W1,p versus C1: The nonsmooth case involving critical growth
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 elemen...
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doaj-1a681b767ec444f09873780017c45dc42020-12-10T07:34:19ZengWorld Scientific PublishingBulletin of Mathematical Sciences1664-36071664-36152020-12-011032050009-12050009-1510.1142/S166436072050009510.1142/S1664360720500095W1,p versus C1: The nonsmooth case involving critical growthYunru Bai0Leszek Gasiński1Patrick Winkert2Shengda Zeng3Guangxi Colleges and Universities Key Laboratory of Complex System, Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, ChinaPedagogical University of Cracow, Department of Mathematics, Podchorazych 2, 30-084 Cracow, PolandTechnische Universität Berlin, Institut für Mathematik, Strasse des 17.Juni 136, 10623 Berlin, GermanyGuangxi Colleges and Universities Key Laboratory of Complex System, Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, ChinaIn 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.http://www.worldscientific.com/doi/epdf/10.1142/S1664360720500095nonhomogeneous partial differential operatorlocal minimizerclarke’s generalized gradientcritical growthneumann problem |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Yunru Bai Leszek Gasiński Patrick Winkert Shengda Zeng |
spellingShingle |
Yunru Bai Leszek Gasiński Patrick Winkert Shengda Zeng W1,p versus C1: The nonsmooth case involving critical growth Bulletin of Mathematical Sciences nonhomogeneous partial differential operator local minimizer clarke’s generalized gradient critical growth neumann problem |
author_facet |
Yunru Bai Leszek Gasiński Patrick Winkert Shengda Zeng |
author_sort |
Yunru Bai |
title |
W1,p versus C1: The nonsmooth case involving critical growth |
title_short |
W1,p versus C1: The nonsmooth case involving critical growth |
title_full |
W1,p versus C1: The nonsmooth case involving critical growth |
title_fullStr |
W1,p versus C1: The nonsmooth case involving critical growth |
title_full_unstemmed |
W1,p versus C1: The nonsmooth case involving critical growth |
title_sort |
w1,p versus c1: the nonsmooth case involving critical growth |
publisher |
World Scientific Publishing |
series |
Bulletin of Mathematical Sciences |
issn |
1664-3607 1664-3615 |
publishDate |
2020-12-01 |
description |
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. |
topic |
nonhomogeneous partial differential operator local minimizer clarke’s generalized gradient critical growth neumann problem |
url |
http://www.worldscientific.com/doi/epdf/10.1142/S1664360720500095 |
work_keys_str_mv |
AT yunrubai w1pversusc1thenonsmoothcaseinvolvingcriticalgrowth AT leszekgasinski w1pversusc1thenonsmoothcaseinvolvingcriticalgrowth AT patrickwinkert w1pversusc1thenonsmoothcaseinvolvingcriticalgrowth AT shengdazeng w1pversusc1thenonsmoothcaseinvolvingcriticalgrowth |
_version_ |
1724387608673386496 |