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CPAA

In this paper, we consider the weighted integral system involving
Wolff potentials in $R^{n}$:
\begin{eqnarray}
u(x) = R_1(x)W_{\beta, \gamma}(\frac{u^pv^q(y)}{|y|^\sigma})(x),
\\
v(x) = R_2(x)W_{\beta,\gamma}(\frac{v^pu^q(y)}{|y|^\sigma})(x).
\end{eqnarray}
where $0< R(x) \leq C$, $1 < \gamma \leq 2$, $0\leq \sigma < \beta \gamma$,
$n-\beta\gamma > \sigma(\gamma-1)$,
$\gamma^{*}-1=\frac{n\gamma}{n-\beta\gamma+\sigma}-1\geq 1$. Due to
the weight $\frac{1}{|y|^\sigma}$, we need more complicated
analytical techniques to handle the properties of the solutions.
First, we use the method of regularity lifting to obtain the
integrability for the solutions of this Wolff type integral
equation. Next, we use the modifying and refining method of moving
planes established by Chen and Li to prove the radial symmetry for
the positive solutions of related integral equation. Based on these
results, we obtain the decay rates of the solutions of (0.1) with
$R_1(x)\equiv R_2(x)\equiv 1$ near infinity. We generalize the
results in the related references.

keywords:
Wolff potential
,
regularity lifting
,
integrability
,
method of moving planes
,
decay rates.
,
radial symmetry

DCDS

This paper is concerned with the properties of solutions for the
weighted Hardy-Littlewood-Sobolev type integral system
\begin{equation}
\left \{
\begin{array}{l}
u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy,\\
v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy
\end{array}
\right. （1）
\end{equation}
and the fractional order partial differential system
\begin{equation} \label{PDE}
\left\{\begin{array}{ll}
(-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\alpha}u(x)) =|x|^{-\beta}
v^{q}(x),
\\
(-\Delta)^{\frac{n-\lambda}{2}}(|x|^{\beta}v(x))
=|x|^{-\alpha} u^p(x).
\end{array} （2）
\right.
\end{equation}
Here $x \in R^n \setminus \{0\}$. Due to $0 < p, q < \infty$, we need more
complicated analytical techniques to handle the case $0< p <1$ or
$0< q <1$.
We first establish the
equivalence of integral system (1) and fractional order partial
differential system (2). For integral system (1), we prove that
the integrable solutions are locally bounded. In addition, we
also show that the positive locally bounded solutions are
symmetric and decreasing about some axis by means of the method of
moving planes in integral forms introduced by Chen-Li-Ou. Thus,
the equivalence implies the positive solutions of the PDE system,
also have the corresponding properties. This paper extends
previous results obtained by other authors to the general case.

DCDS

This paper is concerned with the symmetry results for the
$2k$-order singular Lane-Emden type partial differential system
$$
\left\{\begin{array}{ll}
(-\Delta)^k(|x|^{\alpha}u(x))
=|x|^{-\beta} v^{q}(x),
\\
(-\Delta)^k(|x|^{\beta}v(x))
=|x|^{-\alpha} u^p(x),
\end{array}
\right.
$$
and the weighted Hardy-Littlewood-Sobolev type integral system
$$
\left \{
\begin{array}{l}
u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy\\
v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy.
\end{array}
\right.
$$
Here $x \in R^n \setminus \{0\}$. We first establish the
equivalence of this integral system and an fractional order
partial differential system, which includes the $2k$-order PDE
system above. For the integral system, we prove that the positive
locally bounded solutions are symmetric and decreasing about some
axis by means of the method of moving planes in integral forms
introduced by Chen-Li-Ou. In addition, we also show that the
integrable solutions are locally bounded. Thus, the equivalence
implies the positive solutions of the PDE system, particularly
including the higher integer-order PDE system, also have the
corresponding properties.

DCDS

In this paper, we consider the positive solutions of the following
weighted integral system involving Wolff potential in $R^{n}$:
$$
\left\{ \begin{array}{ll}
u(x) = R_1(x)W_{\beta,\gamma}(\frac{v^q}{|y|^{\sigma}})(x),
\\
v(x) = R_2(x)W_{\beta,\gamma}(\frac{u^p}{|y|^{\sigma}})(x). (0.1)
\end{array} \right.
$$
This system is helpful to understand some nonlinear PDEs and other
nonlinear problems. Different from the case of $\sigma=0$, it is
difficult to handle the properties of the solutions since there is
singularity at origin. First, we overcome this difficulty by
modifying and refining the new method which was introduced to
explore the integrability result establishes by Ma, Chen and Li, and
obtain an optimal integrability. Second, we use the method of moving
planes to prove the radial symmetry for the positive solutions of
(0.1) when $R_{1}(x)\equiv R_{2}(x)\equiv 1$. Based on these
results, by intricate analytical techniques, we obtain the estimate
of the decay rates of those solutions near infinity.

keywords:
Wolff potential
,
integrability
,
method of moving planes
,
radial symmetry
,
regularity lifting
,
decay rates.

CPAA

Let

be either a unit ball or a half space. Consider the following Dirichlet problem involving the fractional Laplacian

$Ω$ |

$\left\{ \begin{array}{*{35}{l}} \begin{align} & {{(-\Delta )}^{\frac{\alpha }{2}}}u=f(u),\ \ \text{in}\ \ \Omega , \\ & u=0, ~~~~~~~~~~~~~~~~~~~~ \text{in}\ \ {{\Omega }^{c}},\ \\ \end{align} & \ & {} \\\end{array} \right.~~~~(1)$ |

where

is any real number between

and

. Under some conditions on

, we study the equivalent integral equation

$α$ |

zhongwenzy$ |

$ |

$f$ |

$ \begin{align}u(x) \ = \ \int{{}}_{ Ω}G(x, y)f(u(y))dy, \end{align}~~~~(2) $ |

here

is the Green's function associated with the fractional Laplacian in the domain

. We apply the method of moving planes in integral forms to investigate the radial symmetry, monotonicity and regularity for positive solutions in the unit ball. Liouville type theorems-non-existence of positive solutions in the half space are also deduced.

$G(x, y)$ |

$Ω$ |

keywords:
The fractional Laplacian
,
semi-linear equations
,
symmetry
,
monotonicity
,
regularity
,
Liouville theorem

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