Bound state solutions for a class of nonlinear Schrödinger equations

Abstract

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form $$ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{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 \\emph{semi-classical limit}, i.e. for \varepsilon\sim 0 , we prove the existence of bound state solutions localized around local minimum points of the auxiliary function \mathcal{A} = V^\theta K^{-\frac{2}{p-1}} , where \theta=(p+1)/(p-1)-N/2 . A special attention is devoted to the qualitative properties of these solutions as \varepsilon$ goes to zero.