Multi-peak solutions for a class of nonlinear Schrödinger equations

Abstract.

In this paper we consider the study of positive solutions of¶¶$-\varepsilon^2\Delta u+\lambda u=f(x,u)\quad {\rm on}\quad \mathbb{R}^N,$¶¶where ε is a small parameter, λ>0 and f is an appropriate function. Here we find multi-peak solutions exhibiting concentration at any prescribed "stable" set of zeroes of the field¶¶${\cal S}(P)=\int\limits_{\mathbb{R}^N}\left[\nabla_xf(P,U_P(y))\cdot y\right]\nabla U_P(y)dy,\quad P\in \mathbb{R}^N,$¶¶where U ^P is the unique radial solution of the limit equation¶¶$-\Delta U_P+\lambda U_P=f(P,U_P)\quad {\rm on} \quad \mathbb{R}^N.$¶¶Conversely, we show that the points at which a sequence of multi-peak solutions concentrate must be zeroes of the field ${\cal S}$.