An integral equation involving Bessel potentials on half space

Yonggang Zhao; Mingxin Wang

doi:10.3934/cpaa.2015.14.527

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

An integral equation involving Bessel potentials on half space

Yonggang Zhao ^1, and Mingxin Wang ^2,

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

Abstract
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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.

Keywords: nonexistence., monotonicity, moving plane method, half space, regularity, Integral equation, Bessel potential.

Mathematics Subject Classification: Primary: 45E10; Secondary: 35G30, 35J6.

Citation: Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains,, arXiv:1309.7499v1 [math.AP]., (). Google Scholar
[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. Google Scholar
[9]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. $&$ Dyn. Sys., 4 (2010). Google Scholar
[10]	W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005), 1-8. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	W. F. Donoghue, Distributions and Fourier Transforms, Vol. 32, Academic Press, Inc., New York, 1969. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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References:

[1]	N. Aronszajn and K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier, 11 (1961), 385-475. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	W. Chen, Y. Fang and R. Yang, Semilinear equations involving the fractional Laplacian on domains,, arXiv:1309.7499v1 [math.AP]., (). Google Scholar
[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. Google Scholar
[9]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. $&$ Dyn. Sys., 4 (2010). Google Scholar
[10]	W. Chen and C. Li, Regularity of solutions for a system of integral equations, Comm. Pure Appl. Anal., 4 (2005), 1-8. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	W. F. Donoghue, Distributions and Fourier Transforms, Vol. 32, Academic Press, Inc., New York, 1969. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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