On solutions of elliptic equations with nonpower nonlinearities in unbounded domains

Abstract

The paper highlighted some class of anisotropic elliptic equations of second order in divergence form with younger members with nonpower nonlinearities $$ \sum\limits_{\alpha=1}^{n}(a_{\alpha}({\bf x},u,\nabla u))_{x_{\alpha}}-a_0({\bf x},u,\nabla u)=0. $$ The condition of total monotony is imposed on the Caratheodory functions included in the equation. Restrictions on the growth of the functions are formulated in terms of a special class of convex functions. These requirements provide limited, coercive, monotone and semicontinuous corresponding elliptic operator. For the considered equations with nonpower nonlinearities the qualitative properties of solutions of the Dirichlet problem in unbounded domains $ \Omega \subset \mathbb {R} _n, \; n \geq 2$ are studied. The existence and uniqueness of generalized solutions in anisotropic Sobolev-Orlicz spaces are proved. Moreover, for arbitrary unbounded domains, the Embedding theorems for anisotropic Sobolev-Orlicz spaces are generalized. It makes possible to prove the global boundedness of solutions of the Dirichlet problem. The original geometric characteristic for unbounded domains along the selected axis is used. In terms of the characteristic the exponential estimate for the rate of decrease at infinity of solutions of the problem with finite data is set.

Keywords

• anisotropic elliptic equations
• sobolev-orlicz space
• nonpower nonlinearity
• the existence of solution
• unbounded domains
• boundedness of solutions
• decay of solution