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.
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