Radial symmetry of solutions for some integral systems of Wolff type

Wenxiong Chen; Congming Li

doi:10.3934/dcds.2011.30.1083

Article Contents

2011, Volume 30, Issue 4: 1083-1093. Doi: 10.3934/dcds.2011.30.1083

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Radial symmetry of solutions for some integral systems of Wolff type

Wenxiong Chen^1, and
Congming Li^2,

1.
Department of Mathematics, Yeshiva University, New York, NY 10033
2.
Department of Applied Mathematics, University of Colorado at Boulder

Received: February 28, 2010

Revised: July 31, 2010

Published: September 2011

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Abstract

We consider the fully nonlinear integral systems involving Wolff potentials:

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

(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.

Keywords:

Wolff potentials,
nonlinear systems,
radial symmetry,
method of moving planes in integral forms,
norm estimates.

Mathematics Subject Classification: Primary: 45G15, 35J60; Secondary: 35J91.

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