Abstract:
The paper highlighted some class of anisotropic elliptic equations of second order in divergence form with younger members with nonpower nonlinearities
$$\sum\limits_{\alpha=1}^{n}(a_{\alpha}({\boldsymbol x},u,\nabla u))_{x_{\alpha}}-a_0({\boldsymbol x},u,\nabla u)=0.$$
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.
Citation:
L. M. Kozhevnikova, A. A. Khadzhi, “On solutions of elliptic equations with nonpower nonlinearities in unbounded domains”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:1 (2015), 44–62
\Bibitem{KozKha15}
\by L.~M.~Kozhevnikova, A.~A.~Khadzhi
\paper On solutions of elliptic equations with nonpower nonlinearities in unbounded domains
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2015
\vol 19
\issue 1
\pages 44--62
\mathnet{http://mi.mathnet.ru/vsgtu1386}
\crossref{https://doi.org/10.14498/vsgtu1386}
\zmath{https://zbmath.org/?q=an:06968947}
\elib{https://elibrary.ru/item.asp?id=23681741}
Linking options:
https://www.mathnet.ru/eng/vsgtu1386
https://www.mathnet.ru/eng/vsgtu/v219/i1/p44
This publication is cited in the following 6 articles:
L. M. Kozhevnikova, “On the entropy solution to an elliptic problem in anisotropic Sobolev–Orlicz spaces”, Comput. Math. Math. Phys., 57:3 (2017), 434–452
L. M. Kozhevnikova, “Existence of entropic solutions of elliptic problem in anisotropic Sobolev–Orlicz spaces”, J. Math. Sci. (N. Y.), 241:3 (2019), 258–284
R. Kh. Karimov, L. M. Kozhevnikova, A. A. Khadzhi, “Behavior of solutions to elliptic equations with non-power nonlinearities in unbounded domains”, Ufa Math. J., 8:3 (2016), 95–108
L. M. Kozhevnikova, A. A. Nikitina, “O skorosti ubyvaniya na beskonechnosti resheniya anizotropnogo ellipticheskogo uravneniyav neogranichennykh oblastyakh”, Aktualnye voprosy universitetskoi nauki, Sbornik nauchnykh trudov, Ufa, 2016, 190–200 pp.
L. M. Kozhevnikova, A. Sh. Kamaletdinov, “Suschestvovanie reshenii anizotropnykh ellipticheskikh uravnenii s peremennymi pokazatelyami nelineinostei v neogranichennykh oblastyakh”, Vestnik VolGU. Seriya 1. Matematika. Fizika, 2016, no. 5 (36), 29–41
L. M. Kozhevnikova, A. Sh. Kamaletdinov, “Suschestvovanie reshenii anizotropnykh ellipticheskikh uravnenii s peremennymi pokazatelyami nelineinostei v neogranichennykh oblastyakh”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2016, no. 5(36), 29–41