Goal of this paper is to study the following singularly perturbed nonlinear Schr\"odinger 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 $\eps>0$ 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 \cite{HeZo}. 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.

Gallo, M., Multiplicity and concentration results for local and fractional NLS equations with critical growth, <<ADVANCES IN DIFFERENTIAL EQUATIONS>>, 2021; 26 (9-10): 397-424. [doi:10.57262/ade026-0910-397] [https://hdl.handle.net/10807/228872]

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

Gallo, Marco
2021

Abstract

Goal of this paper is to study the following singularly perturbed nonlinear Schr\"odinger 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 $\eps>0$ 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 \cite{HeZo}. 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.
2021
Inglese
Gallo, M., Multiplicity and concentration results for local and fractional NLS equations with critical growth, <<ADVANCES IN DIFFERENTIAL EQUATIONS>>, 2021; 26 (9-10): 397-424. [doi:10.57262/ade026-0910-397] [https://hdl.handle.net/10807/228872]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/228872
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