Multiplicity and concentration results for local and fractional NLS equations with critical growth

Abstract

Goal of this paper is to study positive semiclassical solutions of the nonlinear Schrödinger equation $${\epsilon }^{2s}(-\Delta {)}^{s}u+V(x)u=f(u),x\in {\mathbb{R}}^N,$$ where $s\in (0,1)$, $N\ge 2$, $V\in C({\mathbb{R}}^N,\mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. We obtain existence and multiplicity for $\epsilon >0$ small, where the number of solutions is related to the cup-length of a set of local minima of $V$. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. We highlight that these results are new also in the limiting local setting $s=1$ and $N\ge 3$, with an exponential decay of the solutions.

Preprint

References in corpus (2)

• Fractional Quantum Mechanics
• Solitary Wave Dynamics in an External Potential

Cited by in corpus (1)

• On fractional Schrödinger equations with Hartree type nonlinearities