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Multiplicity and concentration results for local and fractional NLS equations with critical growth
- Publication Year :
- 2021
-
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.<br />Comment: Preprint
- Subjects :
- Mathematics - Analysis of PDEs
35A15, 35B25, 35B33, 35Q55, 35R11, 47J30, 58E05
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2101.00448
- Document Type :
- Working Paper