Some Existence and Regularity Results for Unilateral Problems with Degenerate Coercivity

Abstract

In this work we study the existence and regularity of the solutions of unilateral problems 8>><>>: u ‚ 0, a.e.in ›, hAu,u ¡ vi • R› f (u ¡ v), 8v 2 H1 0(›),v ‚ 0, a.e.in ›, (1) Here › is a bounded, open subset of RN , with N ¨ 2, and Au ˘ ¡di v(a(x,u)Du) with a(x,s) : ›£R ! R is a Carathéodory function, satisfying the following conditions: fi (1¯jsj)θ • a(x,s) • fl. (2) for some real number θ such that. 0 • θ ˙ 1 (3) for almost every x 2 ›, for every s 2 R, where fi and fl are positive constants. The objective of this contribution is to study the existence and regularity of the solutions of unilateral problems (1) associated to A under assumptions (2)-(3) and with data f belonging to various Lebesgue space Lm(›) , for some m ¨ 1.