6533b852fe1ef96bd12aaf67

RESEARCH PRODUCT

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

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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 C 1(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 $${\vartheta\in (2-N,2)}$$ .