Revista Matemática Iberoamericana


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Published online first: 2020-01-13
DOI: 10.4171/rmi/1164

Semilinear elliptic equations with Hardy potential and gradient nonlinearity

Konstantinos Gkikas[1] and Phuoc-Tai Nguyen[2]

(1) National and Kapodistrian University of Athens, Greece
(2) Masaryk University, Brno, Czechia

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain, and let $\delta$ be the distance to $\partial \Omega$. In this paper, we study positive solutions of the equation $(\star)\ -L_\mu u+ g(|\nabla u|) = 0$ in $\Omega$, where $L_\mu=\Delta + {\mu}/{\delta^2} $, $\mu \in (0,{1}/{4}]$ and $g$ is a continuous, nondecreasing function on ${\mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition, then there exists a unique solution of $(\star)$ with a prescribed boundary datum $\nu$. When $g(t)=t^q$ with $q \in (1,2)$, we show that equation~$(\star)$ admits a critical exponent $q_\mu$ (depending only on $N$ and $\mu$). In the subcritical case, namely $1 < q < q_\mu$, we establish some a priori estimates and provide a description of solutions with an isolated singularity on $\partial \Omega$. In the supercritical case, i.e., $q_\mu\leq q < 2$, we demonstrate a removability result in terms of Bessel capacities.

Keywords: Hardy potential, Martin kernel, boundary trace, critical exponent, gradient term, isolated singularities, removable singularities

Gkikas Konstantinos, Nguyen Phuoc-Tai: Semilinear elliptic equations with Hardy potential and gradient nonlinearity. Rev. Mat. Iberoam. Electronically published on January 13, 2020. doi: 10.4171/rmi/1164 (to appear in print)