A bifurcation and multiplicity result for a critical growth elliptic problem

TL;DR

This paper establishes the existence of multiple solutions for a critical growth elliptic problem involving the p-Laplacian, using an abstract critical point theorem, with results valid even in the semilinear case.

Contribution

It introduces a new multiplicity result for critical growth p-Laplacian problems, applicable when the parameter or domain volume is small, extending prior work.

Findings

Multiple nontrivial solutions exist under small parameter or domain volume.

Results hold for the critical p-Laplacian case, including the classical semilinear case.

An abstract critical point theorem with a local (PS) condition underpins the proofs.

Abstract

We consider a Br\'ezis-Nirenberg type critical growth $p$-Laplacian problem involving a parameter $\mu > 0$ in a smooth bounded domain $\Omega$. We prove the existence of multiple nontrivial solutions if either $\mu$ or the volume of $\Omega$ is sufficiently small. The proof is based on an abstract critical point theorem that only assumes a local $(\text{PS})$ condition. Our results are new even in the semilinear case $p = 2$.