A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916851812925440 |
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| author | Clapp, Mónica Saldaña, Alberto Szulkin, Andrzej |
| author_facet | Clapp, Mónica Saldaña, Alberto Szulkin, Andrzej |
| contents | We consider the equation $$-Δu+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ where $Q_\varepsilon$ takes the value $1$ on each ball $B_\varepsilon(y)$, $y\in\mathbb{Z}^N$, and the value $-1$ elsewhere. We establish the existence of a least energy solution for each $\varepsilon\in(0,\frac{1}{2})$ and show that their $H^1$ and $L^p$ norms concentrate locally at points of $\mathbb{Z}^N$ as $\varepsilon\to 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_14541 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core Clapp, Mónica Saldaña, Alberto Szulkin, Andrzej Analysis of PDEs 35J61, 35J20, 35B40, 35B44 We consider the equation $$-Δu+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ where $Q_\varepsilon$ takes the value $1$ on each ball $B_\varepsilon(y)$, $y\in\mathbb{Z}^N$, and the value $-1$ elsewhere. We establish the existence of a least energy solution for each $\varepsilon\in(0,\frac{1}{2})$ and show that their $H^1$ and $L^p$ norms concentrate locally at points of $\mathbb{Z}^N$ as $\varepsilon\to 0$. |
| title | A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core |
| topic | Analysis of PDEs 35J61, 35J20, 35B40, 35B44 |
| url | https://arxiv.org/abs/2507.14541 |