Non-homogeneous Dirichlet problems with concave-convex reaction
type: Item
instance of: scholarly article
title: Non-homogeneous Dirichlet problems with concave-convex reaction
zbMATH Open document ID: 1486.35230
DOI: 10.4171/RLM/959
author: Gabriele Bonanno
author: R. Livrea
author: Vicenţiu D. Rădulescu
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2022-03-16T00:00:00Z
review text: Let \(\Omega \subseteq \mathbb{R}^N\) (\(N \geq 2\)) be a bounded domain with a smooth boundary \(\partial \Omega\). The authors consider a Dirichlet problem of the form \[\operatorname{div} A(x,\nabla u)=\lambda f(x,u) \mbox{ in } \Omega,\quad u|_{\partial \Omega}=0, \tag{\(P_\lambda\)}\] where \(A:\Omega \times \mathbb{R}^N \to \mathbb{R}^N\) is a function satisfying suitable properties and covering the behavior of a non-homogeneous differential operator of \(p\)-Laplacian type, \(\lambda >0\) is a positive parameter and \(f:\Omega \times \mathbb{R}\to \mathbb{R}\) is a Caratheodory function which is \((p-1)\)-superlinear at infinity. The main result establishes an exhaustive property, which describes the interval values of the parameter \(\lambda>0\) for which the problem \((P_\lambda)\) has at least two nontrivial weak solutions. Additionally, the positivity of the solutions is obtained when \(f\) satisfies a sign condition. The proofs combine variational methods and critical point theorems.
reviewed by: Calogero Vetro
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 7490886
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: Dirichlet condition
zbMATH Keywords: existence
zbMATH Keywords: variational methods
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/rlm/959
OpenAlex ID: W4213123706
description: scientific article; zbMATH DE number 7490886