Open AccessW1,p versus C1: The nonsmooth case involving critical growth
Author(s) -
Yunru Bai,
Leszek Gasiński,
Patrick Winkert,
Shengda Zeng
Publication year - 2020
Publication title -
bulletin of mathematical sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.407
H-Index - 21
eISSN - 1664-3607
pISSN - 1664-3615
DOI - 10.1142/s1664360720500095
Subject(s) - mathematics , lipschitz continuity , class (philosophy) , differentiable function , boundary (topology) , pure mathematics , argument (complex analysis) , combinatorics , mathematical analysis , philosophy , epistemology , biochemistry , chemistry
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.