W1,p versus C1: The nonsmooth case involving critical growth

Abstract

In this paper, we study a class of generalized and not necessarily differentiable functionals of the form J(u) =∫ΩG(x,∇u)dx −∫Ωj1(x,u)dx −∫∂Ωj2(x,u)dσ with functions j1: Ω × ℝ → ℝ, j2: ∂Ω × ℝ → ℝ that are only locally Lipschitz in the second argument and involving critical growth for the elements of their generalized gradients ∂jk(x,⋅),k = 1, 2 even on the boundary ∂Ω. We generalize the famous result of Brezis and Nirenberg [H1 versus C1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317(5) (1993) 465–472] to a more general class of functionals and extend all the other generalizations of this result which has been published in the last decades.

Keywords

• nonhomogeneous partial differential operator
• local minimizer
• clarke’s generalized gradient
• critical growth
• neumann problem