论文标题
将涉及高阶分数拉普拉斯的方程式的解决方案分类
Classification of solutions to equations involving Higher-order fractional Laplacian
论文作者
论文摘要
在本文中，我们关注的是涉及涉及高阶分数lapalacian \ begin {等式*} \ left \ left \ {\ begin {Aligned}＆（ - δ）^{ ＆\ int _ {\ mathbb {r}^n} u _+^γdx<+\ infty，\ end {aligned} \ right。 \ end {qore*}其中$ p \ geq 1 $是整数，$ 0 <\ alp <2 $，$ n> 2p+α$和$γ\ in（1，\ frac {n} {n-2p- \ alp}）$。我们为任何满足无穷大增长的非构造经典解决方案建立了不可或缺的表示公式。由此我们证明，这些解决方案在$ \ r^n $中的某个点上是径向对称的，而单调通过以整体形式移动平面的方法在径向方向上减小。
In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \begin{equation*} \left\{\begin{aligned} &(-Δ)^{p+{\fracα{2}}}u(x)=u_+^γ~~ \mbox{ in }\mathbb{R}^n,\\ &\int_{\mathbb{R}^n}u_+^γdx<+\infty, \end{aligned}\right. \end{equation*} where $p\geq 1$ is an integer, $0<\alp<2$, $n> 2p+α$ and $γ\in (1,\frac{n}{n-2p-\alp})$. We establish an integral representation formula for any nonconstant classical solution satisfying certain growth at infinity. From this we prove that these solutions are radially symmetric about some point in $\R^n$ and monotone decreasing in the radial direction via method of moving planes in integral forms.
