An integral system and the Lane-Emden conjecture

Abstract

We consider the system of integral equations in $R^n$: <p align="center"> $ u(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} v^q (y) dy$<BR> $ v(x) = \int_{R^{n}} \frac{1}{|x - y|^{n-\mu}} u^p(y) dy$<br> <p align="left" class="times"> with $0 < \mu < n$. Under some integrability conditions, we obtain radial symmetry of positive solutions by using <i>the method of moving planes in integral forms</i>. In the special case when $\mu = 2$, we show that the integral system is equivalent to the elliptic PDE system <p align="center"> -Δ $u = v^q (x)$ <br> -Δ $v = u^p (x)$ <p align="left" class="times"> 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.