New linking theorems with applications to critical growth elliptic problems with jumping nonlinearities

TL;DR

This paper introduces new linking theorems for variational problems with jumping nonlinearities, enabling the existence of solutions where traditional methods fail, and broadening the scope of critical growth elliptic problem analysis.

Contribution

The authors develop generalized linking theorems not based on linear subspaces, facilitating the study of elliptic problems with jumping nonlinearities.

Findings

Established new linking theorems applicable to nonlinear submanifolds.

Proved the existence of nontrivial solutions for critical growth elliptic problems.

Provided a framework for analyzing problems with jumping nonlinearities.

Abstract

We study critical growth elliptic problems with jumping nonlinearities. Standard linking arguments based on decompositions of $H^1_0(\Omega)$ into eigenspaces of $- \Delta$ cannot be used to obtain nontrivial solutions to such problems. We show that the associated variational functional admits certain linking structures based on splittings of $H^1_0(\Omega)$ into nonlinear submanifolds. In order to capture these linking geometries, we prove several generalizations of the classical linking theorem of Rabinowitz that are not based on linear subspaces. We then use these new linking theorems to obtain nontrivial solutions of our problems. Our abstract results are of independent interest and can be used to obtain nontrivial solutions of other types of problems with jumping nonlinearities as well.