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

This paper deals with the existence and multiplicity of solutions for a class Kirchhoff type elliptic systems involving nonlinearities Trudinger-Moser exponential growth. We first study following system: $$\begin{aligned} \left\{ \begin{array}{ll} -\big (a_1+b_1\Vert u\Vert ^{2(\theta _1-1)}\big )\Delta u= \lambda H_u(x,u,v) &{} \quad \text{ in }\ \ \Omega ,\\ (a_2+b_2\Vert v\Vert _2-1)}\big v= H_v(x,u,v) u=0, v=0 on \partial , \end{array}\right. \end{aligned}$$where \(\Omega \) is bounded domain \({\mathbb {R}}^2\) smooth boundary, \(\Vert w\Vert =\big (\int _{\Omega }|\nabla w|^2dx\big )^{1/2}\), \(H_u(x,u,v)\) \(H_v(x,u,v)\) behave like \(e^{\beta |(u,v)|^2}\) when \(|(u,v)|\rightarrow \infty some \(\beta >0\), \(a_1, a_2>0\), \(b_1, b_2> 0\), \(\theta _1, \theta _2> 1\) \(\lambda positive parameter. In later part paper, we also discuss new result above system parameter induced by nonlocal dependence. The term lack compactness associated energy functional due to embedding have be overcome via techniques.