Concentration phenomena for a mixed local/nonlocal Schrödinger equation with Dirichlet datum
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arXiv
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| Format: | Preprint |
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2025
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| author | Dipierro, Serena Su, Xifeng Valdinoci, Enrico Zhang, Jiwen |
| author_facet | Dipierro, Serena Su, Xifeng Valdinoci, Enrico Zhang, Jiwen |
| contents | We consider the mixed local/nonlocal semilinear equation
\begin{equation*}
-ε^{2}Δu +ε^{2s}(-Δ)^s u +u=u^p\qquad \text{in } Ω
\end{equation*} with zero Dirichlet datum, where $ε>0$ is a small parameter, $s\in(0,1)$, $p\in(1,\frac{n+2}{n-2})$ and $Ω$ is a smooth, bounded domain. We construct a family of solutions that concentrate, as $ε\rightarrow 0$, at an interior point of $Ω$ having uniform distance to $\partialΩ$ (this point can also be characterized as a local minimum of a nonlocal functional).
In spite of the presence of the Laplace operator, the leading order of the relevant reduced energy functional in the Lyapunov-Schmidt procedure is polynomial rather than exponential in the distance to the boundary, in light of the nonlocal effect at infinity. A delicate analysis is required to establish some uniform estimates with respect to $ε$, due to the difficulty caused by the different scales coming from the mixed operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration phenomena for a mixed local/nonlocal Schrödinger equation with Dirichlet datum Dipierro, Serena Su, Xifeng Valdinoci, Enrico Zhang, Jiwen Analysis of PDEs We consider the mixed local/nonlocal semilinear equation \begin{equation*} -ε^{2}Δu +ε^{2s}(-Δ)^s u +u=u^p\qquad \text{in } Ω \end{equation*} with zero Dirichlet datum, where $ε>0$ is a small parameter, $s\in(0,1)$, $p\in(1,\frac{n+2}{n-2})$ and $Ω$ is a smooth, bounded domain. We construct a family of solutions that concentrate, as $ε\rightarrow 0$, at an interior point of $Ω$ having uniform distance to $\partialΩ$ (this point can also be characterized as a local minimum of a nonlocal functional). In spite of the presence of the Laplace operator, the leading order of the relevant reduced energy functional in the Lyapunov-Schmidt procedure is polynomial rather than exponential in the distance to the boundary, in light of the nonlocal effect at infinity. A delicate analysis is required to establish some uniform estimates with respect to $ε$, due to the difficulty caused by the different scales coming from the mixed operator. |
| title | Concentration phenomena for a mixed local/nonlocal Schrödinger equation with Dirichlet datum |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.14483 |