On singular equations with critical and supercritical exponents

TL;DR

This paper classifies the singularity behavior of solutions to a nonlinear elliptic PDE with critical and supercritical exponents, introducing a transformation to analyze solutions and establishing existence and asymptotic properties.

Contribution

It provides the first comprehensive analysis of singularities and solution behavior for equations with supercritical exponents and Hardy potential, including a variational approach and asymptotic characterization.

Findings

Complete classification of singularity types at zero for supercritical cases.

Existence of variational solutions via transformation and variational methods.

Asymptotic behavior and blow-up analysis as epsilon approaches zero.

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 $\mu>0$.