An alternative variational approach to constrained critical points for nonlinear Schrödinger equations with combined nonlinearities

In this paper we are motivated by the following Schrödinger type equation

where $\psi(t,x):[0,T)\times\mathbb{R}^{d}\to\mathbb{C}$, $d\ge3$, is a complex wave function and the exponents $q,p$ are of mass subcritical type, meaning that $1<q<p<1+\frac{4}{d}$. Given an initial datum $\psi_0\in H^1(\mathbb{R}^d)$, the Cauchy problem is (globally) well-posed and satisfies conservation of mass and energy, defined respectively by

and

Particular solutions to (0.1) are the so-called standing wave solutions, which are solutions to (0.1) of the form

with $\omega\in\mathbb{R}$, and $u(x)\in\mathbb{C}$ is a time-independent function belonging to $H^1(\mathbb{R}^d)$ which satisfies

As the mass is a physical quantity which is preserved along the flow of (0.1), in order to find solutions $u$ to (0.4), a natural way to study the problem is to look for critical points of the energy functional

constrained on the $L^{2}$-spheres in $H^{1}(\mathbb{R}^{d};\mathbb{C})$ described by

So, by a solution of (0.4) we mean a pair $(\omega_{\rho},u_{\rho})\in\mathbb{R}\times H^{1}(\mathbb{R}^{d};\mathbb{C})$ where $\omega_{\rho}$ appears as the Lagrange multiplier associated to the critical point $u_{\rho}$ on $S_{\rho}$. For a fixed mass $\rho>0$, we define the ground state energy the quantity $I_{\rho^{2}}$, defined as

If a minimizer $u_{\rho}$ to (0.5) exists, we call it ground state solution. Concerning the existence of ground states everything is well understood in arbitrary dimension thanks to the following result:

In order to state our main results it is convenient to introduce the notation

such that

and any standing wave solution $u$ solves

The case $\omega= 0$ corresponds to states that are time independent, i.e. static. Such kind of solution exists and it corresponds to the unique positive radially symmetric static solution $W$ that solves

whose asymptotic decay at infinity is explicitly given by a power-like tail that decays as $|x|^{-\frac{2}{q-1}}$, see [4]. The assumed range of exponents ensures that $W \in L^2(\mathbb{R}^d)$.

The classical Gagliardo-Nirenberg inequality with sharp constant $S$

where $\theta$ is defined by the relation

can be written in terms of $A(u)$, $B(u)$, $C(u)$ as

where

and

Note that the following (crucial) identity holds

Now we introduce two relevant quantitites that are relevant for our first result. We define

such that

Our first result characterizes the energy level of the unique positive static solution that solves (0.7).

then $u$ is a critical point of $E$ at the level $\lambda_\star$ and it attains the sharp constant in GN inequality, i.e.

Moreover, up to translations and multiplication by a constant phase, $u$ coincides with the unique positive radial solution $W$.

Our second result introduces an alternative variational construction of critical points of the energy functional with a mass constraint. Here, instead to minimize the non convex functional $E$ on the $L^2$ sphere $\{D=\rho\}$, we choose to minimize the convex functional $D$ on a non convex constraint set given by the conditions $E\leq\lambda$, $C\geq C_\star$, $\lambda<\lambda_\star$. Of course, for $u$ with $D(u)=\rho$ and $E(u)=\lambda$, minimizing $D$ on the set $\{E\leq\lambda,\ C\geq C_\star\}$ is the same as minimizing $E$ on the set $\{D\leq\rho,\ C\geq C_\star\}$, both facts meaning

As will become clear later, the characterization of the set $E\leq\lambda$, $C\geq C_\star$ given by Theorem 0.2 will be crucial in the proof of the following theorem.

