Existence of solutions to quasilinear elliptic equations in the Musielak–Orlicz–Sobolev spaces for unbounded domains

The paper considers the existence of solutions of the Dirichlet problem for nonlinear elliptic equations of the second order in unbounded domains. Restrictions on the structure of quasilinear equations are formulated in terms of a special class of convex functions (generalized $N$-functions). Namely, nonlinearities are determined by the Musilak–Orlicz functions such that the complementaries functions obeys the condition $ \Delta_2 $. The corresponding Musielak–Orlicz–Sobolev space does not have to be reflexive. This fact is a significant problem, since the theorem for pseudomonotone operators is not applicable here.
For the class of equations under consideration, the proof of the existence theorem is based on an abstract theorem for additional systems. An important tool which allowed to generalize available results on the existence of solutions of the considered equations for bounded domains to the case of unbounded domains is an embedding theorem for Musielak–Orlicz–Sobolev spaces. Thus, in this paper, we find conditions on the structure of quasilinear equations in terms of the Musielak–Orlicz functions sufficient for the solvability of the Dirichlet problem in unbounded domains. In addition, we provide examples of equations which demonstrate that the class of nonlinearities considered in the paper is wider than non-power nonlinearities and variable exponent nonlinearities.