On solutions for a class of Kirchhoff systems involving critical growth in R 2

Abstract

In this work we study the existence of solutions for the following class of elliptic systems involving Kirchhoff equations in the plane: $$\begin{array}{c}\left\{\begin{array}{cc}\mathit{m}\mathrm{(}\Vert \mathit{u}{\Vert }^2\mathrm{)}[-\mathrm{\Delta }\mathit{u}+\mathit{u}]=\mathit{\lambda }\mathit{f}\mathrm{(}\mathit{u}\mathrm{,}\mathit{v}\mathrm{)}\mathrm{,}\hfill & \mathit{x}\in {\mathbb{R}}^2\mathrm{,}\hfill \\ \ell \mathrm{(}\Vert \mathit{v}{\Vert }^2\mathrm{)}[-\mathrm{\Delta }\mathit{v}+\mathit{v}]=\mathit{\lambda }\mathit{g}\mathrm{(}\mathit{u}\mathrm{,}\mathit{v}\mathrm{)}\mathrm{,}\hfill & \mathit{x}\in {\mathbb{R}}^2\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $\mathit{\lambda }\mathrm{>}0$ is a parameter, $\mathit{m}\mathrm{,}\ell :[0\mathrm{,}+\infty \mathrm{)}\to [0\mathrm{,}+\infty \mathrm{)}$ are Kirchhoff-type functions, $\Vert ·\Vert$ denotes the usual norm of the Sobolev space ${\mathit{H}}^1\mathrm{(}{\mathbb{R}}^2\mathrm{)}$ and the nonlinear terms f and g have exponential critical growth of Trudinger–Moser type. Moreover, when f and g are odd functions, we prove that the number of solutions increases when the parameter λ becomes large.