A Complete Classification of the Isolated Singularities for Nonlinear Elliptic Equations with Inverse Square Potentials

In this paper, the author considers semilinear elliptic equations of the form $-\Delta u- \frac{\lambda}{|x|^2}u +b(x)\,h(u)=0$ in $\Omega\setminus\{0\}$, where $\lambda$ is a parameter with $-\infty<\lambda\leq (N-2)^2/4$ and $\Omega$ is an open subset in $\mathbb{R}^N$ with $N\geq 3$ such that $0\in \Omega$. Here, $b(x)$ is a positive continuous function on $\overline \Omega\setminus\{0\}$ which behaves near the origin as a regularly varying function at zero with index $\theta$ greater than $-2$. The nonlinearity $h$ is assumed continuous on $\mathbb{R}$ and positive on $(0,\infty)$ with $h(0)=0$ such that $h(t)/t$ is bounded for small $t>0$. The author completely classifies the behaviour near zero of all positive solutions of equation (0.1) when $h$ is regularly varying at $\infty$ with index $q$ greater than $1$ (that is, $\lim_{t\to \infty} h(\xi t)/h(t)=\xi^q$ for every $\xi>0$). In particular, the author's results apply to equation (0.1) with $h(t)=t^q (\log t)^{\alpha_1}$ as $t\to \infty$ and $b(x)=|x|^\theta (-\log |x|)^{\alpha_2}$ as $|x|\to 0$, where $\alpha_1$ and $\alpha_2$ are any real numbers.