Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence

Abstract

We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }Ω\subseteq \mathbb{R}^{N},\ N\geq 3, \] where $\Omega$ is a radial domain (bounded or unbounded) and $u$ satisfies $u=0$ on $\partial \Omega$ if $\Omega \ne {\mathbb{R}}^N$ and $u\to 0$ as $\left|x\right|\to \infty$ if $\Omega$ is unbounded. The potential $V$ may be vanishing or unbounded at zero or at infinity and the nonlinearity $g$ may be superlinear or sublinear. If $g$ is sublinear, the case with $g\left(\left|\cdot \right|,0\right)\ne 0$ is also considered.

29 pages, 8 figures

References in corpus (2)

• Compactness and existence results in weighted Sobolev spaces of radial functions, Part I: Compactness
• Nonlinear Schrödinger equations without compatibility conditions on the potentials

Cited by in corpus (5)

• Compactness and existence results for the $p$-Laplace equation
• Nonlinear Schrödinger equations without compatibility conditions on the potentials
• Radial quasilinear elliptic problems with singular or vanishing potentials
• Compactness results for the $p$-Laplace equation
• Multiple nonradial solutions for a nonlinear elliptic radial problem: an improved result