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Mat. Sb., 2015, Volume 206, Number 8, Pages 99–126 (Mi msb8482)  

This article is cited in 11 scientific papers (total in 11 papers)

Existence of solutions of anisotropic elliptic equations with nonpolynomial nonlinearities in unbounded domains

L. M. Kozhevnikovaab, A. A. Khadzhic

a Sterlitamak branch of Bashkir State University
b Elabuga Branch of Kazan State University
c Tyumen State University

Abstract: The paper is concerned with the solvability of the Dirichlet problem for a certain class of anisotropic elliptic second-order equations in divergence form with low-order terms and nonpolynomial nonlinearities
$$ \sum_{\alpha=1}^{n}(a_{\alpha}(x,u,\nabla u))_{x_{\alpha}}-a_0(x,u,\nabla u)=0, \qquad x \in \Omega. $$
The Carathéodory functions $a_{\alpha}(x,s_0,s)$, $\alpha=0,1,…,n$, are assumed to satisfy a joint monotonicity condition in the arguments $s_0\in\mathbb{R}$, $s\in\mathbb{R}_n$. Constraints on their growth in $s_0,s$ are formulated in terms of a special class of convex functions. The solvability of the Dirichlet problem in unbounded domains $\Omega\subset \mathbb{R}_n$, $n\geq 2$, is investigated. An existence theorem is proved without making any assumptions on the behaviour of the solutions and their growth as $|x|\to \infty$.
Bibliography: 26 titles.

Keywords: anisotropic elliptic equation, nonpolynomial nonlinearities, Orlicz-Sobolev space, existence of a solution, unbounded domain.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00081-a
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 13-01-00081-a).

Author to whom correspondence should be addressed


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English version:
Sbornik: Mathematics, 2015, 206:8, 1123–1149

Bibliographic databases:

UDC: 517.956.25
MSC: 35J47, 35J60
Received: 26.01.2015 and 19.06.2015

Citation: L. M. Kozhevnikova, A. A. Khadzhi, “Existence of solutions of anisotropic elliptic equations with nonpolynomial nonlinearities in unbounded domains”, Mat. Sb., 206:8 (2015), 99–126; Sb. Math., 206:8 (2015), 1123–1149

Citation in format AMSBIB
\by L.~M.~Kozhevnikova, A.~A.~Khadzhi
\paper Existence of solutions of anisotropic elliptic equations with nonpolynomial nonlinearities in unbounded domains
\jour Mat. Sb.
\yr 2015
\vol 206
\issue 8
\pages 99--126
\jour Sb. Math.
\yr 2015
\vol 206
\issue 8
\pages 1123--1149

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    This publication is cited in the following articles:
    1. Kozhevnikova L.M., Karimov R.Kh., Khadzhi A.A., “O povedenii reshenii ellipticheskikh uravnenii s nestepennymi nelineinostyami v neogranichennykh oblastyakh”, Aktualnye problemy gumanitarnykh i estestvennykh nauk, 2015, 13–17  elib
    2. L. M. Kozhevnikova, A. A. Nikitina, “Qualitative properties of solutions of elliptic equations with non-power nonlinearities in $\mathbb{R}_n$”, J. Math. Sci., 228:4 (2018), 395–408  mathnet  crossref
    3. 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
    4. F. Kh. Mukminov, “Uniqueness of the renormalized solutions to the Cauchy problem for an anisotropic parabolic equation”, Ufa Math. J., 8:2 (2016), 44–57  mathnet  crossref  isi  elib
    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
    6. A. Sh. Kamaletdinov, L. M. Kozhevnikova, L. Yu. Melnik, “Existence of solutions of anisotropic elliptic equations with variable exponents in unbounded domains”, Lobachevskii J. Math., 39:2, 3, SI (2018), 224–235  crossref  mathscinet  zmath  isi  scopus
    7. F. Kh. Mukminov, “Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents”, Sb. Math., 209:5 (2018), 714–738  mathnet  crossref  crossref  adsnasa  isi  elib
    8. A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    10. L. M. Kozhevnikova, A. Sh. Kamaletdinov, “Suschestvovanie reshenii anizotropnykh ellipticheskikh uravnenii s peremennymi pokazatelyami nelineinostei v $\mathbb{R}^n$”, Materialy mezhdunarodnoi konferentsii«InternationalConference onMathematicalModellinginAppliedSciences, ICMMAS-17», Sankt-Peterburgskii politekhnicheskii universitet,2428 iyulya2017 g., Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 160, VINITI RAN, M., 2019, 49–60  mathnet
    11. A. K. Gushchin, “On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65  mathnet  crossref  crossref  mathscinet
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