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 $-\infty0$. 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.