Elliptic equations with Hardy potentials and gradient-dependent absorption: existence and refined asymptotics

TL;DR

This paper investigates positive radial solutions of elliptic equations with Hardy potentials and gradient-dependent absorption, establishing existence, detailed asymptotic behaviors near zero and infinity, and identifying new profiles influenced by potential-absorption competition.

Contribution

It provides the first analysis of local properties of solutions with arbitrary m>0 and nonzero λ, identifying all profiles and higher order asymptotics under optimal conditions.

Findings

Identified blow-up and bounded profiles near zero influenced by Hardy potential and gradient absorption.

Established existence of infinitely many radial solutions with prescribed asymptotic behaviors.

Derived higher order terms in asymptotic expansions for solutions near zero and at infinity.

Abstract

Under sharp conditions, we prove the existence and refined asymptotic behaviour near zero (resp., at infinity) for all positive radial solutions to elliptic equations such as \begin{equation}\label{eq11} \tag{*} \mathbb L_{\rho,\lambda}(u)=\Delta u+ (2-N-2\rho)\, \frac{x\cdot \nabla u}{|x|^2}+ \frac{\lambda}{|x|^2}u=|x|^{\theta}\,u^q\, |\nabla u|^m\quad \mbox{in } \Omega\setminus\{0\}, \end{equation} where $\Omega=B_R(0)$ (resp., $\Omega=\mathbb R^{N}\setminus B_{1/R}(0)$) for $R>0$ and $N\geq 2$. The dynamics of such solutions is very rich since $\rho, \lambda,\theta\in \mathbb R$ are arbitrary, $ m>0$, $q\geq 0$ and $\kappa:=m+q-1>0$. To our knowledge, this is the first study of the local properties of the positive solutions of \eqref{eq11} with arbitrary $m>0$ and $\lambda\not=0$. We identify all profiles near zero (and at infinity via a modified Kelvin transform) under optimal conditions, depending on how $\Theta:=(\theta+2-m)/\kappa$ relates to $0$ or the roots $\Theta_\pm$ of $t^2+2\rho t+\lambda$ when $\lambda\leq \rho^2$. For each profile, we advance new methods that unearth the higher order terms in the asymptotic expansion. We highlight two new asymptotic profiles near zero due to the competition between the Hardy potential with $\lambda>0$ and the gradient-dependent absorption: (i) a blow-up profile $\left[ \lambda \left( \frac{\kappa}{m} \right)^m \right]^{\frac{1}{\kappa}} | \log |x||^{\frac{m}{\kappa}} $ if $\Theta=0$ and (ii) a bounded profile if $\Theta<0$. Any radial solution of \eqref{eq11} with $\lim_{r\to 0^+} u(r)=\gamma\in \mathbb R_+$ satisfies $(P_\pm)$ $u(r)=\gamma\pm \lambda^{1/m} \gamma^{1-\kappa/m} (1/\sigma)\, r^\sigma(1+o(1))$ as $r\to 0^+$, where $\sigma=-\kappa \Theta/m$. For any $\gamma\in \mathbb R_+$, there is $R>0$ such that \eqref{eq11} has a radial solution (infinitely many) satisfying $(P_-)$ ($(P_+)$).