Elliptic equations strongly degenerate at a point

In this paper we will mainly propose some problems for a class of degenerate elliptic equations, either linear or nonlinear. We will study some special cases of these problems and reveal some phenomena which may not have been noticed previously.Our problems originated from the self-similar solutions of the heat flow of harmonic maps. We will prove that the self-similar solutions or the so-called quasi-harmonic spheres are discontinuous at infinity for the equivariant case. In other words, the equivariant quasi-harmonic spheres are not continuous images of topological spheres.