Ein Malmquistscher Satz für algebraische Differentialgleichungen zweiter Ordnung. (A theorem of Malmquist for algebraic differential equations of second order)
The following assertion is known as a theorem of Malmquist: if \(w'=R(z,w)\) \(('=d/dt\), R is rational) has a transcendental meromorphic solution in \({\mathbb{C}}\), then the equation must be a Riccati equation. This paper studies the equation (i) \(w''=P(z,w)w'+Q(z,w)\) where P and Q are rational functions of z and w. First, the following two theorems are obtained. Theorem 1. If (i) has a one valued transcendental meromorphic solution in \({\mathbb{C}}\) which is not a solution of a Riccati equation, then (i) must be of the form \[ (ii)\quad w''=p_ 0w'+p_ 1ww'+q_ 0+q_ 1w+q_ 2w^ 2+q_ 3w^ 3+q(w'-p)/(w-s) \] where \(p_ i,q_ i,,p,q,s\) are rational functions of z. Theorem 2. Assume in (i) that \(s=0\), \(q\neq 0\), then every transcendental meromorphic solution is of finite order. Moreover, if (i) has such a solution which is not a solution of a Riccati equation, of the type (**) \(\nabla \cdot (A(| \nabla u|)\nabla u+f(u)=0,\) \(x\in {\mathbb{R}}^ n\); \(\lim_{x\to \infty}u(x)=0\); where A is a positive continuous function such that the product pA(p) is monotone. The class of equations (**) includes the mean curvature equation.
scientific article; zbMATH DE number 4063526