Abstract:

In this paper, we consider the Existence of Solutions of an Elliptic System with Critical Sobolev Exponents $\left\{\begin{array}{l} -\Delta u+a(x)u=\frac{2\alpha }{\alpha +\beta }u^{\alpha -1}v^\beta +f(x), \quad & x\in \Omega, \\ -\Delta v+b(x)v=\frac{2\beta }{\alpha +\beta }u^\alpha v^{\beta -1} +g(x), & x\in \Omega, \\ u>0, v>0, &x\in\Omega, \\ u=v=0, &x\in\partial \Omega, \end{array}\right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(*)$ Where $\Omega$ is a bounded smooth domain of $\mathbb{R} ^N$, $N=3, 4, a\geq 2, \beta\geq 2,$ $\alpha +\beta=2^*=\frac{2N}{N-2},$ $f(x)\geq 0,$ $g(x)\geq 0,$ $f(x),$ $g(x)\in H^{-1}(\Omega), a(x)\geq 0, b(x)\geq0.$ We obtain that under some assumptions the problem $(*)$ has two positive solutions with energy larger than zero.

Key words: Critical Sobolev exponent, Palais-Smale condition, Ljusternlik-Schnirelman category