Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

Abstract

We consider the following class of nonlinear elliptic equations, where q > 1 and A is a positive C 1(0,1] function which is regularly varying at zero with index v in (2-N,2). We prove that all isolated singularities at zero for the positive solutions are removable if and only if Φ ∉ Lq(B_1(0)), where Φ denotes the fundamental solution of -div (A({pipe}x{pipe})∇ u)=δ0 in D'(B_1(0)) and δ0 is the Dirac mass at 0. Moreover, we give a complete classification of the behaviour near zero of all positive solutions in the more delicate case that Φ ∈ Lq(B1(0)). We also establish the existence of positive solutions in all the categories of such a classification. Our results apply in particular to the model case A({pipe}x{pipe})={pipe}x{pipe}θ with θ ∈ (2-N,2). © 2012 Springer-Verlag Berlin Heidelberg.