Symmetry of non-negative solutions of a semilinear elliptic equation with singular nonlinearity

Abstract

We use the method of the moving plane (MMP) to obtain necessary and sufficient conditions for the radial symmetry of positive solutions of the following semi-linear elliptic equation with singular nonlinearity:

$$ \Delta u-\frac{1}{u^\nu}=0\quad\text{in~}\mathbb{R}^n,\quad n\geq2, $$

where $\nu>0$. In order to apply the MMP, it is crucial to obtain the asymptotic expansion of $u$ at $\infty$.