| Contents |
| In this talk, we consider the following fractional Laplace equation
\begin{eqnarray}\left\{\begin{array}{ll}
(- \Delta)^{\alpha/2}u(x)=0,& \mbox {in\,\,\,} R^n,\\
u(x)\geq 0,& \mbox { in\,\,\,} R^n,
\end{array} \right.\label{a1} \end{eqnarray}
where $n \geq 2$ and $\alpha$ is any real number between $0$ and $2$. We prove that the only solution
for (\ref{a1}) is constant. Or equivalently,
{\em Every $\alpha$-harmonic function bounded either above or below in all of $R^n$ must be constant.}
This extends the classical Liouville Theorem from
Laplacian to the fractional Laplacian.
As an immediate application, we use it to obtain an equivalence between a semi-linear pseudo-differential
\begin{equation}
(-\Delta)^{\alpha/2} u = u^p (x) , \;\; x \in R^n
\label{b1}
\end{equation}
and the corresponding integral equation
$$ u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} d y , \;\; x \in R^n. $$ Combining this with the existing results on
the integral equation, one can obtained much more general results on the qualitative properties of the solutions for (\ref{b1}). |
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