2014 / September Volume 9 No.3
Higher and Fractional Order Hardy-Sobolev Type Equations
| Published Date |
2014 / September
|
|---|---|
| Title | Higher and Fractional Order Hardy-Sobolev Type Equations |
| Author | |
| Keyword | |
| Download | |
| Pagination | 317-349 |
| 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).
|
| AMS Subject Classification |
35J60, 45G05.
|
| Received |
2014-04-16
|
| Accepted |
2014-04-16
|