Singular solutions of a nonlinear elliptic equation and an infinite dimensional dynamical system

Abstract

Some classes of partial differential equations of elliptic type can be converted into autonomous time-evolution equations by choosing one of the space variables as t ime variable. The resulting evolution equations are, of course, still of elliptic type, hence are ill-posed. Nonetheless, there are cases in which one can deal with the problem in the framework of dynamical systems or their analogues. Note that the dynamical systems relevant to those equations are of infinite dimensions, since none of those partial differential equations can be converted into a finite system of ordinary differential equations. In this paper we deal with singular solutions of some nonlinear elliptic equation and s tudy their behavior near the singular point and near the infinity. The main techniques used here come from the theory of infinite dimensional dynamical systems. The advantage of using the dynamical systems point of view is that one can get a geometrical insight into the problem. Thanks to this approach, the results we present in this paper are somewhat of global nature and are perhaps difficult to obtain by other conventional methods of analysis. We consider the following semilinear elliptic equation: