On W^1,p versus C¹(Ω) local minimizers for functionals with critical growth

Abstract

Let Ω be a bounded domain in IR ^N , N ≥ 2 with C ² boundary and let f(x, t), g(x, t) : Ω × IR → IR be Carathéodory functions satisfying sf(x, s) ≥ 0, sg(x, s) ≥ 0 for all (x, s) ∈ Ω × IR and . Consider the functional defined as

where and Φ(t) satisfies the asymptotic conditions as t → 0⁺ and as t → ∞ with p ₀, p ₁ > 1. In this article, we show that a local minimum of J is also a local minimum of J.

Keywords:

• quasilinear equations
• nonlinear Neumann boundary condition
• critical exponent

AMS Subject Classifications::

• 34B15
• 35J60