Compactness and existence results for the $p$-Laplace equation

TL;DR

This paper establishes new compactness results for weighted Sobolev spaces involving the p-Laplace operator, and applies these to prove existence and multiplicity of solutions for nonlinear p-Laplace equations with variable potentials.

Contribution

It provides novel conditions for compact embeddings of weighted radial Sobolev spaces without compatibility constraints on potentials, and applies these to nonlinear p-Laplace equations with complex potential behaviors.

Findings

Derived new compactness criteria for weighted Sobolev embeddings.

Proved existence of solutions under broad potential conditions.

Analyzed solutions for super and sub p-linear nonlinearities.

Abstract

Given $1<p<N$ and two measurable functions $V\left( r\right) \geq 0$ and $K\left( r\right) >0$, $r>0$, we define the weighted spaces \[ W=\left\{ u\in D^{1,p}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V\left( \left| x\right| \right) \left| u\right| ^{p}dx<\infty \right\} ,\quad L_{K}^{q}=L^{q}(\mathbb{R}^{N},K\left( \left| x\right| \right) dx) \] and study the compact embeddings of the radial subspace of $W$ into $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. We consider exponents $q_{1},q_{2},q$ that can be greater or smaller than $p$. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. We then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear $p$-Laplace equations of the form \[ -\triangle _{p}u+V\left( \left| x\right| \right) |u|^{p-1}u=g\left( \left| x\right| ,u\right) \quad \text{in }\mathbb{R}^{N},\ 1<p<N, \] where $V$ and $g\left( \left| \cdot \right| ,u\right) $ with $u$ fixed may be vanishing or unbounded at zero or at infinity. Both the cases of $g$ super and sub $p$-linear in $u$ are studied and, in the sub $p$-linear case, nonlinearities with $g\left( \left| \cdot \right| ,0\right) \neq 0$ are also considered.