A general perturbation theorem with applications to nonhomogeneous critical growth elliptic problems

TL;DR

This paper introduces a versatile perturbation theorem applicable to various nonhomogeneous elliptic problems, enabling the derivation of multiple solutions for complex critical growth equations, including nonlocal and fractional cases.

Contribution

It presents a new general perturbation theorem that extends solution existence results to a broad class of nonhomogeneous elliptic problems, including fractional and critical exponent cases.

Findings

Established a perturbation theorem applicable to diverse elliptic problems.

Derived multiple solutions for critical $p$-Laplacian and fractional problems.

Extended results to nonlocal and critical Hardy-Sobolev exponent cases.

Abstract

We prove a general perturbation theorem that can be used to obtain pairs of nontrivial solutions of a wide range of local and nonlocal nonhomogeneous elliptic problems. Applications to critical $p$-Laplacian problems, $p$-Laplacian problems with critical Hardy-Sobolev exponents, critical fractional $p$-Laplacian problems, and critical $(p,q)$-Laplacian problems are given. Our results are new even in the semilinear case $p = 2$.