Trichotomy of solutions of second-order elliptic equationswith a decreasing potential in the plane

A. V. Neklyudov

Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 1, pp. 37-50

Résumé

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 Schrödinger 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 $\int_Q q(x)\ln|x|\,dx\infty$.