OnW^1,pversusC¹(Ω) local minimizers for functionals with critical growth

Abstract

Let be a bounded domain in IR N , N ! 2 with C 2 boundary and let f(x, t), g(x, t) :  IR ! IR be Carathe´odory functions satisfying sf(x, s) ! 0, sg(x, s) ! 0 for all (x, s) 2  IR and j f ðx, tÞj C 1 ð1 þ jtj p à 1 À1 Þ, j gðx, tÞj C 2 jtj q , 0 q p 1 À 1. Consider the functional J : W 1,p1 ðÞ ! IR defined asgðx, sÞds and È(t) satisfies the asymptotic conditions ÈðtÞ $ t p0 as t ! 0 þ and ÈðtÞ $ t p1 as t ! 1 with p 0 , p 1 41. In this article, we show that a C 1 ðÞ local minimum of J is also a W 1,p1 ðÞ local minimum of J.