On singular equations with critical and supercritical exponents

Abstract

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)\cap L^{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 classification of singularity at $0$ in the supercritical case. We also obtain gradient estimate. 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& \in H^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 super critical exponents for $\mu>0$.