On Kirchhoff type equations with critical Sobolev exponent and Naimen's open problems

TL;DR

This paper investigates Kirchhoff type equations with critical Sobolev exponent, providing new existence and nonexistence results based on parameter values, and analyzing the asymptotic behavior of solutions in a bounded domain.

Contribution

It offers novel existence and nonexistence results for Kirchhoff equations with critical Sobolev exponent, extending previous findings and exploring solution behaviors.

Findings

New existence results depending on parameter ranges

Nonexistence results under certain conditions

Analysis of asymptotic behaviors of solutions

Abstract

We study the following Brezis-Nirenberg problem of Kirchhoff type $$ \left\{\aligned &-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u = \lambda|u|^{q-2}u + \delta |u|^{2}u, &\quad \text{in}\ \Omega, \\ &u=0,& \text{on}\ \partial\Omega, \endaligned \right. $$ where $\Omega\subset \bbr^4$ is a bounded domain with the smooth boundary $\partial\Omega$, $2\leq q<4$ and $a$, $b$, $\lambda$, $\delta$ are positive parameters. We obtain some new existence and nonexistence results, depending on the values of the above parameters, which improves some known results. The asymptotical behaviors of the solutions are also considered in this paper.