Normalized solutions of $L^2$-supercritical Kirchhoff equations in bounded domains

TL;DR

This paper proves the existence of mountain pass-type normalized solutions for a class of $L^2$-supercritical Kirchhoff equations in bounded domains, using variational methods and blow-up analysis, and studies their asymptotic behavior as a parameter tends to zero.

Contribution

It introduces a novel variational approach with Morse index considerations to establish solutions in the supercritical regime and analyzes their asymptotic limits.

Findings

Existence of mountain pass solutions in the supercritical regime

Development of a blow-up analysis for Kirchhoff equations

Asymptotic behavior of solutions as the parameter $b$ approaches zero

Abstract

In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_{\Omega}\vert\nabla u\vert^2dx)\Delta u+\lambda u=\vert u\vert^{p-2}u & \text{ in }\Omega,\\ u=0 & \text{ on }\partial\Omega \end{cases} \end{equation*} subject to the constraint $\int_{\Omega}\vert u\vert^2dx=c$. Here, $a$ and $b$ are positive constants, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ with $1\leq N\leq3$, $c>0$ is a prescribed value, and $\lambda\in \mathbb{R}$ is a Lagrange multiplier. In the $L^2$-supercritical regime $2+\frac{8}{N}<p<2^*$, we establish the existence of mountain pass-type normalized solutions. Our approach relies on utilizing a parameterized version of the minimax theorem with Morse index information for constraint functionals, and developing a blow-up analysis for the nonlinear Kirchhoff equations. Furthermore, we explore the asymptotic behavior of these solutions as $b\rightarrow0$.