A system with weights and with critical Sobolev exponent

Abstract

In this paper, we investigate the minimization problem : $$\underset{\begin{array}{l}u\in H_0^1(\Omega ),v\in H_0^1(\Omega ),\\ \Vert u{\Vert }_{L^q}=1,\Vert v{\Vert }_{L^q}=1\end{array}}{\inf }\left[\frac{1}{2}{\int }_{\Omega }a(x)|\nabla u(x){|}^{2}dx+\frac{1}{2}{\int }_{\Omega }b(x)|\nabla v(x){|}^{2}dx-\lambda {\int }_{\Omega }u(x)v(x)dx\right]$$ where $q=\frac{2N}{N-2}$, $N\ge 4$, $a$ 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 $\lambda$.