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

L. M. Kozhevnikova ; A. A. Khadzhi

Sbornik. Mathematics, Tome 206 (2015) no. 8, pp. 1123-1149

Résumé

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.

Détail

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