Multiple solutions for a class of nonlinear equations involving a duality mapping.
type: Item
instance of: scholarly article
title: Multiple solutions for a class of nonlinear equations involving a duality mapping.
zbMATH Open document ID: 1224.35085
published in: Differential and Integral Equations
publication date: 2011-02-02T00:00:00Z
review text: The paper studies the existence of multiple solutions to abstract equation \(J_{p}u=N_{f}u\), where \(J_{p}\) is the duality mapping on a real reflexive and smooth Banach space \(X\), corresponding to the gauge function \(\varphi (t)=t^{p-1}\), \(1<p<+\infty \). It is assumed, that \(X\) is compactly imbedded in a Lebesgue space \(L^{q}(\Omega )\), \(p\leq q< p^{*}\) and continuously imbedded in \(L^{p^{*}}(\Omega )\), \(p^{*}\) being the Sobolev conjugate exponent, \(N_{f}\: L^{q}(\Omega )\to L^{q'}(\Omega )\), \(1/q+1/q'=1\), being the Nemytskii operator generated by a function \(f\in \mathcal {C}(\bar \Omega \times \mathbf {R},\mathbf {R})\), which satisfies some appropriate conditions. These assumptions allow the use of many procedures appearing essentially in the paper by \textit{G. B. Li} and \textit{H. S. Zhou} [J. Lond. Math. Soc., II. Ser. 65, No. 1, 123--138 (2002; Zbl 1171.35384)]. The results for the abstract problem are applied to Dirichlet and Neumann problems for \(p\)-Laplacian.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 5844180
zbMATH Keywords: equation in Banach space
zbMATH Keywords: multiple solutions
zbMATH Keywords: \(p\)-Laplacian
author: Jenică Crînganu
author: George Dinca
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5844180