Singularity and regularity of solutions to a certain class of degenerate elliptic equations

In this note we consider the solution of the degenerate elliptic system equation(1){div(F(|x|)∇u)=0,x∈B1/{0}u|∂B1=ϕ∈C(∂B1) where B1B1 denotes the unit ball in RnRn and FF is smooth and increasing on [0,1][0,1] with F(0)=0,F(r)>0(r>0)F(0)=0,F(r)>0(r>0). It is known that there exists a unique solution u∈C∞(B1/{0})∩L∞(B1)u∈C∞(B1/{0})∩L∞(B1) to this elliptic system.Here we will study the property of uu at the origin. At first we give the necessary and sufficient condition such that uu can be extended to a continuous function in BB. Furthermore, we also give the necessary and sufficient condition such that uu is a Hölder continuous function in B1B1.