν ) ( ∂ ν ∂ u ) 2 d σ , Using the assumption on γ \gamma γ [ u ] s > 0 [u]_s>0 [ u ] s > 0 C e m b C_{\rm emb} C emb (2.9) , we obtain
0 < [ C e m b ( n − 2 2 − n p + 1 ) − γ ( n − 2 s 2 − n p + 1 ) ] [ u ] s 2 ≤ ( n − 2 2 − n p + 1 ) ∫ Ω ∣ ∇ u ∣ 2 d x − γ ( n − 2 s 2 − n p + 1 ) [ u ] s 2 = − 1 2 ∫ ∂ Ω ( x ⋅ ν ) ( ∂ u ∂ ν ) 2 d σ ≤ 0. \begin{equation}
\begin{aligned}
0
&<\left[
C_{\rm emb}\left(\frac{n-2}{2}-\frac{n}{p+1}\right)
-\gamma\left(\frac{n-2s}{2}-\frac{n}{p+1}\right)
\right][u]_s^2\\
&\le
\left(\frac{n-2}{2}-\frac{n}{p+1}\right)
\int_\Omega |\nabla u|^2\,dx
-\gamma\left(\frac{n-2s}{2}-\frac{n}{p+1}\right)[u]_s^2\\
&=-\frac12\int_{\partial\Omega}(x\cdot\nu)
\left(\frac{\partial u}{\partial\nu}\right)^2\,d\sigma
\le0.
\end{aligned} \tag*{(6.6)}
\end{equation} 0 < [ C emb ( 2 n − 2 − p + 1 n ) − γ ( 2 n − 2 s − p + 1 n ) ] [ u ] s 2 ≤ ( 2 n − 2 − p + 1 n ) ∫ Ω ∣∇ u ∣ 2 d x − γ ( 2 n − 2 s − p + 1 n ) [ u ] s 2 = − 2 1 ∫ ∂ Ω ( x ⋅ ν ) ( ∂ ν ∂ u ) 2 d σ ≤ 0. (6.6) This contradiction proves the theorem. ◻
