Existence and multiplicity of solutions for nonlocal systems with Kirchhoff type

TL;DR

This paper investigates the existence and multiplicity of positive solutions for nonlocal Kirchhoff-type elliptic systems using variational methods, including the Nehari manifold and cohomological index, and addresses the critical case with large parameters.

Contribution

It introduces new existence and multiplicity results for nonlocal Kirchhoff systems using variational techniques and extends analysis to the critical case with large parameters.

Findings

Existence of positive solutions via Nehari manifold and Mountain Pass Lemma.

Multiplicity results established through cohomological index theory.

Existence of least energy solutions in the critical case for large parameters.

Abstract

Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter $\beta$ is sufficiently large.