Existence and multiplicity of solutions for a Kirchhoff system with critical growth

Abstract

We consider the system $$\{\begin{array}{ll}-m(\Vert u{\Vert }^2)\mathrm{\Delta }u=\lambda F_u(x,u,v)+\frac{1}{2^{\ast }}{G}_u(u,v),& in\mathrm{\Omega },\\ -l(\Vert v{\Vert }^2)\mathrm{\Delta }v=\lambda F_v(x,u,v)+\frac{1}{2^{\ast }}{G}_v(u,v),& in\mathrm{\Omega },\\ u,v\in H_0^1(\mathrm{\Omega }),\end{array}$$ where $\mathrm{\Omega }\subset {\mathbf{R}}^N$, $N\ge 3$, is a bounded smooth domain, $\Vert \cdot {\Vert }^2={\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\cdot {|}^2\mathrm{d}x$, $\lambda >0$ is a parameter, the functions $m$ and $l$ are positive and increasing, the function $F$ is superlinear both at origin and at infinity, the function $G$ is $2^{\ast }$-homogeneous. In our first result, we obtain a nonzero nonnegative solution for large values of $\lambda$. We also prove that, for any $k\in \mathbf{N}$, there exists ${\lambda }_k^{\ast }>0$ such that the problem has at least $k$ pairs of nonzero solutions if $\lambda \ge {\lambda }_k^{\ast }$.