# American Institute of Mathematical Sciences

March  2015, 14(2): 527-548. doi: 10.3934/cpaa.2015.14.527

## An integral equation involving Bessel potentials on half space

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150080, China 2 Science Research Center, Harbin Institute of Technology, Harbin, 150080

Received  February 2014 Revised  September 2014 Published  December 2014

In this article we consider the following integral equation involving Bessel potentials on a half space $\mathbb{R}^n_+$: \begin{eqnarray} u(x)=\int_{ \mathbb{R}^n_+ }\{g_\alpha(x-y)-g_\alpha(\bar x-y)\} u^\beta(y) dy,\;\;x\in \mathbb{R}^n_+, \end{eqnarray} where $\alpha>0$, $\beta>1$, $\bar x$ is the reflection of $x$ about $x_n=0$, and $g_\alpha(x)$ denotes the Bessel kernel. We first enhance the regularity of positive solutions for the integral equation by regularity-lifting-method, which has been extensively used by many authors. Then, employing the method of moving planes in integral forms, we demonstrate that there is no positive solution for the integral equation.
Citation: Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure and Applied Analysis, 2015, 14 (2) : 527-548. doi: 10.3934/cpaa.2015.14.527
##### References:
 [1] N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. [2] G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ or $\mathbbR^n_+$ through the method of moving planes, Comm. partial differential Equations, 22 (1997), 1671-1690. doi: 10.1080/03605309708821315. [3] C. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3=0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. [4] L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937. [5] L. Cao and Z, Dai, A Liouville-type theorem for an integral equation on a half-space $\mathbbR^n_+$, J. Math. Anal. Appl., 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015. [6] W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems in $\mathbbR^n_+$, J. Funct. Anal., 256 (2013), 1522-1555. doi: 10.1016/j.jfa.2013.06.010. [7] W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv:1309.7499v1 [math.AP]. [8] W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Disc. Cont. Dyn. Sys., S (2005), 164-172. [9] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010). [10] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005), 1-8. [11] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [12] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [13] W. Chen and C. Li, A sup + inf inequality near $R = 0$, Advances in Math, 220 (2009), 219-245. doi: 10.1016/j.aim.2008.09.005. [14] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [15] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [16] W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753. doi: 10.1016/j.jmaa.2010.11.035. [17] L. Damascelli and F. Gladiali, Some nonexistence results for positive soluions of elliptic equantions in unbounded domain, Rev. Mat. Iberoam., 20 (2004), 67-86. [18] W. F. Donoghue, Distributions and Fourier Transforms, Vol. 32, Academic Press, Inc., New York, 1969. [19] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [20] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 34 (1981), 525-598. doi: 10.1080/03605308108820196. [21] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. [22] X. Han and G. Lu, Regularity of solutions to an integral equation associated with bessel potential, Comm. Pure Appl. Anal., 10 (2011), 1111-1119. doi: 10.3934/cpaa.2011.10.1111. [23] M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [24] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 4231-4240. doi: 10.1016/j.na.2009.01.014. [25] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453. [26] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [27] Y. Li and M. Zhu, Uniqueness theorems through the mothod of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8. [28] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064. [29] L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [30] L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type system, Math. Comput. Modelling, 49 (2009), 379-385. doi: 10.1016/j.mcm.2008.06.010. [31] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [32] L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space, Adv. Math., 225 (2010), 3052-3063. doi: 10.1016/j.aim.2010.05.022. [33] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $R^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145. doi: 10.1007/BF00275874. [34] W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827. doi: 10.1007/s00209-008-0352-3. [35] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. [36] Y. Zhao, Regularity and symmetry for solutions to a system of weighted integral equations, J. Math. Anal. Appl., 391 (2012), 209-222. doi: 10.1016/j.jmaa.2012.02.016. [37] R. Zhuo, F. Li and B. Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 13 (2014), 977-990. doi: 10.3934/cpaa.2014.13.977. [38] W. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, in: Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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##### References:
 [1] N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. [2] G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in $R^N$ or $\mathbbR^n_+$ through the method of moving planes, Comm. partial differential Equations, 22 (1997), 1671-1690. doi: 10.1080/03605309708821315. [3] C. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3=0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. [4] L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937. [5] L. Cao and Z, Dai, A Liouville-type theorem for an integral equation on a half-space $\mathbbR^n_+$, J. Math. Anal. Appl., 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015. [6] W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems in $\mathbbR^n_+$, J. Funct. Anal., 256 (2013), 1522-1555. doi: 10.1016/j.jfa.2013.06.010. [7] W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains, arXiv:1309.7499v1 [math.AP]. [8] W. Chen, C. Jin, C. Li and C. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Disc. Cont. Dyn. Sys., S (2005), 164-172. [9] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. & Dyn. Sys., 4 (2010). [10] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005), 1-8. [11] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 4 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167. [12] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [13] W. Chen and C. Li, A sup + inf inequality near $R = 0$, Advances in Math, 220 (2009), 219-245. doi: 10.1016/j.aim.2008.09.005. [14] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [15] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [16] W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753. doi: 10.1016/j.jmaa.2010.11.035. [17] L. Damascelli and F. Gladiali, Some nonexistence results for positive soluions of elliptic equantions in unbounded domain, Rev. Mat. Iberoam., 20 (2004), 67-86. [18] W. F. Donoghue, Distributions and Fourier Transforms, Vol. 32, Academic Press, Inc., New York, 1969. [19] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [20] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 34 (1981), 525-598. doi: 10.1080/03605308108820196. [21] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. [22] X. Han and G. Lu, Regularity of solutions to an integral equation associated with bessel potential, Comm. Pure Appl. Anal., 10 (2011), 1111-1119. doi: 10.3934/cpaa.2011.10.1111. [23] M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [24] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 4231-4240. doi: 10.1016/j.na.2009.01.014. [25] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453. [26] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Comm. Pure Appl. Anal., 6 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [27] Y. Li and M. Zhu, Uniqueness theorems through the mothod of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8. [28] L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl., 342 (2008), 943-949. doi: 10.1016/j.jmaa.2007.12.064. [29] L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [30] L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type system, Math. Comput. Modelling, 49 (2009), 379-385. doi: 10.1016/j.mcm.2008.06.010. [31] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [32] L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space, Adv. Math., 225 (2010), 3052-3063. doi: 10.1016/j.aim.2010.05.022. [33] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $R^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145. doi: 10.1007/BF00275874. [34] W. Reichel and T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827. doi: 10.1007/s00209-008-0352-3. [35] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. [36] Y. Zhao, Regularity and symmetry for solutions to a system of weighted integral equations, J. Math. Anal. Appl., 391 (2012), 209-222. doi: 10.1016/j.jmaa.2012.02.016. [37] R. Zhuo, F. Li and B. Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 13 (2014), 977-990. doi: 10.3934/cpaa.2014.13.977. [38] W. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, in: Graduate Texts in Mathematics, Vol. 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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