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

Abstract

We consider the following class of nonlinear elliptic equations $$\begin{array}{ll}{-}{\rm div}(\mathcal{A}(|x|)\nabla u) +u^q=0\quad {\rm in}\; B_1(0)\setminus\{0\}, \end{array}$$ where q > 1 and \({\mathcal{A}}\) is a positive C1(0,1] function which is regularly varying at zero with index \({\vartheta}\) in (2−N,2). We prove that all isolated singularities at zero for the positive solutions are removable if and only if \({\Phi\not\in L^q(B_1(0))}\) , where \({\Phi}\) denotes the fundamental solution of \({-{\rm div}(\mathcal{A}(|x|)\nabla u)=\delta_0}\) in \({\mathcal D'(B_1(0))}\) 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 \({\Phi\in L^q(B_1(0))}\) . 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 \({\mathcal{A}(|x|)=|x|^\vartheta}\) with \({\vartheta\in (2-N,2)}\) .