Discontinuous semilinear problems in vector-valid function spaces
type: Item
instance of: scholarly article
title: Discontinuous semilinear problems in vector-valid function spaces
zbMATH Open document ID: 0848.35051
published in: Differential and Integral Equations
publication date: 1996-05-09T00:00:00Z
review text: The existence of solutions to the discontinuous semilinear problem in vector-valued function spaces: \[ a(u, v)- \langle f, v\rangle_V+ \int_\Omega j^0(u, v) d\Omega\geq 0,\quad \forall v\in V \] is studied. The notation \(j^0(\cdot, \cdot)\) stands for the Clark's directional differential which satisfies the directional growth condition \((*)\) \(j^0(\xi, \eta- \xi)\leq \alpha(r) (1+ |\xi|^s)\). In the case \(1\leq s< 2\), the existence of solutions is established based on the finite-dimensional space regularization, Brouwer's fixed-point theorem and the Dunford-Pettis compactness criterion in \(L^1\). For \(s\geq 2\), some further hypotheses strengthening \((*)\) are imposed to guarantee that the solution can arise as a critical point of a locally Lipschitz functional. Then a nonsmooth minimax method in critical point theory due to K. C. Chang is applied to obtain the existence result in this case.
reviewed by: Ding Zhengzhong
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 856939
zbMATH Keywords: Brouwer's fixed-point theorem
zbMATH Keywords: nonsmooth minimax method
author: Dumitru Motreanu
author: Z. Naniewicz
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 856939