Pairs of nontrivial solutions to concave-linear-convex type elliptic problems

TL;DR

This paper establishes the existence of a pair of nontrivial solutions for a class of elliptic problems with concave-linear-convex nonlinearities, using advanced critical point theory techniques.

Contribution

It introduces a novel approach to find higher critical points that are not local minimizers or mountain pass solutions for these elliptic problems.

Findings

Existence of two nontrivial solutions in critical/subcritical cases

Solutions have nontrivial higher critical groups

Intermediate results on localization and homotopy invariance of critical groups

Abstract

We obtain a pair of nontrivial solutions for a class of concave-linear-convex type elliptic problems that are either critical or subcritical. The solutions we find are neither local minimizers nor of mountain pass type in general. They are higher critical points in the sense that they each have a higher critical group that is nontrivial. This fact is crucial for showing that our solutions are nontrivial. We also prove some intermediate results of independent interest on the localization and homotopy invariance of critical groups of functionals involving critical Sobolev exponents.