Trichotomy of Solutions of Second-Order Elliptic Equations with a Decreasing Potential in the Plane

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2019.1.27733 Trichotomy of Solutions of Second-Order Elliptic Equations with a Decreasing Potential in the Plane Neklyudov, A. V. Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 1. Abstract: We consider a uniformly elliptic second-order divergent equation with measurable coefficients in two-dimensional domain \(Q\) external to the circle. An equation contains the lower nonnegative coefficient \(q(x)=q(x_1, x_2)\) of potential type in the stationary Schrodinger equation. Weak solutions in the Sobolev space \(W_2^1\) in any bounded subdomain are studied. The possible rate of solutions at infinity is considered. It is established that if the lower coefficient decreases with a sufficient rate then the positive solution exists and has the same rate at infinity as the fundamental solution of respective elliptic equation without lower term. The rate is logarithmic. This solution has uniformly bounded "heat flow'' on circles of radius \(R\). It is established Sen-Venan type inequality for Dirichlet integral of solution of power rate. Sen-Venan inequality leads to the evaluation of Dirichlet integral in a ring domain via average value of solution on the circle. It means that the solution has the same rate on the circle as its average value. Maximum principle implies that any tending to infinity solution has the logarithmic rate. The main result of paper is the trichotomy of solutions: The solution is either bounded, or tends to infinity with a logarithmic rate, preserving the sign, or oscillates and has a~power-law rate of the maximum of the modulus. The basic condition for the decrease of the lower coefficient is formulated in integral form \(\int_Q q(x)\ln\|x\|\,dx<\infty\). Keywords: elliptic equation, unbounded domain, lower coefficient, asymptotic behaviour of solutions, trichotomy of solutions. Language: Russian Download the full text For citation: Neklyudov, A. V. Trichotomy of Solutions of Second-Order Elliptic Equations with a Decreasing Potential in the Plane, Vladikavkaz Math. J., 2013, vol. 21, no. 4, pp. 37-50 (in Russian). DOI 10.23671/VNC.2019.1.27733 + References 1. Landis, Е. М., Panasenko, G. P. A Variant of a Theorem of Phragmen-Lindelof Type for Elliptic Equations with Coefficients That Are Periodic in All Variables But One, Trudy Seminara imeni I.G. Petrovskogo [Proceedings of the Petrovskiy Seminar], 1979, vol. 5, pp. 105-136 (in Russian). 2. Oleinik, O. A., Iosif'yan, G. A. On the Behavior at Infinity of Solutions of Second Order Elliptic Equations in Domains with Noncompact Boundary, Mathematics of the USSR-Sbornik, 1981, vol. 40, no. 4, pp. 527-548. DOI: 10.1070/SM1981v040n04ABEH001849. 3. Landis, E. M. Ibragimov, A. I. Neumann Problems in Unbounded Domains, Doklady Akademii Nauk [Reports of Akademy of Science], 1995, vol. 343, no. 1, pp. 17-18 (in Russian). 4. Kondrat'ev, V. A. and Oleinik, O. A. Asymptotics in a Neighborhood of Infinity of Solutions with Finite Dirichlet Integral of Second-Order Elliptic Equations, Journal of Soviet Mathematics, 1989, vol. 47, no. 4, pp. 2596-2607. DOI: 10.1007/BF01105913. 5. Nekludov, A. V. On Solutions of Second Order Elliptic Equations in Cylindrical Domains, Ufa Mathematical Journal, 2016, vol. 8, no. 4, pp. 131-143. DOI: 10.13108/2016-8-4-131. 6. Nekludov, A. V. On the Robin Problem for Second-Order Elliptic Equations in Cylindrical Domains, Mathematical Notes, 2018, vol. 103, no. 3-4, pp. 430-446. DOI: 10.1134/S0001434618030094. 7. Nekludov, A. V. Asymptotic of Solutions of Two-Dimesional Gauss-Bierbach-Rademacher Equation with Variable Coefficients in External Area, Sibirskie Electronnie Matematicheskie Izvestiya [Syberian Electronic Mathematical Reports], 2018, vol. 15, pp. 338-354 (in Russian). 8. Littman, W., Stampacchia, G. and Weinberger, H. F. Regular Points for Elliptic Equations with Discontinuous Coefficients, Ann. Scuola Norm. Sup. Pisa. Ser. 3, 1963, vol. 17, no. 1-2, pp. 43-77. 9. Gilbarg, D. and Trudinger, N. Elliptic Partial Differential Equations of Second Order, Berlin, N.Y., Springer Verlag, 1977, 401 p. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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