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We study the problem \begin{equation*} (I_{\epsilon}) \left\{\begin{aligned}  -\Delta u- \frac{\mu u}{|x|^2}&=u^p -\epsilon u^q \quad\text{in }\quad\Omega, \\ u&>0 \quad\text{in }\quad \Omega, \\ u &\in H^1_0(\Omega)\capL^{q+1}(\Omega),  \end{aligned}  \right. \end{equation*} where $q>p\geq 2^*-1$, $\epsilon>0$ is a parameter,$\Omega\subseteq\mathbb{R}^N$ is a bounded domain with smooth boundary, $0\in\Omega$, $N\geq 3$ and $0<\mu<\bar\mu:=\big(\frac{N-2}{2}\big)^2$. We prove at$0$, any solution of $(I_{\epsilon})$ has the singularity of order $|x|^{-\nu}$when $q<\frac{2+\nu}{\nu}$ and of the order $|x|^{-\frac{2}{q-1}}$, when$q>\frac{2+\nu}{\nu}$, where $\nu=\sqrt{\bar\mu}-\sqrt{\bar\mu-\mu}$. Moreover,we show that when $q=\frac{2+\nu}{\nu}$ and $u$ is radial, $u\sim|x|^{-\nu}|\log|x||^{-\frac{\nu}{2}}$. This gives the complete classificationof singularity at $0$ in the supercritical case. We also obtain gradientestimate. Using the transformation $v=|x|^{\nu}u$, we reduce the problem$(I_{\epsilon})$ to $(J_{\epsilon})$ \begin{equation*} (J_{\epsilon})\left\{\begin{aligned}  -div(|x|^{-2\nu} \nabla v)&=|x|^{-(p+1)\nu} v^p -\epsilon |x|^{-(q+1)\nu} v^q\quad\text{in }\quad \Omega, \\ v&>0 \quad\text{in }\quad \Omega, \\ v& \inH^1_0(\Omega, |x|^{-2\nu} )\cap L^{q+1}(\Omega, |x|^{-(q+1)\nu} ),  \end{aligned}  \right. \end{equation*} and then formulating a variational problem for$(J_{\epsilon})$, we establish the existence of a variational solution$v_{\epsilon}$. Furthermore, we characterize the asymptotic behavior of$v_{\epsilon}$ as $\epsilon\to 0$ by variational arguments and when $p=2^*-1$,we show how the solution $v_{\epsilon}$ blows-up at $0$.  This is the first paper where the results have been established with supercritical exponents for $\mu>0$.

On singular equations with critical and supercritical exponents