A generalized version of the Lions-type lemma

Magdalena Chmara

Published: 2023-08-28

Vol. 37 No. 2 (2023)

A generalized version of the Lions-type lemma

Magdalena Chmara

Abstract

In this short paper, I recall the history of dealing with the lack of compactness of a sequence in the case of an unbounded domain and prove the vanishing Lions-type result for a sequence of Lebesgue-measurable functions. This lemma generalizes some results for a class of Orlicz-Sobolev spaces. What matters here is the behavior of the integral, not the space.

Keywords:

Lions-type result , concentration-compactness , unbounded domains

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Chmara, M. (2023). A generalized version of the Lions-type lemma. Annales Mathematicae Silesianae, 37(2), 240–247. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/15880

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References

R.A. Adams and J.J.F. Fournier, Sobolev Spaces, second edition, Pure Appl. Math. (Amst.), 140, Elsevier/Academic Press, Amsterdam, 2003.
Google Scholar

C.O. Alves and M.L.M. Carvalho, A Lions type result for a large class of Orlicz-Sobolev space and applications, Mosc. Math. J. 22 (2022), no. 3, 401–426.
Google Scholar

C.O. Alves, G.M. Figueiredo, and J.A. Santos, Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications, Topol. Methods Nonlinear Anal. 44 (2014), no. 2, 435–456.
Google Scholar

S. Bahrouni, H. Ounaies, and O. Elfalah, Problems involving the fractional g-Laplacian with lack of compactness, J. Math. Phys. 64 (2023), no. 1, Paper No. 011512, 18 pp.
Google Scholar

G. Barletta and A. Cianchi, Dirichlet problems for fully anisotropic elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 147 (2017), no.1, 25–60.
Google Scholar

Ph. Clément, M. García-Huidobro, R. Manásevich, and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 11 (2000), no. 1, 33–62.
Google Scholar

D.G. Costa, An Invitation to Variational Methods in Differential Equations, Birkhäuser Boston, Inc., Boston, MA, 2007.
Google Scholar

M. Lewin, Describing lack of compactness in Sobolev spaces, lecture notes on Variational Methods in Quantum Mechanics, University of Cergy-Pontoise, 2010. Avaliable at HAL: hal-02450559.
Google Scholar

E.H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), no. 3, 441–448.
Google Scholar

P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145.
Google Scholar

E.D. Silva, M.L. Carvalho, J.C. de Albuquerque, and S. Bahrouni, Compact embedding theorems and a Lions’ type lemma for fractional Orlicz-Sobolev spaces, J. Differential Equations 300 (2021), 487–512.
Google Scholar

M. Struwe, Variational Methods, fourth edition, Ergeb. Math. Grenzgeb. (3), 34 [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Springer–Verlag, Berlin, 2008.
Google Scholar

K. Wroński, Quasilinear elliptic problem in anisotrpic Orlicz-Sobolev space on unbounded domain, arXiv preprint, 2022. Avaliable at arXiv: 2209.10999.
Google Scholar

Vol. 37 No. 2 (2023)
Published: 2023-09-21

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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