Multiplicity of positive solutions for semilinear elliptic systems

We study the effect of the coefficient h(x) of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems. Under suitable assumptions for f(x), g(x), and h(x), we should prove that for sufficiently small ,>0, there are at least k+1 positive solutions of the semilinear elliptic systems -Δu=f(x)|u|^q-2u+(α/ (α+β))h(x)|u|^α-2u|v|^β, -Δv=g(x)|v|^q-2v+(β/(α+β))h(x)|u| ^α|v|^β-2v, where 0εΩ⊂R ^N is a bounded domain, α>1, β>1, and N/(N-2)<q<2<α+β=2* for N>4.