W1, N versus C1 local minimizers for elliptic functionals with critical growth in RN

Abstract

Let Ω ⊂ RN be a bounded smooth domain, f : Ω × R → R be a Caratheodory function with s f (x, s) ≥ 0 ∀ (x, s) ∈ Ω × R and supx ∈ Ω | f (x, s) | ≤ C (1 + | s |)p e| s |frac(N, (N - 1)), ∀ s ∈ R, for some C > 0. Consider the functional J : W1, N (Ω) → R, Ω defined asJ (u) over(=, def) frac(1, N) {norm of matrix} u {norm of matrix}W1, N (Ω)N - under(∫, Ω) F (x, u) - frac(1, q + 1) {norm of matrix} u {norm of matrix}Lq + 1 (∂ Ω)q + 1, where F (x, u) = ∫0u f (x, s) d s and q > 0. We show that if u0 ∈ C1 (over(Ω, -)) is a local minimum of J in the C1 (over(Ω, -)) topology, then it is also a local minimum of J in W1, N (Ω) topology. To cite this article: J. Giacomoni et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2009 Académie des sciences.