Radial reductions of variable exponent Laplacian equations on warped products with compact symmetry

Abstract

We study nonlinear elliptic equations of variable exponent type, $-Δ_{q(x)}u+|u|^{q(x)-2}u=f(x,u),$ on non-compact Riemannian manifolds with symmetry. We consider complete warped products $M=\mathcal R\times_ϕY,\qquad g_ϕ=dr^2\oplusϕ(r)^2g_Y,$ where $\mathcal R$ is a non-compact interval, possibly with boundary, and $Y$ is a compact homogeneous manifold. We establish sufficient conditions on the warping function ensuring a Ricci lower bound, a uniform non-collapsing volume condition, and the orbit-growth hypothesis of the abstract existence theorem proved in \cite{JFA}. We also obtain comparison bounds for the admissible warping functions. We analyze the corresponding radial equations for the admissible exponential and linear models $ϕ(r)=e^{ar}$ and $ϕ(r)=ar$, with $r\geq r_0>0$. For a constant diffusion exponent $q=k$, we consider explicit decaying profiles $u=ϕ^{-σ}$, which belong to $L_1^k(M)$ provided $σk>\dim Y$ in the exponential case and $σk>\dim Y+1$ in the linear case. Under additional explicit parameter restrictions, these profiles solve a resonant equation with a positive bounded coefficient, a construction distinct from the superlinear regime of the abstract existence theorem. Finally, we establish logarithmic reductions for constant and variable radial exponents, and justify the passage from radial identities to weak formulations against arbitrary compactly supported interior test functions.