Nonlinear Robin problems with a reaction of arbitrary growth
type: Item
instance of: scholarly article
title: Nonlinear Robin problems with a reaction of arbitrary growth
zbMATH Open document ID: 1348.35066
author: Nikolaos S. Papageorgiou
author: Patrick Winkert
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2016-08-11T00:00:00Z
review text: The Robin problem in question is the nonlinear elliptic boundary value problem \(-\operatorname{div} a(\nabla u)=f(x,u)\) in \(\Omega\), \(\partial u/\partial n_a=-\beta(x)|u|^{p-2}\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with a \(C^2\) boundary \(\partial\Omega\), \(a:\mathbb R^N\to\mathbb R^N\) is a continuous strictly monotone function satisfying a set of regularity and growth conditions, and \(\partial u/\partial n_a=(a(\nabla u),n)_{\mathbb R^N}\) with the outward unit normal \(n(x)\) at \(x\in\partial\Omega\). The boundary value function \(\beta\) is a non-negative \(\alpha\)-Hölder continuous function on \(\partial\Omega\) with \(0<\alpha<1\). The reaction \(f\) is a Carathéodory function which, instead of being assumed to satisfy any global growth condition, admits \(x\)-dependent zeros of constant sign. Examples of admissible functions \(a\) include \(a(\xi)=|\xi|^{p-2}\xi\) with \(1<p<\infty\) for the well-known \(p\)-Laplacian, \(a(\xi)=|\xi|^{p-2}\xi+|\xi|^{q-2}\xi\) with \(1<q<p<\infty\), \(a(\xi)=(1+|\xi|^2)^{(p-2)/2}\xi\) with \(1<p<\infty\) for the generalized \(p\)-mean curvature operator, or \(a(\xi)=|\xi|^{p-2}\xi[1+(1+|\xi|^p)^{-1}]\) with \(1<p<\infty\) for an operator arising in plasticity theory. The authors prove multiplicity theorems for the solutions of the Robin problem. Using variational methods based on critical point theory, suitable truncation and perturbation techniques and the Morse theory, they show that the problem has at least three nontrivial solutions, two of them having constant opposite sign and the third one being nodal.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 6614272
zbMATH Keywords: Robin problem
zbMATH Keywords: constant sign
zbMATH Keywords: nodal solution
zbMATH Keywords: reaction with zeros
zbMATH Keywords: nonlinear regularity
zbMATH Keywords: nonlinear maximum principle
zbMATH Keywords: superlinear near zero
zbMATH Keywords: critical groups
reviewed by: Jiří Rákosník
MaRDI profile type: Publication
full work available at URL: s10231-015-0512-0
OpenAlex ID: W1047374684
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DOI: 10.1007/S10231-015-0512-0
description: scientific article; zbMATH DE number 6614272