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An integral system and the Lane-Emden conjecture

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  • 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.

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

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