Nodal solutions of an NLS equation concentrating on lower dimensional spheres

Abstract

In this work we deal with the following nonlinear Schrödinger equation: {−^ϵ2Δu+V(x)u=f(u)in ^RNu∈^H1(^RN),(Formula presented.) where N≥3, f is a subcritical power-type nonlinearity and V is a positive potential satisfying a local condition. We prove the existence and concentration of nodal solutions which concentrate around a k-dimensional sphere of R^N, where (Formula presented.). The radius of such a sphere is related with the local minimum of a function which takes into account the potential V. Variational methods are used together with the penalization technique in order to overcome the lack of compactness.