Radial symmetry of solutions for some integral systems of Wolff type
Wenxiong Chen Congming Li
We consider the fully nonlinear integral systems involving Wolff potentials:

$\u(x) = W_{\beta, \gamma}(v^q)(x)$, $\x \in R^n$;
$\v(x) = W_{\beta, \gamma} (u^p)(x)$, $\x \in R^n$;



$ \W_{\beta,\gamma} (f)(x) = \int_0^{\infty}$ $[ \frac{\int_{B_t(x)} f(y) dy}{t^{n-\beta\gamma}} ]^{\frac{1}{\gamma-1}} \frac{d t}{t}.$

   After modifying and refining our techniques on the method of moving planes in integral forms, we obtain radial symmetry and monotonicity for the positive solutions to systems (1).        
   This system includes many known systems as special cases, in particular, when $\beta = \frac{\alpha}{2}$ and $\gamma = 2$, system (1) reduces to

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


The solutions $(u,v)$ of (2) are critical points of the functional associated with the well-known Hardy-Littlewood-Sobolev inequality. We can show that (2) is equivalent to a system of semi-linear elliptic PDEs

$(-\Delta)^{\alpha/2} u = v^q$, in $R^n$,
$(-\Delta)^{\alpha/2} v = u^p$, in $R^n$


which comprises the well-known Lane-Emden system and Yamabe equation.
keywords: radial symmetry Wolff potentials nonlinear systems method of moving planes in integral forms norm estimates.
Harmonic maps on complete manifolds
Wenxiong Chen Congming Li
In this article, we study harmonic maps between two complete noncompact manifolds M and N by a heat flow method. We find some new sufficient conditions for the uniform convergence of the heat flow, and hence the existence of harmonic maps.
Our condition are: The Ricci curvature of M is bounded from below by a negative constant, M admits a positive Green’s function and

$ \int_M G(x, y)|\tau(h(y))|dV_y $ is bounded on each compact subset. $\qquad$ (1)

Here $\tau(h(x))$ is the tension field of the initial data $h(x)$.
Condition (1) is somewhat sharp as is shown by examples in the paper.

keywords: Harmonic maps between complete heat flow method noncompact manifolds uniform convergence of heat flows.
Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations
Wenxiong Chen Chao Jin Congming Li Jisun Lim
In this paper, we consider systems of integral equations related to the weighted Hardy-Littlewood-Sobolev inequality. We present the symmetry, monotonity, and regularity of the solutions. In particular, we obtain the optimal integrability of the solutions to a class of such systems. We also present a simple method for the study of regularity, which has been extensively used in various forms. The version we present here contains some new developments. It is much more general and very easy to use. We believe the method will be helpful to both experts and non-experts in the field.
keywords: Weighted Hardy-Littlewood-Sobolev inequalities integral equations and systems moving planes in integral forms. radial symmetry monotonicity
A priori estimate for the Nirenberg problem
Wenxiong Chen Congming Li
We establish a priori estimate for solutions to the prescribing Gaussian curvature equation

$ - \Delta u + 1 = K(x) e^{2u}, x \in S^2,$    (1)

for functions $K(x)$ which are allowed to change signs. In [16], Chang, Gursky and Yang obtained a priori estimate for the solution of (1) under the condition that the function K(x) be positive and bounded away from 0. This technical assumption was used to guarantee a uniform bound on the energy of the solutions. The main objective of our paper is to remove this well-known assumption. Using the method of moving planes in a local way, we are able to control the growth of the solutions in the region where K is negative and in the region where K is small and thus obtain a priori estimate on the solutions of (1) for general functions K with changing signs.

keywords: semi-linear elliptic equations method of moving planes in a local way. Gaussian curvature Nirenberg problem a priori estimate
Super polyharmonic property of solutions for PDE systems and its applications
Wenxiong Chen Congming Li
In this paper, we prove that all the positive solutions for the PDE system \begin{eqnarray} (- \Delta)^k u_i = f_i(u_1, \cdots, u_m), \ x \in R^n, \ i = 1, 2, \cdots, m \ \ \ \ \ (1) \end{eqnarray} are super polyharmonic, i.e. \begin{eqnarray} (- \Delta)^j u_i > 0, \ j=1, 2, \cdots, k-1; \ i =1, 2, \cdots, m. \end{eqnarray}
To prove this important super polyharmonic property, we introduced a few new ideas and derived some new estimates.

