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September/October 2021 Multiplicity and concentration results for local and fractional NLS equations with critical growth
Marco Gallo
Adv. Differential Equations 26(9/10): 397-424 (September/October 2021).

Abstract

Goal of this paper is to study the following singularly perturbed nonlinear Schrödinger equation $$ \varepsilon^{2s}(- \Delta)^s v+ V(x) v= f(v), \quad x \in \mathbb{R}^N, $$ 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. When $\varepsilon>0$ is small, we obtain existence and multiplicity of semiclassical solutions, relating the number of solutions to the cup-length of a set of local minima of $V$; in particular, we improve the result in [37]. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. Finally, we prove the previous results also in the limiting local setting $s=1$ and $N\geq 3$, with an exponential decay of the solutions.

Citation

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Marco Gallo. "Multiplicity and concentration results for local and fractional NLS equations with critical growth." Adv. Differential Equations 26 (9/10) 397 - 424, September/October 2021.

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Published: September/October 2021
First available in Project Euclid: 12 August 2021

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Rights: Copyright © 2021 Khayyam Publishing, Inc.

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Vol.26 • No. 9/10 • September/October 2021
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