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

Yunru Bai,Leszek Gasiński,Patrick Winkert,Shengda Zeng

doi:10.1142/s1664360720500095

Abstract

In this paper, we study a class of generalized and not necessarily differentiable functionals of the form [Formula: see text] with functions [Formula: see text], [Formula: see text] that are only locally Lipschitz in the second argument and involving critical growth for the elements of their generalized gradients [Formula: see text] even on the boundary [Formula: see text]. We generalize the famous result of Brezis and Nirenberg [[Formula: see text] versus [Formula: see text] 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.

Highlights

Consider the following functional Φ : H01(Ω) → R defined by 1 Φ(u) = |∇u|2dx −F (x, u)dx, where F (x, s) =s 0 f (x, t)dt with a Caratheodory function f: ×R →R that satisfies the growth condition|f (x, u)| ≤ C(1 + |u|p) with p ≤ N N
We study a class of generalized and not necessarily differentiable functionals of the form
A first result concerning local minimizers and nonhomogeneous operators was presented in the work of Motreanu and Papageorgiou [18] who studied functionals of the form φ0(u) = G(x, ∇u)dx − F0(x, u)dx, u ∈ Wn1,p(Ω), Ω

Summary

Introduction

S 0 f (x, t)dt with a Caratheodory function f:. It is well known that a local C01(Ω)-minimizer of Φ is a local H01(Ω)-minimizer of Φ. A first result concerning local minimizers and nonhomogeneous operators was presented in the work of Motreanu and Papageorgiou [18] who studied functionals of the form φ0(u) = G(x, ∇u)dx − F0(x, u)dx, u ∈ Wn1,p(Ω), where G is the potential of a general nonhomogeneous operator. Papageorgiou and Radulescu [19] studied functionals that are related to nonhomogeneous operator and have a boundary term and the potential term in the domain is related to a Carathedory function that has critical growth. They considered the functional φ0 : W 1,p(Ω) → R defined by φ0(u) =. An element u ∈ R is said to be a critical point of a locally Lipschitz function f : X → R if there holds f ◦(x; y) ≥ 0 for all y ∈ X or, equivalently, 0 ∈ ∂f (x) (see [4])

Preliminaries and Hypotheses

Main Result