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.

arXiv commentPreprint

References in corpus (10)

• Fractional Schrodinger equation
• Fractional Quantum Mechanics
• Multiplicity results for some nonlinear Schroedinger equations with potentials
• Solitary Wave Dynamics in an External Potential
• Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes
• Fractional elliptic problems with critical growth in the whole of ${\mathbb{R}}^n$
• Long time motion of NLS solitary waves in a confining potential
• Existence and concentration results for some fractional Schrödinger equations in ${\mathbb{R}}^N$ with magnetic fields
• Multiple Semiclassical Standing Waves for Fractional Nonlinear Schrödinger Equations
• On the Dynamics of solitons in the nonlinear Schroedinger equation

Cited by in corpus (1)

• On fractional Schrödinger equations with Hartree type nonlinearities