Existence result for semilinear elliptic systems involving critical exponents

where ⊂RN (N ≥ ) is a smooth bounded domain such that ξi ∈ , i = , , . . . ,k, k ≥ , are different points, ≤ μi < μ̄ := (N–  ), L := – · – ∑k i=μi · |x–ξi| , η,λ,σ ≥ , a,a,a ∈ R,  < α, β < ∗ – , α + β = ∗. We work in the product space H ×H , where the space H :=H ( ) is the completion of C∞  ( ) with respect to the norm ( ∫ |∇ · | dx)   . In resent years many publications [–] concerning semilinear elliptic equations involving singular points and the critical Sobolev exponent have appeared. Particularly in the last decade or so, many authors used the variational method and analytic techniques to study the existence of positive solutions of systems of the form of (.) or its variations; see, for example, [–]. Before stating the main result, we clarify some terminology. Since our method is variational in nature, we need to define the energy functional of (.) on H ×H