DCDS

This paper is concerned with the integral system
$$\left \{
\begin{array}{ll}
&u(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{v^q(y)},\quad u>0~in~R^n,\\
&v(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{u^p(y)},\quad v>0~in~R^n,
\end{array} \right.
$$
where $n \geq 1$, $p,q,\lambda \neq 0$.
Such an integral system appears in the study of the conformal
geometry. We obtain several necessary conditions for the existence of
the $C^1$ positive entire solutions, particularly including the
critical condition
$$
\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n},
$$
which is the necessary and sufficient condition for the invariant
of the system and some energy functionals under the scaling
transformation. The necessary condition
$\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}$ can be relaxed to
another weaker one $\min\{p,q\}>\frac{n+\lambda}{\lambda}$ for the
system with double bounded coefficients. In addition, we classify
the radial solutions in the case of $p=q$ as the form
$$
u(x)=v(x)=a(b^2+|x-x_0|^2)^{\frac{\lambda}{2}}
$$
with $a,b>0$ and $x_0 \in R^n$. Finally, we also deduce some analogous
necessary conditions of existence for the weighted system.

DCDS

In this paper, we are concerned with the positive continuous entire
solutions of the Wolff type integral equation
$$
u(x)=c(x)W_{\beta,\gamma}(u^{-p})(x), \quad u>0 ~in~ R^n,
$$
where $n \geq 1$, $p>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma \neq n$.
In addition, $c(x)$ is a double bounded function.
Such an integral equation is related to the study of the conformal
geometry and nonlinear PDEs, such as $\gamma$-Laplace equations and
$k$-Hessian equations with negative exponents. By some Wolff type potential
integral estimates, we obtain the asymptotic rates and the integrability
of positive solutions, and discuss the existence and nonexistence
results of the radial solutions.

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.

CPAA

This paper is concerned with integral
systems involving the Bessel potentials. Such integral systems are
helpful to understand the corresponding PDE systems,
such as some static Shrödinger systems with the critical and the supercritical exponents.
We use the lifting lemma on regularity
to obtain an integrability interval of solutions. Since the Bessel kernel does not
have singularity at infinity, we extend the integrability
interval to the whole $[1,\infty]$. Next, we use the
method of moving planes to prove the radial symmetry for the
positive solution of the system. Based on these results, by an iteration we obtain the
estimate of the exponential decay of those solutions near infinity. Finally,
we discuss the uniqueness of the positive solution of PDE system under some assumption.

CPAA

In this paper we study the asymptotic behavior of the positive
solutions of the following system of Euler-Lagrange equations of
the Hardy-Littlewood-Sobolev type in $R^n$
$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy $,

$ v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy.
$

We obtain the growth rate of the solutions around the origin and
the decay rate near infinity. Some new cases beyond the work of C.
Li and J. Lim [17] are studied here. In particular, we
remove some technical restrictions of [17], and thus
complete the study of the asymptotic behavior of the solutions for
non-negative $\alpha$ and $\beta$.

DCDS

In this paper, we study the properties of the positive solutions
of a $\gamma$-Laplace equation in $R^n$
-div$(|\nabla u|^{\gamma-2}\nabla u) =K u^p$,

Here $1<\gamma<2$, $n>\gamma$,
$p=\frac{(\gamma-1)(n+\gamma)}{n-\gamma}$ and $K(x)$ is a smooth
function bounded by two positive constants. First, the positive
solution $u$ of the $\gamma$-Laplace equation above satisfies an
integral equation involving a Wolff potential. Based on this, we
estimate the decay rate of the positive solutions of the
$\gamma$-Laplace equation at infinity. A new method is introduced
to fully explore the integrability result established recently by
Ma, Chen and Li on Wolff type integral equations to derive the
decay estimate.

DCDS

In this paper, we establish the sharp criteria for the
nonexistence of positive solutions to the Hardy-Littlewood-Sobolev
(HLS) system of nonlinear equations and the corresponding
nonlinear differential systems of Lane-Emden type. These
nonexistence results, known as Liouville theorems, are
fundamental in PDE theory and applications. A special iteration
scheme, a new shooting method and some Pohozaev identities in
integral form as well as in differential form are created.
Combining these new techniques with some observations and some
critical asymptotic analysis, we establish the sharp criteria of
Liouville type for our systems of nonlinear equations. Similar
results are also derived for the system of Wolff type of integral
equations and the system of $\gamma$-Laplace equations. A
dichotomy description in terms of existence and nonexistence for
solutions with finite energy is also obtained.