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...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
World Scientific Publishing
2020-12-01
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| Series: | Bulletin of Mathematical Sciences |
| Subjects: | |
| Online Access: | http://www.worldscientific.com/doi/epdf/10.1142/S1664360720500095 |
| Summary: | 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. |
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| ISSN: | 1664-3607 1664-3615 |