Multiple positive solutions for Kirchhoff type problems involving concave and critical nonlinearities in ${R}^3$

TL;DR

This paper proves the existence of multiple solutions for a class of Kirchhoff problems with concave and critical nonlinearities in three-dimensional space, using variational and concentration compactness methods.

Contribution

It establishes the multiplicity of solutions for Kirchhoff problems involving sub-linear and critical nonlinearities on unbounded domains, which was not previously known.

Findings

At least two solutions exist for the considered Kirchhoff problem.

Application of Ekeland's variational principle and concentration compactness principle.

Results extend understanding of nonlinear Kirchhoff problems with critical terms.

Abstract

In this paper, we consider the multiplicity of solutions for a class of Kirchhoff type problems with sub-linear and critical terms on an unbounded domain. With the aid of Ekeland's variational principle and the concentration compactness principle we prove that the Kirchhoff problem has at least two solutions.