An abstract critical point theorem with applications to elliptic problems with combined nonlinearities

TL;DR

This paper introduces an abstract critical point theorem using cohomological index theory, enabling the discovery of multiple solutions for nonlinear elliptic problems with combined nonlinearities, including nonlocal and fractional cases.

Contribution

The paper presents a novel critical point theorem based on cohomological index theory, applicable to a broad class of nonlinear elliptic problems with combined nonlinearities.

Findings

Produces pairs of nontrivial critical points with nontrivial higher critical groups

Finds solutions that are neither local minimizers nor mountain pass type

Applies to subcritical, critical, nonlocal, and fractional p-Laplacian problems

Abstract

We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications are given to subcritical and critical $p$-Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional $p$-Laplacian problems.