Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms
type: Item
instance of: scholarly article
title: Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms
zbMATH Open document ID: 1390.35094
DOI: 10.12775/TMNA.2017.029
author: Nikolaos S. Papageorgiou
author: Calogero Vetro
author: Francesca Vetro
published in: Topological Methods in Nonlinear Analysis
publication date: 2018-03-16T00:00:00Z
review text: Given a bounded domain \(\Omega \subseteq \mathbb{R}^N\) with a \(C^2\)-boundary \(\partial\Omega\), the authors study the following semilinear Robin problem \[ \begin{alignedat}{2} -\Delta u(z) +\xi(z) u(z) & = \vartheta(z) |u(z)|^{q-2}u(z)+f(z,u(z))\quad \text{in}\quad \Omega,\\ \frac{\partial u}{\partial n}+\beta(z)u & = 0\quad \text{on}\quad \partial \Omega, \end{alignedat}\tag{1} \] where \(1<q<2\), \(\xi \in L^s(\Omega)\) with \(s>N\) being sign-changing, \(\vartheta \in L^\infty(\Omega)\), \(\vartheta(z)>0\) for almost all \(z\in\Omega\) and \(f:\Omega\times\mathbb{R}\to\mathbb{R}\) is a Carathéodory function of arbitrary growth. By applying a variant of the symmetric mountain pass theorem, the authors show the existence of smooth nodal (i.e., sign-changing) solutions which converge to zero in \(C^1(\overline{\Omega})\). Furthermore, if the coefficient of the concave term (that is, \(\vartheta(z) |u(z)|^{q-2}u(z)\), \(1<q<2\)) is sign changing, then it is shown again the existence of a sequence of smooth solutions converging to zero in \(C^1(\overline{\Omega})\) but without any knowledge about the sign.
reviewed by: Patrick Winkert
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 6851000
zbMATH Keywords: semilinear Robin problem
zbMATH Keywords: indefinite potential
zbMATH Keywords: smooth nodal solutions
MaRDI profile type: Publication
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full work available at URL: tmna.2017.029
OpenAlex ID: W2747940542
description: scientific article; zbMATH DE number 6851000