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 [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.