Bound state solutions for a class of nonlinear Schrödinger equations

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrodinger equations of the form -epsilon(2)Delta u + V(x)u = K(x)u(p), x is an element of R-N, where V, K are positive continuous functions and p > 1 is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential V is allowed to vanish at infinity and the competing function K does not have to be bounded. In the semi-classical limit, i.e. for epsilon similar to 0, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function A = (VK-2/p-1)-K-theta, where theta = (p + 1)/(p - 1) - N/2. A special attention is devoted to the qualitative properties of these solutions as e goes to zero.