A system with weights and with critical Sobolev exponent

Abstract

In this paper, we investigate the minimization problem : $$ \inf_{ \displaystyle{\begin{array}{lll} u \in H_0^1(\Omega), v \in H_0^1(\Omega),\\ \quad \| u \|_{L^{q}} =1, \quad \| v \|_{L^{q}} = 1 \end{array}}} \left[ \frac{1}{2} \int_{\Omega} a(x) \vert \nabla u(x) \vert^2dx + \displaystyle{ \frac{1}{2} \int_{\Omega} b(x) \vert \nabla v (x)\vert^2dx } - \lambda \displaystyle{\int_{\Omega} u(x)v (x)dx} \right] $$ where $q=\frac{2N}{N-2}$, $ N \geq 4$, $a$ and $b$ are two continuous positive weight functions. We show the existence of solutions of the previous minimizing problem under some conditions on $a$, $b$, the dimension of the space and the parameter $\lambda$.