×

A classification of positive solutions of some integral systems. (English) Zbl 1227.45005

The authors examine the following system of integral equations:
\[ \begin{aligned} & u(x)=\int_{\mathbb R^n}|x-y|^{\alpha-n}e^{v(y)}dy,\\ & v(x)=\int_{\mathbb R^n}|x-y|^{\beta-n}e^{u(y)}dy, \end{aligned} \]
where \(0<\alpha,\beta<n\). The main result of this paper states that if \(\{u,v\}\) is a pair of positive solutions to the above system such that \(e^{u(y)}\in L^r(\mathbb R^n)\) and \(e^{v(y)}\in L^s(\mathbb R^n)\), where \(r,s>1\), \(\frac{n}{s}+\frac{n}{r}=\alpha+\beta\), then \(u\) and \(v\) are radially symmetric and monotone decreasing about some point \(x_0\in\mathbb R^n\). The proofs are based on the method of moving planes.

MSC:

45G15 Systems of nonlinear integral equations
45M20 Positive solutions of integral equations
45G05 Singular nonlinear integral equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alexandroff A.D.: A characteristic property of the spheres. Ann. Math. Pura. Appl. 58, 303–354 (1962) · Zbl 0107.15603
[2] Adams R.: Sobolev Spaces, Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975) · Zbl 0314.46030
[3] Chen D., Ma L.: Radial symmetry and monotonicity for an integral equation. J. Math. Anal. Appl. 342, 943–949 (2008) · Zbl 1136.68048
[4] Chen D., Ma L.: Radial symmetry and uniqueness for positive solutions of a Schröinger type system. Math. Comput. Model. 49, 379–385 (2009) · Zbl 1176.34069
[5] Caffarelli L., Gidas B., Spruck J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42, 271–297 (1989) · Zbl 0702.35085
[6] Chen W., Li C.: Regularity of solutions for a system of integral equations. Commun. Pure Appl. Anal. 4, 1–8 (2005) · Zbl 1073.45004
[7] Chen W., Li C., Ou B.: Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59, 330–343 (2006) · Zbl 1093.45001
[8] Chen W., Li C., Ou B.: Classification of solutions for a system of integral equations. Commun. Partial Differ. Equ. 30, 59C65 (2005) · Zbl 1073.45005
[9] Chen W., Li C., Ou B.: Qualitative properties of solutions for an integral equation. Disc. Cont. Dyn. Syst. 12, 347–354 (2005) · Zbl 1081.45003
[10] Gidas B., Ni W.M., Nirenberg L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979) · Zbl 0425.35020
[11] Jin C., Li C.: Symmetry of solutions to some systems of integral equations. Proc. Am. Math. Sci. 134, 1661–1670 (2005) · Zbl 1156.45300
[12] Li C., Lim J.: The singularity analysis of solutions to some integral equations. Commun. Pure Appl. Anal. 6, 453–464 (2007) · Zbl 1141.42024
[13] Lin C.: A classification of solutions of a conformally invariant fourth order equation in R n . Comment. Math. Helv. 73, 206–231 (1998) · Zbl 0933.35057
[14] Li D., Ströhmer G., Wang L.: Symmetry of integral equations on bounded domains. Proc. Am. Math. Soc. 137, 3695–3702 (2009) · Zbl 1188.45001
[15] Li Y.: Prescribing scalar curvature on S n and related problems, part 1. J. Differ. Equ. 120, 319–410 (1995) · Zbl 0827.53039
[16] Serrin J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318 (1971) · Zbl 0222.31007
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.