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On solutions of elliptic equations with nonpower nonlinearities in unbounded domains. (Russian. English summary) Zbl 1413.35213
Summary: The paper highlighted some class of anisotropic elliptic equations of second order in divergence form with younger members with nonpower nonlinearities \[ \sum\limits_{\alpha=1}^{n}(a_{\alpha}({\mathbf x},u,\nabla u))_{x_{\alpha}}-a_0({\mathbf x},u,\nabla u)=0. \] 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.

MSC:
35J62 Quasilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J15 Second-order elliptic equations
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[1] [1] Kozhevnikova L. M., Khadzhi A. A., “On solutions of elliptic equations with nonpower nonlinearities in unbounded domains”, The 4nd International Conference “Mathematical Physics and its Applications”, Book of Abstracts and Conference Materials, eds. I. V. Volovich; V. P. Radchenko, Samara State Technical Univ., Samara, 2014, 199-200 (In Russian)
[2] [2] “Solvability of the first boundary value problem for quasilinear equations with rapidly increasing coefficients in Orlicz classes”, Sov. Math., Dokl., 4 (1963), 1060–1064 · Zbl 0158.12403
[3] [3] Dubinskii Yu. A., “Weak convergence for nonlinear elliptic and parabolic equations”, Mat. Sb. (N.S.), 67(109):4 (1965), 609–642 (In Russian) · Zbl 0145.35202
[4] [4] Donaldson T., “Nonlinear elliptic boundary value problems in Orlicz–Sobolev spaces”, J. Diff. Eq., 10:3 (1971), 507–528 · Zbl 0207.41501
[5] [5] Klimov V. S., “Boundary value problems in Orlicz–Sobolev spaces”, Kachestvennye i priblizhennye metody issledovaniia operatornykh uravnenii [Qualitative methods for investigating operator equation], Yaroslavl State Univ., Yaroslavl, 1976, 75–93 (In Russian)
[6] [6] De Giorgi E., “On the differentiability and the analiticity of extremals of regular multiple integrals”, Selected papers, eds. Luigi Ambrosio, Gianni Dal Maso, Marco Forti, Mario Miranda, and Sergio Spagnolo, Springer-Verlag, Berlin, New York, 2006, 149–166 · Zbl 1096.01015
[7] [7] Moser J., “A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations”, Comm. Pure Appl. Math., 13:3 (1960), 457–468 · Zbl 0111.09301
[8] [8] Kruzhkov S. N., “A priori bounds and some properties of solutions of elliptic and parabolic equations”, Mat. Sb. (N.S.), 65(107):4 (1964), 522–570 (In Russian) · Zbl 0148.35703
[9] [9] Serrin J., “Local behavior of solutions of quasi-linear equations”, Acta Math., 111:1 (1964), 247–302 · Zbl 0128.09101
[10] [10] Landis E. M., “A new proof of E. De Giorgi’s theorem”, Tr. Mosk. Mat. Obs., 16, MSU, Moscow, 1967, 319–328 (In Russian) · Zbl 0182.43403
[11] [11] Kolodij I. M., “The boundedness of generalized solutions of elliptic differential equations”, Mosc. Univ. Math. Bull., 25:5-6 (1970), 31–37 · Zbl 0245.35029
[12] [12] Kozhevnikova L. M., Khadzhi A. A., “Boundedness of solutions to anisotropic second order elliptic equations in unbounded domains”, Ufa Math. Journal, 6:2 (2014), 66–76 · Zbl 1374.35171
[13] [13] Korolev A. G., “On boundedness of generalized solutions of elliptic differential equations with nonpower nonlinearities”, Math. USSR-Sb., 66:1 (1990), 83–106 · Zbl 0706.35030
[14] [14] Ladyzhenskaya O. A., Ural’tseva N. N., Linear and quasilinear elliptic equations, Mathematics in Science and Engineering, 46, Academic Press, New York, London, 1968, xviii+495 pp. · Zbl 0164.13002
[15] [15] Klimov V. S., “Imbedding theorems for Orlicz spaces and their applications to boundary value problems”, Sib. Math. J., 13 (1972), 231-240 · Zbl 0246.46022
[16] [16] Korolev A. G., “Boundedness of the generalized solutions of elliptic equations with nonpower nonlinearities”, Math. Notes, 42:2 (1987), 639–645 · Zbl 0681.35022
[17] [17] Oleinik O. A., Iosif’yan G. A., “On the behavior at infinity of solutions of second order elliptic equations in domains with noncompact boundary”, Math. USSR-Sb., 40:4 (1981), 527–548 · Zbl 0469.35045
[18] [18] Kondrat’ev V. A., Kopáček J., Lekveishvili D. M., Oleĭnik O. A., “Sharp estimates in Hölder spaces and precise Saint-Venant principle for solutions of the biharmonic equation”, Proc. Steklov Inst. Math., 166 (1986), 97–116 · Zbl 0584.35038
[19] [19] Kozhevnikova L. M., “Behaviour at infinity of solutions of pseudodifferential elliptic equations in unbounded domains”, Sb. Math., 199:8 (2008), 1169–1200 · Zbl 1261.35157
[20] [20] Gilimshina V. F., “On the decay of solutions of non-uniformly elliptic equations”, Izv. Math., 75:1 (2011), 53–71 · Zbl 1210.35131
[21] [21] Karimov R. Kh., Kozhevnikova L. M., “Behavior on infinity of decision quasilinear elliptical equations in unbounded domain”, Ufimsk. Mat. Zh., 2:2 (2010), 53–66 (In Russian) · Zbl 1240.35043
[22] [22] Kozhevnikova L. M., Khadzhi A. A., “Solutions of anisotropic elliptic equations in unbounded domains”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. & Math. Sci.], 1(30) (2013), 90–96 (In Russian)
[23] [23] Krasnosel’skij M. A.; Rutitskij Ya. B., Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961, ix+249 pp. · Zbl 0095.09103
[24] [24] “Korolev A. G.”, Mosc. Univ. Math. Bull., 38:1 (1983), 37–42 · Zbl 0518.46023
[25] [25] Lions J. L., Some methods in the mathematical analysis of systems and their control, Science Press, Beijing, China; Gordon and Breach, Science Publishers, Inc, New York, 1981, xxiii+542 pp. · Zbl 0542.93034
[26] [26] Andriyanova E. R., “Estimates of decay rate for solution to parabolic equation with non-power nonlinearities”, Ufa Math. Journal, 6:2 (2014), 3–24 · Zbl 1374.35111
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