On a singular elliptic equation

Abstract:

In this paper, we study the singular elliptic equation $Lu + K(x){u^p} = 0$, where $L$ is a uniformly elliptic operator of divergence form, $p > 1$ and $K(x)$ has a singularity at the origin. We prove that this equation does not possess any positive (local) solution in any punctured neighborhood of the origin if there exist two constants ${C_1}$, ${C_2}$ such that ${C_1}|x{|^\sigma } \geqslant K(x) \geqslant {C_2}|x{|^\sigma }$ near the origin for some $\sigma \leqslant - 2$ (with no other condition on the gradient of $K$ ). In fact, an integral condition is derived.