Oscillating and proper solutions of singular quasi-linear differential equations

Abstract

We study the behaviour of solutions of quasi-linear differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. We consider solutions entering a singular point either with definite tangential direction (proper solutions) or without definite tangential direction (oscillating solutions). Equations of this type appear in many problems arising in analysis, geometry, dynamical systems theory, and physics. First, we prove that a generic equation of the considered type has no oscillating solutions. Then we concentrate on proper solutions, which can enter a singular point in admissible tangential directions only. Great attention is paid to second-order differential equations, whose right-hand sides are cubic polynomials by the first-order derivative.
We obtain local representations for solutions of such equations in a form similar to Newton--Puiseux series -- series with fractional exponents (and, in a special case, with logarithmic terms).