Radial symmetry of solutions for some integral systems of Wolff type

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.