We study a nonlinear equation in the half-space with a Hardy potential, specifically, \[ \displaystyle - \Delta_p u= \lambda \frac{u ^{p-1}}{x_1^p}-x_1^\theta f(u)\ \ {\rm in}\ T,\] where $\Delta_p$ stands for the $p$-Laplacian operator defined by $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $\theta > -p$, and $T$ is a half-space $\{x_1>0\}$. When $\lambda > \Theta$ (where $\Theta$ is the Hardy constant), we show that under suitable conditions on $f$ and $\theta$, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as $x_1\to 0^+$, and the symmetric property of the positive solution are obtained.
Yujuan CHEN
,
Lei WEI
,
Yimin ZHANG
. THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE[J]. Acta mathematica scientia, Series B, 2022
, 42(5)
: 2149
-2164
.
DOI: 10.1007/s10473-022-0524-y
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