Compactness results for the $p$-Laplace equation

TL;DR

This paper investigates the compactness of embeddings for weighted Sobolev spaces involving the p-Laplace operator, providing new conditions that do not depend on the behavior of potentials at zero or infinity.

Contribution

It establishes novel compact embedding results for radial functions in weighted Sobolev spaces with minimal assumptions on potential behaviors.

Findings

Derived conditions for compact embeddings of weighted Sobolev spaces

Applicable to a range of exponents relative to p

Results do not depend on potential behavior at zero or infinity

Abstract

Given $1<p<N$ and two measurable functions $V(r)\geq 0$ and $K(r)>0$, $r>0$, 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. Both exponents $q_1,q_2,q$ greater and lower than $p$ are considered. 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.