On the Brezis-Nirenberg problem for a Kirchhoff type equation in high dimension

TL;DR

This paper investigates the existence, non-existence, and multiplicity of solutions for a high-dimensional Kirchhoff type equation with critical nonlinearity, using variational methods and analysis of fiber maps.

Contribution

It extends the analysis of Kirchhoff problems with critical nonlinearities by incorporating subcritical perturbations and exploring parameter ranges via Nehari manifold techniques.

Findings

Existence of solutions under certain parameter conditions

Non-existence results in specific regimes

Multiplicity of solutions in high dimensions

Abstract

The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.