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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2015, Volume 19, Number 1, Pages 44–62 (Mi vsgtu1386)  

This article is cited in 5 scientific papers (total in 5 papers)

Differential Equations and Mathematical Physics

On solutions of elliptic equations with nonpower nonlinearities in unbounded domains

L. M. Kozhevnikova, A. A. Khadzhi

Sterlitamak branch of Bashkir State University, Sterlitamak, 453103, Russian Federation

Abstract: The paper highlighted some class of anisotropic elliptic equations of second order in divergence form with younger members with nonpower nonlinearities
$$\sum\limits_{\alpha=1}^{n}(a_{\alpha}({\boldsymbol x},u,\nabla u))_{x_{\alpha}}-a_0({\boldsymbol x},u,\nabla u)=0.$$
The condition of total monotony is imposed on the Caratheodory functions included in the equation. Restrictions on the growth of the functions are formulated in terms of a special class of convex functions. These requirements provide limited, coercive, monotone and semicontinuous corresponding elliptic operator. For the considered equations with nonpower nonlinearities the qualitative properties of solutions of the Dirichlet problem in unbounded domains $ \Omega \subset \mathbb {R} _n, \; n \geq 2$ are studied. The existence and uniqueness of generalized solutions in anisotropic Sobolev–Orlicz spaces are proved. Moreover, for arbitrary unbounded domains, the Embedding theorems for anisotropic Sobolev–Orlicz spaces are generalized. It makes possible to prove the global boundedness of solutions of the Dirichlet problem. The original geometric characteristic for unbounded domains along the selected axis is used. In terms of the characteristic the exponential estimate for the rate of decrease at infinity of solutions of the problem with finite data is set.

Keywords: anisotropic elliptic equations, Sobolev–Orlicz space, nonpower nonlinearity, the existence of solution, unbounded domains, boundedness of solutions, decay of solution

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00081-а
This work has been supported by the Russian Foundation for Basic Research (project no. 13–01–00081-a).


DOI: https://doi.org/10.14498/vsgtu1386

Full text: PDF file (853 kB) (published under the terms of the Creative Commons Attribution 4.0 International License)
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Bibliographic databases:

UDC: 517.956.25
MSC: 35J62, 35J25, 35J15
Original article submitted 15/XII/2014
revision submitted – 13/II/2015

Citation: L. M. Kozhevnikova, A. A. Khadzhi, “On solutions of elliptic equations with nonpower nonlinearities in unbounded domains”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:1 (2015), 44–62

Citation in format AMSBIB
\Bibitem{KozKha15}
\by L.~M.~Kozhevnikova, A.~A.~Khadzhi
\paper On solutions of elliptic equations with nonpower nonlinearities in unbounded domains
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2015
\vol 19
\issue 1
\pages 44--62
\mathnet{http://mi.mathnet.ru/vsgtu1386}
\crossref{https://doi.org/10.14498/vsgtu1386}
\zmath{https://zbmath.org/?q=an:06968947}
\elib{http://elibrary.ru/item.asp?id=23681741}


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    This publication is cited in the following articles:
    1. R. Kh. Karimov, L. M. Kozhevnikova, A. A. Khadzhi, “Behavior of solutions to elliptic equations with non-power nonlinearities in unbounded domains”, Ufa Math. J., 8:3 (2016), 95–108  mathnet  crossref  mathscinet  isi  elib
    2. L. M. Kozhevnikova, A. A. Nikitina, “O skorosti ubyvaniya na beskonechnosti resheniya anizotropnogo ellipticheskogo uravneniyav neogranichennykh oblastyakh”, Aktualnye voprosy universitetskoi nauki, Sbornik nauchnykh trudov, Ufa, 2016, 190–200 pp.  elib
    3. L. M. Kozhevnikova, A. Sh. Kamaletdinov, “Suschestvovanie reshenii anizotropnykh ellipticheskikh uravnenii s peremennymi pokazatelyami nelineinostei v neogranichennykh oblastyakh”, Vestnik VolGU. Seriya 1. Matematika. Fizika, 2016, no. 5 (36), 29–41  elib
    4. L. M. Kozhevnikova, A. Sh. Kamaletdinov, “Suschestvovanie reshenii anizotropnykh ellipticheskikh uravnenii s peremennymi pokazatelyami nelineinostei v neogranichennykh oblastyakh”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 5(36), 29–41  mathnet  crossref
    5. L. M. Kozhevnikova, “On the entropy solution to an elliptic problem in anisotropic Sobolev–Orlicz spaces”, Comput. Math. Math. Phys., 57:3 (2017), 434–452  mathnet  crossref  crossref  isi  elib
  • Вестник Самарского государственного технического университета. Серия: Физико-математические науки
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