In this paper, we discuss the integral equation on a half space R+n
where is the reflection of the point x about the . We study the regularity for the positive solutions of (0.1). A regularity lifting method by contracting operators is used in proving the boundedness of solutions, and the Lipschitz continuity is derived by combinations of contracting and shrinking operators introduced by Ma-Chen-Li ().
 L. Ma and D. Chen, “Radial Symmetry and Monotonicity Results for an Integral Equation,” Journal of Mathematical Analysis and Applications, Vol. 342, No. 2, 2008, pp. 943-949. doi:10.1016/j.jmaa.2007.12.064
 L. Ma and D. Chen, “Radial Symmetry and Uniqueness of Non-Negative Solutions to an Integral System,” Mathematical and Computer Modelling, Vol. 49, No. 1-2, 2009, pp. 379-385. doi:10.1016/j.mcm.2008.06.010