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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.00806 |
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| _version_ | 1866911031963418624 |
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| author | Medina, Maria Wu, Jing |
| author_facet | Medina, Maria Wu, Jing |
| contents | Let $s\in (0,1)$, $\varepsilon>0$ and let $Ω$ be a bounded smooth domain. Given the problem $$\varepsilon^{2s}(-Δ)^{s} u + V(x)u = |u|^{p-1}u \quad \mbox{in }\; Ω,$$ with Dirichlet boundary conditions and $1<p<(n+2s)/(n-2s)$, we analyze the existence of positive multi-peak solutions concentrating, as $\varepsilon\to 0$, to one or several points of $Ω$. Under suitable conditions on $V$, we construct positive solutions concentrating at any prescribed set of its non degenerate critical points. Furthermore, we prove existence and non existence of clustering phenomena around local maxima and minima of $V$, respectively. The proofs rely on a Lyapunov-Schmidt reduction where three effects need to be controlled: the potential, the boundary and the interaction among peaks. The slow decay of the associated {\it ground-state} demands very precise asymptotic expansions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00806 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-peak solutions for the fractional Schrödinger equation with Dirichlet datum Medina, Maria Wu, Jing Analysis of PDEs Let $s\in (0,1)$, $\varepsilon>0$ and let $Ω$ be a bounded smooth domain. Given the problem $$\varepsilon^{2s}(-Δ)^{s} u + V(x)u = |u|^{p-1}u \quad \mbox{in }\; Ω,$$ with Dirichlet boundary conditions and $1<p<(n+2s)/(n-2s)$, we analyze the existence of positive multi-peak solutions concentrating, as $\varepsilon\to 0$, to one or several points of $Ω$. Under suitable conditions on $V$, we construct positive solutions concentrating at any prescribed set of its non degenerate critical points. Furthermore, we prove existence and non existence of clustering phenomena around local maxima and minima of $V$, respectively. The proofs rely on a Lyapunov-Schmidt reduction where three effects need to be controlled: the potential, the boundary and the interaction among peaks. The slow decay of the associated {\it ground-state} demands very precise asymptotic expansions. |
| title | Multi-peak solutions for the fractional Schrödinger equation with Dirichlet datum |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.00806 |