Positive solutions for a Kirchhoff problem of Brezis-Nirenberg type in dimension four

TL;DR

This paper establishes new existence results for positive solutions to a Kirchhoff problem of Brezis-Nirenberg type in four dimensions, addressing the interaction between higher order terms and critical nonlinearity.

Contribution

It introduces a novel approximation approach to handle the interaction between Kirchhoff and critical nonlinearities in four dimensions, providing explicit parameter bounds for solution existence.

Findings

Derived several existence results for positive solutions

Provided explicit bounds for parameters b and λ

Improved upon previous results in the literature

Abstract

We consider a Kirchhoff problem of Brezis-Nirenberg type in a smooth bounded domain of $\mathbb{R}^4$ with Dirichlet boundary conditions. Our approach, novel in this framework and based upon approximation arguments, allows us to cope with the interaction between the higher order Kirchhoff term and the critical nonlinearity, typical of the dimension four. We derive several existence results of positive solutions, complementing and improving earlier results in the literature. In particular, we provide explicit bounds of the parameters $b$ and $\lambda$ coupled, respectively, with the higher order Kirchhoff term and the subcritical nonlinearity, for which the existence of solutions occurs.