THE ASYMPTOTIC BEHAVIOR AND SYMMETRY OF POSITIVE SOLUTIONS TO p-LAPLACIAN EQUATIONS IN A HALF-SPACE

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doi:10.1007/s10473-022-0524-y

Abstract: We study a nonlinear equation in the half-space with a Hardy potential, specifically,

- Δ_{p} u = λ \frac{u^{p - 1}}{x_{1}^{p}} - x_{1}^{θ} f (u) i n T,

$\displaystyle - \Delta_p u= \lambda \frac{u ^{p-1}}{x_1^p}-x_1^\theta f(u)\ \ {\rm in}\ T,$ where

Δ_{p}

$\Delta_p$ stands for the

p

$p$ -Laplacian operator defined by

Δ_{p} u = d i v (| \nabla u |^{p - 2} \nabla u)

$\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$ ,

p > 1

$p>1$ ,

θ > - p

$\theta > -p$ , and

T

$T$ is a half-space

{x_{1} > 0}

$\{x_1>0\}$ . When

λ > Θ

$\lambda > \Theta$ (where

Θ

$\Theta$ is the Hardy constant), we show that under suitable conditions on

f

$f$ and

θ

$\theta$ , the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as

x_{1} \to 0^{+}

$x_1\to 0^+$ , and the symmetric property of the positive solution are obtained.

Key words: p-Lapacian, Hardy potential, symmetry, uniqueness, asymptotic behavior

CLC Number:

35J20

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.

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