Regularity of Solutions to an Integral Equation on a Half-Space *R*_{+}^{n}

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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 ([1]).

References

[1] C. Ma, W. Chen and C. Li, “Regularity of Solutions for an Integral System of Wolff Type,” Advances in Mathematics, Vol. 226, No. 3, 2011, pp. 2676-2699.

[2] D. Li and R. Zhuo, “An Integral Equation on Half Space,” Proceedings of the American Mathematical Society, Vol. 138, 2010, pp. 2779-2791.
doi:10.1090/S0002-9939-10-10368-2

[3] W. Chen and C. Li, “The Equivalence between Integral Systems and PDE Systems,” Preprint, 2010.

[4] W. Chen and C. Li, “Regularity of Solutions for a System of Integral Equations,” Communications on Pure and Applied Analysis, Vol. 4, No. 1, 2005, pp. 1-8.

[5] 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

[6] 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

[7] X. Han and G. Lu, “Regularity of Solutions to an Integral Equation Associated with Bessel Potential,” Communications on Pure and Applied Analysis, Vol. 10, No. 4, 2011, pp. 1111-1119.

[8] W. Chen and C. Li, “Methods on Nonlinear Elliptic Equations. AIMS Series on Differential Equations and Dynamical Systems,” American Institute of Mathematical Science (AIMS), Springfield, 2010.