Non-removable singularities of a fourth-order nonlinear partial differential equation

The aim of this paper is to give a classification of the solutions ϕ=ϕ(x,y) of the fourth-order nonlinear equation ϕyyρxx−2ϕxyρxy+ϕxxρyy=0, ρ=(det(∇2ϕ))−3/4, in a punctured domain which are C2 at the singularity. We prove that this kind of solutions either have a removable singularity or are asymptotic to rotationally symmetric solutions near the singularity. A geometric description of solutions which are not C1 at the singularity is also given.