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首页> 外文期刊>Proceedings of the American Mathematical Society. Series B >A note on the asymptotic behavior of radial solutions to quasilinear elliptic equations with a Hardy potential
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A note on the asymptotic behavior of radial solutions to quasilinear elliptic equations with a Hardy potential

机译：径向溶液与硬势径向椭圆型方程的渐近行为的说明

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摘要

The quasilinear elliptic equation with a Hardy potential egin{equation*} {mathrm {div}}( x ^lpha abla u ^{p-2}abla u) + rac {mu }{ x ^{p-lpha }} u ^{p-2}u = 0 quad ext {in} {mathbf {R}}^N-{0} end{equation*} is considered, where $Nin {mathbf {N}}$, $p1$ and $lpha in {mathbf {R}}$, $mu in {mathbf {R}}-{0}$. In this note, the asymptotic behaviors of radial solutions are obtained divided into three case $mu (N-p+lpha )/p ^p$. This equation also appears as the Euler-Lagrange equation related to the weighted Hardy inequality egin{equation*} int _Omega abla u(x) ^p x ^lpha dx ge Biggl rac {N-p+lpha }{p} Biggr ^p int _Omega u(x) ^p x ^{lpha -p} dx end{equation*} for $u in C_c^infty ({mathbf {R}}^N)$ and $N-p+lpha e 0$, where $Omega$ is a domain of ${mathbf {R}}^N$. The rectifiability of oscillatory solutions to the ordinary differential equation with one-dimensional $p$-Laplacian is also studied, and an answer to an open problem is given.
机译：具有硬潜力的Quasilinear椭圆方程 begin {armation *} { mathrm {div}}（x ^ alpha nabla u ^ {p-2} nabla u）+ frac { mu} {x ^ { u ^ {p-2} u = 0 quad text {} { mathbf {r}} ^ n - {0 } 结束{等式*}在其中$ n in { mathbf {n}} $，$ p＆ 1 $和$ alpha 在{ mathbf {r}} $，$ mu 中{ mathbf {r}} - {0 } $。在本说明中，获得径向解决方案的渐近行为分为三个案例$ mu（n-p + alpha）/ p ^ p $。该方程式也显示为与加权硬性不等式相关的欧拉拉格朗日方程 begin {公式*} int _ oomega nabla u（x）^ px ^ alpha dx ge biggl frac {n-p + alpha} {p} biggr ^ p int _ oomega u（x）^ px ^ { alpha -p} dx end {aralmation *}在c_c ^ intty（{ mathbf {r}中} ^ n）$和$ n-p + alpha ne 0 $，其中$ omega $是$ { mathbf {r}} ^ n $的域名。还研究了振荡解决方案的振荡解决方案的差异性，并进行了一维的$ P $ -laplacian，给出了一个打开问题的答案。

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