Chen, Wenxiong; Li, Congming Radial symmetry of solutions for some integral systems of Wolff type. (English) Zbl 1221.45006 Discrete Contin. Dyn. Syst. 30, No. 4, 1083-1093 (2011). Summary: We consider the fully nonlinear integral systems involving Wolff potentials:\[ \begin{cases} u(x) = W_{\beta,\gamma}(v^q)(x), \quad & x \in\mathbb R^n; \\ v(x) = W_{\beta,\gamma}(u^p)(x), & x \in\mathbb R^n;\end{cases}\tag{1} \]where\[ W_{\beta,\gamma}(f)(x)=\int^\infty_0\left[\frac{\int_{B_t(x)}f(y)\,dy}{t^{n-\beta\gamma}}\right]^{\frac{1}{\gamma-1}}\frac{dt}{t}\,. \]After modifying and refining our techniques on the method of moving places in integral forms, we obtain radial symmetry and monotonicity for the positive solutions to system (1). This system includes many known systems as special cases, in particular, when \(\beta = \frac{\alpha}{2}\) and \(\gamma=2\) , system (1) reduces to\[ \begin{cases} u(x)=\int_{R^{n}} \frac{1}{|x-y|^{n-\alpha}}\;v(y)^q\,dy],\quad & x\in\mathbb R^n,\\ v(x)=\int_{R^{n}} \frac{1}{|x-y|^{n-\alpha}}\;u(y)^p\,dy], & x\in\mathbb R^n.\end{cases}\tag{2} \]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 partial differential equations\[ \begin{cases} (-\Delta)^{\alpha/2} \,u = v^q,\quad & \text{in }\mathbb R^n,\\ (-\Delta)^{\alpha/2} \,v = u^p, & \text{in }\mathbb R^n,\end{cases} \]which comprises the well-known Lane-Emden system and Yamabe equation. Cited in 1 ReviewCited in 33 Documents MSC: 45G15 Systems of nonlinear integral equations 45M20 Positive solutions of integral equations Keywords:Wolff potentials; radial symmetry; method of moving planes in integral forms; norm estimates; nonlinear integral systems; positive solutions; critical points; Hardy-Littlewood-Sobolev inequality; Lane-Emden system; Yamabe equation PDFBibTeX XMLCite \textit{W. Chen} and \textit{C. Li}, Discrete Contin. Dyn. Syst. 30, No. 4, 1083--1093 (2011; Zbl 1221.45006) Full Text: DOI