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

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.