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arXiv · 2504.18877

Solutions for autonomous semilinear elliptic equations

Alexis Molino·Salvador Villegas
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Abstract

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N\geq 1$, $\lambda \in \mathbb{R}$ 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 $N\geq 2$.

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Alexis Molino, Salvador Villegas. 2025-04-26. Solutions for autonomous semilinear elliptic equations. https://arxiv.org/abs/2504.18877

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