Multiplicity and concentration results for local and fractional NLS equations with critical growth
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866915353735462912 |
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| author | Gallo, Marco |
| author_facet | Gallo, Marco |
| contents | Goal of this paper is to study positive semiclassical solutions of the nonlinear Schrödinger equation $$ \varepsilon^{2s}(- Δ)^s u+ V(x) u= f(u), \quad x \in \mathbb{R}^N,$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_00448 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Multiplicity and concentration results for local and fractional NLS equations with critical growth Gallo, Marco Analysis of PDEs 35A15, 35B25, 35B33, 35Q55, 35R11, 47J30, 58E05 Goal of this paper is to study positive semiclassical solutions of the nonlinear Schrödinger equation $$ \varepsilon^{2s}(- Δ)^s u+ V(x) u= f(u), \quad x \in \mathbb{R}^N,$$ 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. |
| title | Multiplicity and concentration results for local and fractional NLS equations with critical growth |
| topic | Analysis of PDEs 35A15, 35B25, 35B33, 35Q55, 35R11, 47J30, 58E05 |
| url | https://arxiv.org/abs/2101.00448 |