Discontinuous semilinear problems in vector-valued function spaces
Dumitru Motreanu, Z. Naniewicz · Differential and Integral Equations · 1996
The aim of this paper is to study the existence of solutions to the following discontinuous semilinear problem:where V is a reflexive Banach space compactly imbedded into L p (⌦; R N ) with p > 2 and N 1, for a bounded domain ⌦ in R m , m 1.Throughout Section 2 it will be supposed thatto denote the dual space of V , the pairing over V ⇤ ⇥ V , the norm in V and the norm in L p (⌦; R N ), respectively.The data entering (1.1) are the following: a(•, •) is a continuous symmetric bilinear form on V satisfying the ellipticity condition:with the associated operatorThe notation j 0 (•, •) stands for the Clarke's directional di↵erential defined by[10]) by means of which the Clarke's generalized gradient is introduced as