Nonlinear, nohonogeneous parametric Neumann problems
type: Item
instance of: scholarly article
title: Nonlinear, nohonogeneous parametric Neumann problems
zbMATH Open document ID: 1367.35083
DOI: 10.12775/TMNA.2016.035
author: Sergiu Aizicovici
author: Nikolaos S. Papageorgiou
author: Vasile Staicu
published in: Topological Methods in Nonlinear Analysis
publication date: 2017-04-04T00:00:00Z
review text: The authors consider a parametric nonlinear Neumann problem \[ \begin{cases} -\text{div} \, a(Du(z))+\lambda|u(z)|^{p-2}u(z)=f(z, u(z)) \;\text{ in } \;\Omega\\ \frac{\partial u}{\partial n}=0 \qquad\qquad\qquad\qquad\qquad\qquad \quad \text{ on } \;\partial\Omega, \;\lambda>0 \end{cases} \] driven by a nonlinear nonhomogeneous differential operator, with a Carathéodory reaction \(f\) which is \(p\)-superlinear in the second variable, but not necessarily satisfying the usual in such cases Ambrosetti-Rabinowitz condition. A bifurcation type result is obtained describing the dependence of positive solutions on the parameter \(\lambda>0\), the existence of a smallest positive solution \(\bar u_\lambda\) is shown, and the properties of the map \(\bar\lambda\to \bar u_\lambda\) are investigated. In addition the existence of nodal solutions is shown.
Mathematics Subject Classification ID: 35J66
zbMATH DE Number: 6700548
zbMATH Keywords: nonlinear nonhomogeneous differential operator
zbMATH Keywords: Neuman problem
zbMATH Keywords: positive solutions
zbMATH Keywords: nodal solutions
reviewed by: Lubomira Softova
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6700548