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DOI: 10.23671/VNC.2019.1.27733 Trichotomy of Solutions of SecondOrder 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 secondorder divergent equation with measurable coefficients in twodimensional 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 SenVenan type inequality for Dirichlet integral of solution of power rate. SenVenan 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~powerlaw rate of the maximum of the modulus. The basic condition for the decrease of the lower coefficient is formulated in integral form \(\int_Q q(x)\lnx\,dx<\infty\).
Keywords: elliptic equation, unbounded domain, lower coefficient, asymptotic behaviour of solutions, trichotomy of solutions.
Language: Russian
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For citation: Neklyudov, A. V. Trichotomy of Solutions of SecondOrder Elliptic Equations with a Decreasing Potential in the Plane, Vladikavkaz Math. J., 2013, vol. 21, no. 4, pp. 3750 (in Russian). DOI 10.23671/VNC.2019.1.27733 ← Contents of issue 
 

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