Remark 0.1. The restriction to the radial space in Theorem 0.3 is used only to recover compactness. By restricting the minimization problem to $H^1_{\mathrm{rad}}(\mathbb{R}^d)$, we can use the compact embedding

This yields the strong convergence of a minimizing sequence in $L^{q+1}(\mathbb{R}^d)$ and $L^{p+1}(\mathbb{R}^d)$, which is needed to pass to the limit in the nonlinear functionals $B$ and $C$. This compactness is crucial to ensure that the weak limit of a minimizing sequence remains in the constraint set.

a contradiction. Thus $C(u_0) \ge C_*$. The equality $C(u_0) = C_*$ is also impossible. Indeed, since $E(u_0) = 0 < \lambda_*$, Theorem 0.2 would again imply $u_0 = W$, yielding

which is absurd. Consequently,

It follows that $u_0 \in\Omega(\lambda)$ for every $\lambda> 0$. Hence for $0 < \lambda< \lambda_*$, the mass $\rho(\lambda)$ of the minimizer $u_\lambda$ fulfills

Since every minimizer $u_\lambda$ of $\rho(\lambda)$ satisfies $E(u_\lambda) = \lambda$, the function $u_0$ cannot realize the minimum. Therefore,

for every $\lambda> 0$, showing that the standing waves provided by Theorem 0.3 have mass strictly smaller than the threshold mass $\rho_0$ of Theorem 0.1.

Remark 0.4. Since the seminal work by Tao, Vișan, and Zhang [11], the NLS with combined nonlinearities has attracted a lot of attention. Concerning the existence of standing waves when the leading nonlinearity is mass super-critical it is worth mentioning at least the following recent works [2, 9, 10], and references therein.

Remark 0.5. A further motivation for the present work comes from recent scattering results for nonlinear Schrödinger equations with combined mass-subcritical nonlinearities. In the recent paper [1] Bellazzini, Forcella and Georgiev proved a scattering result for datum belonging to $\Sigma$, i.e. the subspace of $H^1(\mathbb{R}^d)$ with finite variance. More precisely the authors proved that solutions scatter whenever the initial mass is sufficiently small, without requiring a smallness assumption on the whole $\Sigma$-norm. The sharp size of this threshold mass for scattering remains an open problem. If $d \ge3$ and taken $\lambda\in(-\infty, \lambda_*)$ and calling $u_\lambda$ the minimizer for the mass given by Theorem 0.3 we proved that exist non scattering states with mass strictly smaller then the threshold mass $\rho_0$ given by Theorem 0.1.

Proof of Theorem 0.2

We recall an elementary inequality.

Proof. Calling $\tilde{x} = \frac{x}{x+y}$, $\tilde{y} = \frac{y}{x+y}$, $\tilde{\alpha} = \frac{\alpha}{\alpha+\beta}$, $\tilde{\beta} = \frac{\beta}{\alpha+\beta}$, we have $\tilde{x}+\tilde{y}=1$, $\tilde{\alpha}+\tilde{\beta}=1$. The concave function $f(\tilde{x}) = \tilde{x}^{\tilde{\alpha}}(1-\tilde{x})^{1-\tilde{\alpha}}$ attains the maximum in $[0,1]$ at $\tilde{x}=\tilde{\alpha}$ such that $f(\tilde{x}) \leq\tilde{\alpha}^{\tilde{\alpha}}(1-\tilde{\alpha})^{1-\tilde{\alpha}}$. Hence we have

which implies

with equality iff $\frac{x}{\alpha}=\frac{y}{\beta}$.

Now let us consider $u \in H^1(\mathbb{R}^d) \cap L^{q+1}(\mathbb{R}^d)$ fulfilling

Then, by GN inequality (0.8) and Lemma 1.1, hiding the dependence on $u$, we obtain

where we have used in the last line the relation $\lambda_\star=\frac{2\alpha}{d}C_\star$.

