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

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

(1)

where$ \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

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

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

(2)

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 $R^n$

(3)

which comprises the well-known Lane-Emden system and Yamabe equation.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.

$ - \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.

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.

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

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

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 $1 < p < {n+2}/{n-2}$.

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

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

for all even dimensions n.

Let

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

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.

$ 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.

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