Existence of entropy solution to the Neumann problem for elliptic equation with measure-valued potential
- Authors: Vildanova V.F.1, Mukminov F.K.1
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Affiliations:
- Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa
- Issue: Vol 89, No 3 (2025)
- Pages: 45-79
- Section: Articles
- URL: https://journal-vniispk.ru/1607-0046/article/view/303959
- DOI: https://doi.org/10.4213/im9565
- ID: 303959
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Abstract
In a bounded or unbounded domain in $\mathbb{R}^n$,the Neumann problem for a non-linear second order elliptic equation with measure-valued potential is considered.The assumptions on the structure of the equation are stated in terms of a generalized $N$-function.The existence of an entropy solution to the problem is proved.
About the authors
Venera Fidarisovna Vildanova
Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa
Author for correspondence.
Email: gilvenera@mail.ru
Candidate of physico-mathematical sciences, Associate professor
Farit Khamzaevich Mukminov
Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa
Email: mfkh@rambler.ru
Doctor of physico-mathematical sciences, Professor
References
- Ph. Benilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre, J. L. Vazquez, “An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22:2 (1995), 241–273
- N. Saintier, L. Veron, “Nonlinear elliptic equations with measure valued absorption potential”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22:1 (2021), 351–397
- V. F. Vil'danova, F. Kh. Mukminov, “Perturbations of nonlinear elliptic operators by potentials in the space of multiplicators”, J. Math. Sci. (N.Y.), 257:5 (2021), 569–578
- S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holder, Solvable models in quantum mechanics, With an appendix by P. Exner, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005, xiv+488 pp.
- М. И. Нейман-заде, А. А. Шкаликов, “Операторы Шрeдингера с сингулярными потенциалами из пространств мультипликаторов”, Матем. заметки, 66:5 (1999), 723–733
- A. Malusa, M. M. Porzio, “Renormalized solutions to elliptic equations with measure data in unbounded domains”, Nonlinear Anal., 67:8 (2007), 2370–2389
- N. Aguirre, “$p$-harmonic functions in $mathbb{R}^N_+$ with nonlinear Neumann boundary conditions and measure data”, Adv. Nonlinear Stud., 19:4 (2019), 797–825
- Л. М. Кожевникова, А. П. Кашникова, “Эквивалентность энтропийных и ренормализованных решений нелинейной эллиптической задачи в пространствах Музилака–Орлича”, Дифференц. уравнения, 59:1 (2023), 35–50
- Л. М. Кожевникова, “Существование энтропийного решения нелинейной эллиптической задачи в неограниченной области”, ТМФ, 218:1 (2024), 124–148
- S. Ouaro, N. Sawadogo, “Nonlinear elliptic $p(u)$-Laplacian problem with Fourier boundary condition”, Cubo, 22:1 (2020), 85–124
- A. Kristaly, M. Mihăilescu, V. Rădulescu, “Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz–Sobolev space setting”, Proc. Roy. Soc. Edinburgh Sect. A, 139:2 (2009), 367–379
- G. Bonanno, G. Molica Bisci, V. Rădulescu, “Existence of three solutions for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces”, Nonlinear Anal., 74:14 (2011), 4785–4795
- B. K. Bonzi, S. Ouaro, F. D. Y. Zongo, “Entropy solutions for nonlinear elliptic anisotropic homogeneous Neumann problem”, Int. J. Differ. Equ., 2013 (2013), 476781, 14 pp.
- M. B. Benboubker, S. Ouaro, U. Traore, “Entropy solutions for nonlinear nonhomogeneous Neumann problems involving the generalized $p(x)$-Laplace operator and measure data”, J. Nonlinear Evol. Equ. Appl., 2014:5 (2014), 53–76
- A. Siai, “Nonlinear Neumann problems on bounded Lipschitz domains”, Electron. J. Differential Equations, 2005 (2005), 09, 16 pp.
- A. Siai, “A fully nonlinear nonhomogeneous Neumann problem”, Potential Anal., 24:1 (2006), 15–45
- P. Gassiat, B. Seeger, The Neumann problem for fully nonlinear SPDE
- J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Math., 1034, Springer-Verlag, Berlin, 1983, iii+222 pp.
- Y. Ahmida, I. Chlebicka, P. Gwiazda, A. Youssfi, “Gossez's approximation theorems in Musielak–Orlicz–Sobolev spaces”, J. Funct. Anal., 275:9 (2018), 2538–2571
- P. Gwiazda, P. Wittbold, A. Wroblewska, A. Zimmermann, “Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces”, J. Differential Equations, 253:2 (2012), 635–666
- А. Н. Колмогоров, С. В. Фомин, Элементы теории функций и функционального анализа, 4-е перераб. изд., Наука, М., 1976, 543 с.
- J.-P. Gossez, “Some approximation properties in Orlicz–Sobolev spaces”, Studia Math., 74:1 (1982), 17–24
- M. B. Benboubker, E. Azroul, A. Barbara, “Quasilinear elliptic problems with nonstandard growth”, Electron. J. Differential Equations, 2011 (2011), 62, 16 pp.
- Н. Данфорд, Дж. Т. Шварц, Линейные операторы. Общая теория, ИЛ, М., 1962, 895 с.
- Г. И. Лаптев, “Слабые решения квазилинейных параболических уравнений второго порядка с двойной нелинейностью”, Матем. сб., 188:9 (1997), 83–112
- Ж.-Л. Лионс, Некоторые методы решения нелинейных краевых задач, Мир, М., 1972, 587 с.
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