Existence of Solutions for the Critical Elliptic System with Inverse Square Potentials

Let Ω■0 be an open bounded domain in RN ( N ≥ 3) and 2 * ( s ) = 2( N-s ) /N -2 , 0 < s < 2. We consider the following elliptic system of two equations in H 1 0 ( Ω) × H 1 0 ( Ω):-△u-tu/| x | 2 = 2 α /(α + β) (| u | α-2 u | v | β) /| x | s + λu1, -△v-t v /| x | 2 = 2 β/( α + β) (| u | α | v |( β-2))/ v | x | s + μv1,where λ1, μ > 0 and α1, β > 1 satisfy α + β = 2 *( s ). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.