Calculus of Variations and Geometric Measure Theory

M. Gallo

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

created by gallo on 10 May 2022
modified on 13 Nov 2022

[BibTeX]

Published Paper

Inserted: 10 may 2022
Last Updated: 13 nov 2022

Journal: Advances in Differential Equations
Volume: 26
Number: 9-10
Pages: 397-424
Year: 2021

ArXiv: 2101.00448 PDF
Links: Project Euclid

Abstract:

Goal of this paper is to study positive semiclassical solutions of the nonlinear Schr\"odinger equation $$ \varepsilon{2s}(- \Delta)s u+ V(x) u= f(u), \quad x \in \mathbb{R}N, $$ where $s \in (0,1)$, $N \geq 2$, $V \in C(\mathbb{R}^N,\mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. We obtain existence and multiplicity for $\varepsilon>0$ 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 $N\geq 3$, with an exponential decay of the solutions.