Multiplicity and asymptotics of positive solutions for critical-concave Kirchhoff equation

TL;DR

This paper investigates the number and behavior of positive solutions to a critical Kirchhoff equation with concave perturbation, using variational methods, and explores their asymptotic limits as certain parameters tend to zero.

Contribution

It establishes the existence of multiple positive solutions for small parameters without requiring the smallness condition on the Kirchhoff coefficient, extending previous results.

Findings

Multiple positive solutions exist for small mbda.

Positive solutions exhibit specific asymptotic behavior as b and mbda.

The work extends prior results by removing the smallness condition on b.

Abstract

This paper focuses on the critical Kirchhoff equation with concave perturbation \begin{align*} \begin{cases} \displaystyle -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=|u|^4u+\lambda|u|^{q-2}u\ \ &\mbox{in}\ \Omega, \displaystyle u=0\ \ &\mbox{on}\ \partial\Omega, \end{cases} \end{align*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$, $a,b,\lambda>0$ and $1<q<2$. By the constrained minimization methods, the mountain pass theorem and the concentration-compactness principle, we verify the multiplicity of positive solutions for $\lambda>0$ small enough. Moreover, we analyse the asymptotic behaviour of positive solutions as $b\rightarrow0$ and $\lambda\rightarrow0$, respectively. This work is a counterpart of [A. Ambrosetti et al., J.~Funct.~Anal. 1994] for the Kirchhoff equation. It is noteworthy that we don't require that $b>0$ is small enough here, which is imposed in the existing literatures to make refined estimates for the mountain pass level.