On singular equations with critical and supercritical exponents

Abstract

We study the problem \begin{equation*} (I_ε) \left\{\begin{aligned} -Δu- \frac{μu}{|x|^2}&=u^p -εu^q \quad\text{in }\quad Ω, \\ u&>0 \quad\text{in }\quad Ω, \\ u &\in H^1_0(Ω)\cap L^{q+1}(Ω), \end{aligned} \right. \end{equation*} where $q>p\ge 2^{\ast }-1$, $\epsilon >0$ is a parameter, $\Omega \subseteq {\mathbb{R}}^N$ is a bounded domain with smooth boundary, $0\in \Omega$, $N\ge 3$ and $0<\mu <\bar{\mu }:=(\frac{N-2}{2}{)}^2$. 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_ε) \left\{\begin{aligned} -div(|x|^{-2ν} \nabla v)&=|x|^{-(p+1)ν} v^p -ε|x|^{-(q+1)ν} v^q \quad\text{in }\quad Ω, \\ v&>0 \quad\text{in }\quad Ω, \\ v& \in H^1_0(Ω, |x|^{-2ν} )\cap L^{q+1}(Ω, |x|^{-(q+1)ν} ), \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 $p=2^{\ast }-1$, 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 $\mu >0$.

52 pages