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.