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

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 $(P_-)$ ($(P_+)$).