This article is cited in 17 papers

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

L. M. Kozhevnikova^ab, A. A. Khadzhi<sup>c

a</sup> 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 $a_{\alpha}(x,s_0,s)$, $\alpha=0,1,\dots,n$, 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 $\Omega\subset \mathbb{R}_n$, $n\geqslant 2$, 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.

UDC: 517.956.25

MSC: 35J47, 35J60

Received: 26.01.2015 and 19.06.2015

DOI: 10.4213/sm8482

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

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