Existence and multiplicity of solutions for resonant-superlinear problems
type: Item
instance of: scholarly article
title: Existence and multiplicity of solutions for resonant-superlinear problems
zbMATH Open document ID: 1540.3522
DOI: 10.1080/17476933.2023.2164888
author: Nikolaos S. Papageorgiou
author: Calogero Vetro
author: Francesca Vetro
published in: Complex Variables and Elliptic Equations
publication date: 2024-05-14T00:00:00Z
review text: The authors study existence and multiplicity of nontrivial solutions for semilinear Dirichlet problems of the form\N\begin{align*}\N-\Delta u(z)=\hat{\lambda}_1u(z)+f(z,u^+(z))+\theta(z) \quad \text{in }\Omega, \quad u|_{\partial\Omega}=0,\N\end{align*}\Nwhere \(\Omega\subseteq \mathbb{R}^N\) with \(N\geq 2\) is a bounded domain with a \(C^2\)-boundary \(\partial\Omega\), \(\hat{\lambda}_1>0\) is the principal eigenvalue of \((-\Delta,H^1_0(\Omega))\), \(u^+=\max\{u,0\}\), \(\theta \in L^\infty(\Omega)\) with \(\theta(z) \leq 0\) for a.a.\,\(z\in\Omega\) and \(f\colon \Omega \times \mathbb{R}\to\mathbb{R}\) is a Carathéodory function which exhibits superlinear growth. By applying variational tools from critical point theory together with truncation and comparison techniques as well as critical groups, the authors prove two multiplicity theorems producing two and three nontrivial solutions. In addition, it is shown that the problem cannot have negative solutions.
reviewed by: Patrick Winkert
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 7846665
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7846665