Existence problems for the $p$-Laplacian

TL;DR

This paper investigates boundary value problems involving the p-Laplacian, establishing necessary conditions for solutions' existence, which involve sharp inequalities related to Sobolev constants and applications to nonlinear eigenvalue problems.

Contribution

It derives optimal necessary conditions for the existence of solutions to p-Laplacian boundary value problems, including applications to nonlinear eigenvalue problems.

Findings

Necessary conditions involve lower bounds with Sobolev constants.

Most inequalities derived are sharp and optimal.

Applications to nonlinear eigenvalue problems are discussed.

Abstract

We consider a number of boundary value problems involving the $p$-Laplacian. The model case is $-\Delta_p u=V|u|^{p-2}u$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R}^n$. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of $V$, the measure of $D$, and a sharp Sobolev constant. In most cases, these inequalities are best possible. Applications to non-linear eigenvalue problems are also discussed.