Solutions for autonomous semilinear elliptic equations
Alexis Molino, Salvador Villegas
TL;DR
This paper investigates the existence of nontrivial solutions to the autonomous semilinear elliptic boundary-value problem $- abla^2 u=\lambda u+f(u)$ in a smooth bounded domain $\Omega$ with Dirichlet boundary conditions, under the condition that the primitive $F(s)=\int_0^s f(t)\,dt$ is nonpositive. The authors show a dimension- and threshold-dependent picture: for $\lambda\le 0$ the trivial solution is unique, while for $\lambda\ge \lambda_1$ there exist choices of $f$ with $F\le 0$ yielding nontrivial solutions; in dimensions $N\ge 3$ and on star-shaped domains, Pohožaev identities yield nonexistence for $\lambda\le \frac{N-2}{N}\lambda_1$, whereas in $N=1$ a complete positive result holds with nontrivial solutions for all $\lambda>0$ via a constructive $f$. For $N\ge 2$ the results are partial, with open questions about the regime $0<\lambda<\lambda_1$ (and, in higher dimensions, the interval $(\frac{N-2}{N}\lambda_1,\lambda_1)$). The work highlights the delicate balance between domain geometry, spectral thresholds, and nonlinear primitive constraints in autonomous semilinear elliptic problems, and it outlines clear directions for future research on unresolved dimension- and λ-driven cases.
Abstract
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 $N\geq 2$.