On a semi-linear Schrödinger equation in RN

We are concerned with the semi-linear Schrodinger equation:-Δu+V(x)u=K(x)|u|2*-2u+g(x,u)u∈H1(RN)where N⩾3, V(x) is a continuous function such that the spectrum σ(-Δ+V(x)) of -Δ+V(x) in L2(RN) has a negative part, 2*=2N/(N-2) is the critical Sobolev exponent, K(x) is a bounded positive function, g is of subcritical growth. We prove that under suitable hypotheses the equation has a nontrivial solution.