Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values
type: Item
instance of: scholarly article
title: Constant-sign and sign-changing solutions for nonlinear elliptic equations with Neumann boundary values
zbMATH Open document ID: 1208.35065
author: Patrick Winkert
published in: Advances in Differential Equations
publication date: 2010-07-09T00:00:00Z
review text: From the author's abstract: We study the existence of multiple solutions to the equation \[ -\Delta_p u = f(x,u) - |u|^{p-2}u \] with the nonlinear boundary condition \[ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu} = \lambda |u|^{p-2}u + g(x,u). \] We establish the existence of a smallest positive solution, a greatest negative solution, and a nontrivial sign-changing solution when the parameter \(\lambda\) is greater than the second eigenvalue of the Steklov eigenvalue problem. Our approach is based on truncation techniques and comparison principles for nonlinear elliptic differential inequalities. In particular, we make use of variational and topological tools, such as critical point theory, the mountain-pass theorem, the second deformation lemma and variational characterizations of the second eigenvalue of the Steklov eigenvalue problem.
Mathematics Subject Classification ID: 35J66
Mathematics Subject Classification ID: 35B38
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B09
zbMATH DE Number: 5735001
zbMATH Keywords: nonlinear Neumann problem
zbMATH Keywords: \(p\)-Laplacian
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5735001