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

We consider the following class of nonlinear elliptic equations -div(A(vertical bar x vertical bar)del u) + u(q) = 0 in B-1(0) \ {0}, 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 Phi is an element of L-q(B-1(0)), where Phi denotes the fundamental solution of -div(A(vertical bar x vertical bar)del u) = delta(0) in D'(B-1(0)) and delta(0) 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 Phi is an element of L-q(B-1(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(vertical bar x vertical bar) = vertical bar x vertical bar(v) with v is an element of (2 - N, 2).