Higher and Fractional Order Hardy-Sobolev Type Equations
by Wenxiong Chen   Yanqin Fang  

Vol. 9 No. 3 (2014) P.317~P.349

ABSTRACT

In this paper we consider the following higher order or fractional order Hardy-Sobolev type equation \begin{equation} \label{equ.1a} (-\Delta)^{\frac{\alpha}{2}} u(x)=\frac{u^{p}(x)}{|y|^{t}}, \;x=(y,z)\in (R^{k}\backslash\{0\})\times R^{n-k}, \end{equation} where $0$<$\alpha$<$n$, $0$<$t$<$\min$$\{$$\alpha$,$k$$\}$, and $1$<$p$$\leq$$\tau$:=$\frac{n+\alpha-2t}{n-\alpha}$. In the case when $\alpha$ is an even number, we first prove that the positive solutions of (1) are super poly-harmonic, i.e. \begin{equation} \label{02} (-\Delta)^{i}u>0,\;\;i=1,\cdots,\frac{\alpha}{2}-1. \end{equation} Then, based on (2), we establish the equivalence between PDE (1) and the integral equation $$ u(x)=\int_{R^{n}}G(x,\xi)\frac{u^{p}(\xi)}{|\eta|^{t}}d\xi, $$ where $G(x,\xi)=\frac{c_{n,\alpha}}{|x-\xi|^{n-\alpha}}$ is the Green's function of $(-\Delta)^{\frac{\alpha}{2}} $ in $R^{n}$. By the method of moving planes in integral forms, in the critical case, we prove that each nonnegative solution $u(y,z)$ of (1) is radially symmetric and monotone decreasing in $y$ about the origin in $R^{k}$ and in $z$ about some point $z_{0}$ in $R^{n-k}$. In the subcritical case, we obtain the nonexistence of positive solutions for (1).

KEYWORDS
Hardy-Sobolev inequality, super poly-harmonic properties, moving planes in integral forms, equivalence, integral equations, radial symmetry, monotonicity, nonexistence

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35J60, 45G05.

MILESTONES

Received: 2013-07-28
Revised : 2014-04-16
Accepted: 2014-04-16

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