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Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy–Littlewood–Sobolev system of integral equations

  • Yutian Lei
  • Congming LiEmail author
  • Chao Ma
Article

Abstract

In this article, we study the asymptotics of the positive solutions of the Euler–Lagrange system of the weighted Hardy–Littlewood–Sobolev in R n
$$\begin{array}{ll} u(x) = \frac{1}{|x|^{\alpha}}\int\limits_{R^{n}} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy,\\ v(x) = \frac{1}{|x|^{\beta}}\int\limits_{R^{n}} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}} dy.\end{array}$$
A new iterative method is introduced to obtain the optimal weighted local integrability of u(x). By this new method, we establish the asymptotic estimates of the solutions around the origin and near infinity. With these new estimates, we complete the study of the asymptotic behavior of the solutions. We believe this new iterative method and the new type of the weighted local estimates can be used in many other cases.

Mathematics Subject Classification (2000)

45E10 45G05 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.School of Mathematical SciencesNanjing Normal UniversityNanjingChina
  2. 2.Department of Applied MathematicsUniversity of Colorado BoulderBoulderUSA
  3. 3.Department of MathematicsUniversity of Colorado BoulderBoulderUSA

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