# American Institute of Mathematical Sciences

July  2011, 10(4): 1111-1119. doi: 10.3934/cpaa.2011.10.1111

## Regularity of solutions to an integral equation associated with Bessel potential

 1 Department of Mathematics, Wayne State University, Detroit, MI 48202, United States

Received  July 2010 Revised  December 2010 Published  April 2011

In this paper, we study the regularity of the positive solutions to an integral equation associated with the Bessel potential. The kernel estimates for the Bessel potential plays an essential role in deriving such regularity results. First, we apply the regularity lifting by contracting operators to get the $L^\infty$ estimate. Then, we use the regularity lifting by combinations of contracting and shrinking operators, which was recently developed in [4] and [5], to prove the Lipschitz continuity estimate. Our regularity results here have been recently extended to positive solutions to an integral system associated with Bessel potential [9].
Citation: Xiaolong Han, Guozhen Lu. Regularity of solutions to an integral equation associated with Bessel potential. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1111-1119. doi: 10.3934/cpaa.2011.10.1111
##### References:
 [1] W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations,, Discrete Contin. Dyn. Syst., (2005), 164. Google Scholar [2] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Commun. Pure Appl. Anal., 4 (2005), 1. doi: 10.3934/cpaa.2005.4.1. Google Scholar [3] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167. Google Scholar [4] W. Chen and C. Li, "Methods on Nonliear Elliptic Equations,'', AIMS Series on Differential Equations and Dynamical Systems, (2010). Google Scholar [5] W. Chen, C. Li and C. Ma, Regularity of solutions for an integral system of Wolff type,, preprint., (). Google Scholar [6] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347. doi: 0.3934/dcds.2005.12.347. Google Scholar [7] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Commun. Pure Appl. Math, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar [8] L. Grafakos, "Classical and Modern Fourier Analysis,'', Pearson Education, (2004). Google Scholar [9] X. Han and G. Lu, Regularity of solutions to an integral system of Bessel potential,, preprint., (). Google Scholar [10] Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc. (JEMS), 6 (2004), 153. doi: 10.4171/JEMS/6. Google Scholar [11] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Commun. Pure Appl. Anal., 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453. Google Scholar [12] M. Li and D. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855. Google Scholar [13] M. Li and D. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 342 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064. Google Scholar [14] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'', Princeton Mathematical Series, (1970). Google Scholar

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##### References:
 [1] W. Chen, C. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations,, Discrete Contin. Dyn. Syst., (2005), 164. Google Scholar [2] W. Chen and C. Li, Regularity of solutions for a system of integral equations,, Commun. Pure Appl. Anal., 4 (2005), 1. doi: 10.3934/cpaa.2005.4.1. Google Scholar [3] W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167. Google Scholar [4] W. Chen and C. Li, "Methods on Nonliear Elliptic Equations,'', AIMS Series on Differential Equations and Dynamical Systems, (2010). Google Scholar [5] W. Chen, C. Li and C. Ma, Regularity of solutions for an integral system of Wolff type,, preprint., (). Google Scholar [6] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete Contin. Dyn. Syst., 12 (2005), 347. doi: 0.3934/dcds.2005.12.347. Google Scholar [7] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Commun. Pure Appl. Math, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar [8] L. Grafakos, "Classical and Modern Fourier Analysis,'', Pearson Education, (2004). Google Scholar [9] X. Han and G. Lu, Regularity of solutions to an integral system of Bessel potential,, preprint., (). Google Scholar [10] Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc. (JEMS), 6 (2004), 153. doi: 10.4171/JEMS/6. Google Scholar [11] C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, Commun. Pure Appl. Anal., 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453. Google Scholar [12] M. Li and D. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855. Google Scholar [13] M. Li and D. Chen, Radial symmetry and monotonicity for an integral equation,, J. Math. Anal. Appl., 342 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064. Google Scholar [14] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'', Princeton Mathematical Series, (1970). Google Scholar
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