On degenerate solutions of second order elliptic equations in plane

Abstract

In the work we study conditions, under which a solution to a second order partial differential equation in the unit disk on the plane degenerates. We prove that each degenerate solution is either a polynomial of degree at most $2$ or a linear combination of a constant and the logarithm of a fractional–rational expression. In proof of the main result we use the Taylor series expansion of the degenerate solution of the equation at an arbitrary point and study the dependence of coefficients of resulting series on the coefficients at the lower powers of the same series.