On a Elliptic System Involving Nonhomogeneous Nonlinearities and Critical Growth

Abstract

Let $\Omega \subset {\mathbb {R}}^N$ be a bounded domain with $N\ge 3$. We address the existence and nonexistence of solutions for the following class of elliptic systems:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=a u+b v+ \mu |u|^{p-2}u &{} \text {in}\quad \Omega \\ -\Delta v=bu+a v+ |v|^{2^*-2}v &{} \text {in}\quad \Omega \end{array}\right. } \end{aligned}$$

with Dirichlet boundary condition, where $a,b\in {\mathbb {R}}$, the exponent $p>1$, $\mu \in {\mathbb {R}}$ is a parameter and $2^*=2N/(N-2)$ is the critical Sobolev exponent. By exploiting minimization arguments, minimax techniques and a Pohozaev identity, we obtain various results for the above system by analyzing the interplay between $a,b,\mu$ and the exponent p.