The Toda system and multiple-end solutions of autonomous planar elliptic problems

Abstract

We construct a new class of positive solutions for the classical elliptic problem ¢u ¡ u + up = 0; p > 2; in R2: We establish a deep relation between them and the following Toda system c2f00 j = efj¡1¡fj ¡ efj¡fj+1 in R; j = 1; : : : ; k: We show that these solutions have the approximate form u(x; z) » Pk j=1 w(x ¡ fj(z)) where w is the unique even, positive, asymptotically vanishing solution of w00 ¡ w + wp = 0 in R. Functions fj(z), representing the multiple ends of u(x; z), solve the aforementioned Toda system, they are even, asymptotically linear, with f0 ´ ¡1 < f1 ¿ ¢ ¢ ¢ ¿ fk < fk+1 ´ +1: The solutions of the elliptic problem we construct have their counterpart in the theory of constant mean curvature surfaces. An analogy can also be made between their construction and the gluing of constant scalar curvature Fowler singular metrics in the sphere.