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An integral system and the Lane-Emden conjecture. (English) Zbl 1176.35067
Summary: We consider the system of integral equations in $\Bbb R^n$: $$u(x)= \int_{\Bbb R^n} \frac{1}{|x-y|^{n-\mu}} v^q (y)\,dy, \qquad v(x)= \int_{\Bbb R^n} \frac{1}{|x-y|^{n-\mu}} u^p(y)\, dy,$$ 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 $$-\Delta [u=v^q(x)], \qquad -\Delta [v=u^p(x)]$$ in $\Bbb R^n$. Our symmetry result, together with non-existence of radial solutions by {\it E. Mitidieri} [Commun. Partial Differ. Equations 18, No. 1--2, 125--151 (1993; Zbl 0816.35027], 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.

35J60Nonlinear elliptic equations
45G15Systems of nonlinear integral equations
35J46First-order elliptic systems
35J47Second-order elliptic systems
35J48Higher-order elliptic systems
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