As an interesting application, we establish the equivalence between the integral system \begin{eqnarray} u_i(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} f_i(u_1(y), \cdots, u_m(y)) d y, \ x \in R^n \ \ \ \ \ (2) \end{eqnarray} and PDE system (1) when $\alpha = 2k < n.$

In the last few years, a series of results on qualitative properties for solutions of integral systems (2) have been obtained, since the introduction of a powerful tool--the method of moving planes in integral forms. Now due to the equivalence established here, all these properties can be applied to the corresponding PDE systems.

We say that systems (1) and (2) are equivalent, if whenever $u$ is a positive solution of (2), then $u$ is also a solution of \begin{eqnarray} (- \Delta)^k u_i = c f_i(u_1, \cdots, u_m), \ x \in R^n, \ i= 1,2, \cdots, m \end{eqnarray} with some constant $c$; and vice versa.
keywords: fractional power Laplacians. integral systems super poly-harmonic properties equivalences Poly-harmonic PDE systems
Indefinite elliptic problems in a domain
Wenxiong Chen Congming Li
In this paper, we study the elliptic boundary value problem in a bounded domain $\Omega$ in $R^n$, with smooth boundary:

$-\Delta u = R(x) u^p \quad \quad u > 0 x \in \Omega$

$u(x) = 0 \quad \quad x \in \partial \Omega.$

where $R(x)$ is a smooth function that may change signs. In [2], using a blowing up argument, Berestycki, Dolcetta, and Nirenberg, obtained a priori estimates and hence the existence of solutions for the problem when the exponent $1 < p < {n+2}/{n-1}$. Inspired by their result, in this article, we use the method of moving planes to fill the gap between ${n+2}/{n-1}$ and the critical Sobolev exponent ${n+2}/{n-2}$. We obtain a priori estimates for the solutions for all $1 < p < {n+2}/{n-2}$.

keywords: Indefinite nonlinear elliptic equations method of moving planes. a priori estimates
Some new approaches in prescribing gaussian and salar curvature
Wenxiong Chen Congming Li
Please refer to Full Text.
Regularity of solutions for a system of integral equations
Wenxiong Chen Congming Li
In this paper, we study positive solutions of the following system of integral equations in $R^n$:

$u(x) = \int_{R^{n}} |x-y|^{\alpha -n} v(y)^q dy$, $ v(x) = \int_{R^{n}} |x-y|^{\alpha -n} u(y)^p dy$

with $\frac{1}{q+1}+\frac{1}{p+1}=\frac{n-\alpha}{n}$. In our previous paper, under the natural integrability conditions $u \in L^{p+1} (R^n)$ and $v \in L^{q+1} (R^n)$, we prove that all the solutions are radially symmetric and monotone decreasing about some point. In this paper, we go further to study the regularity of the solutions. We show that the solutions are bounded, and hence continuous and smooth. We also prove that if $p = q$, then $u = v$, and they both must assume the standard form

$ c(\frac{t}{t^2 + |x - x_o|^2})^{(n-\alpha)/2} $

with some constant $c = c(n, \alpha)$, and for some $t > 0$ and $x_o \in R^n$.

keywords: systems of integral equations regularity classification of solutions. boundedness Hardy-Littlewood-Sobolev inequalities
A note on a class of higher order conformally covariant equations
Sun-Yung Alice Chang Wenxiong Chen
In this paper, we study the higher order conformally covariant equation

$(- \Delta )^{\frac{n}{2}} w = (n -1)! e^{n w} x \in R^n$

for all even dimensions n.

$\alpha = \frac{1}{|S^n|} \int_{R^n} e^{n w} dx .$

We prove, for every $0 < \alpha < 1$, the existence of at least one solution. In particular, for $ n = 4$, we obtain the existence of radial solutions.

keywords: non-uniqueness variational method. Higher order semilinear elliptic equations conformally covariant equations
An integral system and the Lane-Emden conjecture
Wenxiong Chen Congming Li
We consider the system of integral equations in $R^n$:

$ u(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} v^q (y) dy$
$ v(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} u^p(y) dy$

with $0 < \mu < n$. Under some integrability conditions, we obtain radial symmetry of positive solutions by using the method of moving planes in integral forms. In the special case when $\mu = 2$, we show that the integral system is equivalent to the elliptic PDE system

-Δ $u = v^q (x)$
-Δ $v = u^p (x)$

in $R^n$. Our symmetry result, together with non-existence of radial solutions by Mitidieri [30], implies that, under our integrability conditions, the PDE system possesses no positive solution in the subcritical case. This partially solved the well-known Lane-Emden conjecture.

keywords: moving planes. symmetry integral systems Lane-Emden conjecture elliptic PDE systems non-existence

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