Hence we obtain

thanks to the relation (0.10) and (0.12). Therefore the above are in fact equalities and in particular

and thus $\frac{A}{\alpha}=\frac{B}{\beta}$. Calling $\frac{A}{\alpha}=\frac{B}{\beta}=c$ we get $A+B=(\alpha+\beta)c$ such that

that implies, thanks to (0.12), $c=C=\frac{A}{\alpha}=\frac{B}{\beta}$. Finally, thanks to the fact that $C=\sigma^{-1}A^\alpha B^\beta$, $u$ attains the sharp constant in Gagliardo-Nirenberg inequality and minimizes the ratio $\frac{A^\alpha B^\beta}{C}$. Thus

Now to conclude that $u=W$ (modulo space and phase shift) it sufficies to notice that

• any solution $u \in\dot{H}^{1}(\mathbb{R}^{d}) \cap L^{q+1}(\mathbb{R}^{d})$ to the equation $\nabla E(u)=0$ shall fulfill the relations

that are equivalent to

• the GN inequality implies

and hence an a priori bound of energy of non trivial critical point of $E$ is given by $E\geq\lambda_{\star}$.

• the optimizers for GN can be obtained also by minimizing (a la Berestycki-Lions [3]) $A(u)$ on the constraint $B(u)-C(u)=1$ and then rescaling to remove the Lagrange multiplier.

• any minimizer for $A(u)$ on the constraint $B(u)-C(u)=1$ is radially symmetric (Lopes [8]) and it is positive (Berestycki-Lions [3]) hence it is unique (Kwong-Zhang [6]).

Proof of Theorem 0.3

Notice that $\Omega(\lambda)\ne\varnothing$ for every $\lambda\in\mathbb{R}$. Indeed, for any $v\in H^{1}(\mathbb{R}^{d})\setminus\{0\}$,

as $t\to\infty$, while $C(tv)\to+\infty$. Therefore $tv\in\Omega(\lambda)$ for $t$ large enough. Now let us consider a minimizing sequence $u_j$ for $D$ on $\Omega_\lambda$ and denote $A_j=A(u_j)$, $B_j=B(u_j)$, $C_j=C(u_j)$, $D_j=D(u_j)$, thus

By the fact that $u_j\in\Omega_\lambda$ we have $A_j<C_j+\lambda$ and hence by GN inequality

The mass subcritical condition $p<1+\frac{4}{d}$ guarantees hence that $C_j$ is bounded as well as $A_j$ and that $u_j\rightharpoonup u$ in $H^{1}_{\mathrm{rad}}(\mathbb{R}^{d})$. Denoting $A=A(u)$, $B=B(u)$, $C=C(u)$, $D=D(u)$ we have, by the compact embedding of radial function,

Thus $u\in\Omega_\lambda$ and $D\le\inf_{\Omega_\lambda}D$, so $D=\delta_\lambda$ and the infimum is achieved. We have also that $C(u)>C_\ast$ thanks to Theorem 0.2 because $\lambda<\lambda_\ast$. Moreover if we assume $E(u)<\lambda$, $u$ is forced to be a free critical point of $D$ which is the trivial function, therefore $E(u)=\lambda$. Moreover $\nabla E(u)\ne0$ by Theorem 0.2 and by Lagrange multiplier principle we have for some $\omega\ne0$

Now, in order to prove that $\omega>0$, we argue by contradiction assuming that $\omega<0$. For $0<t<1$ certainly $tu\notin\Omega_\lambda$ because $D(tu)=t^2D(u)<D(u)=\inf_{\Omega_\lambda}D$, and since $C(tu)>C_\ast$ for $t$ close to $1$ one has $E(tu)>\lambda$ for all $t<1$ close to $1$. Thus $\left.\frac{d}{dt}E(tu)\right|_{t=1}=(\nabla E(u)\cdot u)\le0$, whence from $(\nabla E(u)\cdot u)+\omega(\nabla D(u)\cdot u)=0$ we have $\left.\frac{d}{dt}E(tu)\right|_{t=1}=-\omega(\nabla D(u)\cdot u)>0$.

References

Jacopo Bellazzini

Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy

Pietro Majer

Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5, 56100 Pisa, Italy