We obtained best possible conditions for removable singularity at the point for solutions of nonlinear elliptic second order equations of divergent form. Well known Serrin's results guarantee the removability of the singularity at the point x0 if and for m< nor if m=nwith some positive number δ. We established analogous result with δ=0. The proof is based on new approach connected with point-wise estimates of singular solutions.

We consider a wide class of degenerate quasilinear second order elliptic equations with model representative , whose solutions have singularity at a point. There are established sharp point-wise conditions for removable isolated singularity of solutions of such equations. For solutions with non-removable point singularity (source type or fundamental solution), precise upper and lower estimates near the singularity point are obtained.