Existence and multiplicity of solutions to a Kirchhoff type elliptic system with Trudinger-Moser growth

TL;DR

This paper investigates the existence and multiple solutions of a Kirchhoff type elliptic system with exponential nonlinearities in two dimensions, introducing new techniques to handle nonlocal terms and lack of compactness.

Contribution

It provides new results on solution existence and multiplicity for a Kirchhoff elliptic system with Trudinger-Moser growth, addressing challenges from nonlocality and exponential nonlinearities.

Findings

Existence of solutions under certain conditions.

Multiple solutions established for the system.

Development of new techniques to overcome compactness issues.

Abstract

This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following system \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\big(a_1+b_1\|u\|^{2(\theta_1-1)}\big)\Delta u= \lambda H_u(x,u,v)\ \ \ &\ \mbox{in}\ \ \ \Omega,\\[2mm] -\big(a_2+b_2\|v\|^{2(\theta_2-1)}\big)\Delta v= \lambda H_v(x,u,v)\ \ \ &\ \mbox{in}\ \ \ \Omega,\\[2mm] u=0, v=0\ \ \ \ &\ \mbox{on}\ \ \ \partial\Omega, \end{array} \right. \end{eqnarray*} where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary,\ $\|u\|=\big(\int_{\Omega}|\nabla u|^2dx\big)^{1/2}$, $H_u$ and $H_v$ behave like $e^{\beta |s|^2}$ when $|s|\rightarrow \infty$ for some $\beta>0$, $a_1,\ a_2>0$, $b_1,\ b_2> 0$, $\theta_1,\ \theta_2> 1$ and $\lambda$ is a positive parameter. In the later part of the paper, we also discuss a new multiplicity result for the above system with a positive parameter induced by the nonlocal dependence. The Kirchhoff term and the lack of compactness of the associated energy functional due to the Trudinger-Moser embedding have to be overcome via some new techniques.