Solutions for autonomous semilinear elliptic equations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909595566342144 |
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| author | Molino, Alexis Villegas, Salvador |
| author_facet | Molino, Alexis Villegas, Salvador |
| contents | We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -Δu &=& λu+f(u)&\text{ in }Ω,\\ u&=&0&\text{ on }\partial Ω, \end{array}\right. \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $N\geq 1$, $λ\in \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ is any locally Lipschitz function with nonpositive primitive. A complete description is obtained for $N=1$ and partial results for $N\geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18877 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solutions for autonomous semilinear elliptic equations Molino, Alexis Villegas, Salvador Analysis of PDEs Classical Analysis and ODEs 35J05, 35J15, 35J25 We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -Δu &=& λu+f(u)&\text{ in }Ω,\\ u&=&0&\text{ on }\partial Ω, \end{array}\right. \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $N\geq 1$, $λ\in \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ is any locally Lipschitz function with nonpositive primitive. A complete description is obtained for $N=1$ and partial results for $N\geq 2$. |
| title | Solutions for autonomous semilinear elliptic equations |
| topic | Analysis of PDEs Classical Analysis and ODEs 35J05, 35J15, 35J25 |
| url | https://arxiv.org/abs/2504.18877 |