Existence and symmetry of positive solutions of an integral equation system. (English) Zbl 1206.45007

The main result of this paper is the following existence and uniqueness theorem. Assume \((u,v)\) is a pair of positive \({C}^1\) solutions of the system of integral equations
\[ u(x) = \int_{\mathbb{R}^n} |x-y|^{\alpha -n} v^p(y) dy,\quad v(x) =\int_{\mathbb{R}^n} |x-y|^{\beta -n} u^q(y) dy, \]
where \( 0<\alpha, \beta < n,\) \(1<p\leq \frac{n+\alpha}{n-\beta},\) \(1<q < \frac{n+\beta}{n-\alpha},\) then this system of integral equations has no positive solution unless \(p=\frac{n+\alpha}{n-\beta}\) and \(q=\frac{n+\beta}{n-\alpha}.\) Furthermore, when \(p=\frac{n+\alpha}{n-\beta}\) and \(q=\frac{n+\beta}{n-\alpha},\) \(u(x), v(x)\) are radially symmetric and monotonic about some points with the form
\[ u(x) = \biggl( \frac{a_1}{d_1+|x-x_0|^2} \biggr )^{\frac{n-\alpha}{2}},\quad v(x) = \biggl( \frac{a_2}{d_2+|x-x_0|^2} \biggr )^{\frac{n-\beta}{2}}, \]
where \(a_1, d_1, a_2, d_2 >0\) are constants, and \(x_0\in\mathbb{R}^n\).


45G15 Systems of nonlinear integral equations
45M20 Positive solutions of integral equations
Full Text: DOI


[1] Lieb, E., Sharp constants in the hardy – littlewood – sobolev and related inequalities, Ann. of math., 118, 349-374, (1983) · Zbl 0527.42011
[2] Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[3] Caffarelli, L.A.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear equations with critical Sobolev growth, Comm. pure appl. math., 42, 271-297, (1989) · Zbl 0702.35085
[4] Li, A.; Li, Y.Y., On some conformally invariant fully nonlinear equations, Comm. pure appl. math., 56, 1416-1464, (2003) · Zbl 1155.35353
[5] Li, A.; Li, Y.Y., On some conformally invariant fully nonlinear equations, part II: Liouville, Harnack and Yamabe, Acta math., 195, 117-154, (2005) · Zbl 1216.35038
[6] Viaclovsky, J., Conformal geometry, contact geometry, and the calculus of variations, Duke math. J., 101, 283-316, (2000) · Zbl 0990.53035
[7] Chang, S.Y.A.; Gursky, M.; Yang, P., An a priori estimate for a fully nonlinear equation on four-manifolds, J. anal. math., 87, 151-186, (2002) · Zbl 1067.58028
[8] Alexandrov, A.D., Uniqueness theorems for surfraces in the large. V, Vestn. leningr. univ. ser. mat. mekh. astron., Amer. math. soc. transl., 21, 412-416, (1962) · Zbl 0119.16603
[9] Serrin, J., A symmetry problem in potential theory, Arch. ration. mech. anal., 43, 304-318, (1971) · Zbl 0222.31007
[10] Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), Math. anal. appl., 7A, 369-402, (1981)
[11] Berestycki, H.; Nirenberg, L., On the method of moving planes and the sliding method, Bol. soc. brasil. mat. (NS), 22, 1-37, (1991) · Zbl 0784.35025
[12] Chen, W.X.; Li, C.M., Classification of solutions of some nonlinear elliptic equations, Duke math. J., 63, 615-622, (1991) · Zbl 0768.35025
[13] Li, Y.Y.; Zhu, M., Uniqueness theorems through the method of moving spheres, Duke math., 80, 2, 383-417, (1995) · Zbl 0846.35050
[14] Li, Y.Y., Remark on some conformally invariant integral equations: the method of moving spheres, J. eur. math. soc., 6, 153-180, (2004) · Zbl 1075.45006
[15] Li, Y.Y.; Zhang, L., Liouvelle-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. anal. math., 90, 27-87, (2003)
[16] Zhang, Y.J.; Hao, J.H., A remark on an integral equation via the method of moving spheres, J. math. anal. appl., 344, 682-686, (2008) · Zbl 1148.45005
[17] Chen, W.X.; Li, C.M.; Ou, B., Classification of solutions for an integral equation, Comm. pure appl. math., 59, 330-343, (2006) · Zbl 1093.45001
[18] Chen, W.X.; Li, C.M.; Ou, B., Classification of solutions for a system of integral equations, Comm. partial differential equations, 30, 59-65, (2005) · Zbl 1073.45005
[19] Chen, W.X.; Li, C.M., Regularity of solutions for a system of integral equations, Commun. pure appl. anal., 4, 1-8, (2005) · Zbl 1073.45004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.