Existence and Multiplicity of Solutions to a Kirchhoff Type Elliptic System with Trudinger–Moser Growth

Abstract

This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic systems involving nonlinearities with Trudinger-Moser exponential growth. We first study the existence of solutions for the following system:

$$\begin{aligned} \left\{ \begin{array}{ll} -\big (a_1+b_1\Vert u\Vert ^{2(\theta _1-1)}\big )\Delta u= \lambda H_u(x,u,v) &{} \quad \text{ in }\ \ \ \Omega ,\\ -\big (a_2+b_2\Vert v\Vert ^{2(\theta _2-1)}\big )\Delta v= \lambda H_v(x,u,v) &{} \quad \text{ in }\ \ \ \Omega ,\\ u=0, v=0 &{} \quad \text{ on }\ \ \ \partial \Omega , \end{array}\right. \end{aligned}$$

where $\Omega$ is a bounded domain in ${\mathbb {R}}^2$ with smooth boundary, $\Vert w\Vert =\big (\int _{\Omega }|\nabla w|^2dx\big )^{1/2}$, $H_u(x,u,v)$ and $H_v(x,u,v)$ behave like $e^{\beta |(u,v)|^2}$ when $|(u,v)|\rightarrow \infty$ 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.