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.

中文翻译：

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具有Hardy势和梯度非线性的半线性椭圆方程

假设$ \ Omega \ subset {\ mathbb R} ^ N $（$ N \ geq 3 $）为$ C ^ 2 $有界域，令$ \ delta $为到$ \ partial \ Omega $的距离。在本文中，我们研究方程$（\ star）\ -L_ \ mu u + g（| \ nabla u |）= 0 $在$ \ Omega $中的正解，其中$ L_ \ mu = \ Delta + {\ mu} / {\ delta ^ 2} $，$ \ mu \ in（0，{1} / {4}] $和$ g $是$ {\ mathbb R} _ + $上的连续的，不递减的函数。证明如果$ g $满足奇异积分条件，则存在具有指定边界基准$ \ nu $的$（\ star）$唯一解。当$ g（t）= t ^ q $且$ q \在（1,2）$中，我们显示等式〜$（\ star）$允许一个临界指数$ q_ \ mu $（仅取决于$ N $和$ \ mu $）。在次临界情况下，即$ 1 < q <q_mu $，我们建立一些先验估计，并提供对$ \ partial \ Omega $具有孤立奇点的解决方案的描述。