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 & 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,, \emph{Ann. Inst. Fourier}, 11 (1961), 385. 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,, \emph{Comm. partial differential Equations}, 22 (1997), 1671. 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,, \emph{Arch. Rational Mech. Anal.}, 46 (1972), 81. Google Scholar

[4]

L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 3937. 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 $\mathbbR^n_+$,, \emph{J. Math. Anal. Appl.}, 389 (2012), 1365. 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 $\mathbbR^n_+$,, \emph{J. Funct. Anal.}, 256 (2013), 1522. 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,, \emph{Disc. Cont. Dyn. Sys.}, (2005), 164. 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,, \emph{Comm. Pure Appl. Anal.}, 4 (2005), 1. Google Scholar

[11]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, \emph{Disc. Cont. Dyn. Sys.}, 4 (2009), 1167. 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,, \emph{Disc. Cont. Dyn. Sys.}, 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083. Google Scholar

[13]

W. Chen and C. Li, A sup + inf inequality near $R = 0$,, \emph{Advances in Math}, 220 (2009), 219. 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,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[15]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, \emph{Disc. Cont. Dyn. Sys.}, 12 (2005), 347. Google Scholar

[16]

W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems,, \emph{J. Math. Anal. Appl.}, 377 (2011), 744. 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,, \emph{Rev. Mat. Iberoam.}, 20 (2004), 67. Google Scholar

[18]

W. F. Donoghue, Distributions and Fourier Transforms,, Vol. 32, (1969). Google Scholar

[19]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, \emph{Adv. Math.}, 229 (2012), 2835. 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,, \emph{Comm. Partial Differential Equations}, 34 (1981), 525. doi: 10.1080/03605308108820196. Google Scholar

[21]

L. Grafakos, Classical and Modern Fourier Analysis,, Pearson Education, (2004). Google Scholar

[22]

X. Han and G. Lu, Regularity of solutions to an integral equation associated with bessel potential,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 1111. doi: 10.3934/cpaa.2011.10.1111. Google Scholar

[23]

M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, \emph{Arch. Rational Mech. Anal.}, 105 (1989), 243. doi: 10.1007/BF00251502. Google Scholar

[24]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, \emph{Nonlinear Anal.}, 71 (2009), 4231. 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,, \emph{Comm. Pure Appl. Anal.}, 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453. Google Scholar

[26]

C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, \emph{Comm. Pure Appl. Anal.}, 6 (2009), 1925. doi: 10.3934/cpaa.2009.8.1925. Google Scholar

[27]

Y. Li and M. Zhu, Uniqueness theorems through the mothod of moving spheres,, \emph{Duke Math. J.}, 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8. Google Scholar

[28]

L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation,, \emph{J. Math. Anal. Appl.}, 342 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064. Google Scholar

[29]

L. Ma and D. Chen, A Liouville type theorem for an integral system,, \emph{Commun. Pure Appl. Anal.}, 5 (2006), 855. 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,, \emph{Math. Comput. Modelling}, 49 (2009), 379. 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,, \emph{Adv. Math.}, 226 (2011), 2676. 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,, \emph{Adv. Math.}, 225 (2010), 3052. 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 $R^n$,, \emph{Arch. Rational Mech. Anal.}, 99 (1987), 115. 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,, \emph{Math. Z.}, 261 (2009), 805. doi: 10.1007/s00209-008-0352-3. Google Scholar

[35]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970). Google Scholar

[36]

Y. Zhao, Regularity and symmetry for solutions to a system of weighted integral equations,, \emph{J. Math. Anal. Appl.}, 391 (2012), 209. 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,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 977. 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, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

show all references

References:
[1]

N. Aronszajn and K. T. Smith, Theory of Bessel potentials I,, \emph{Ann. Inst. Fourier}, 11 (1961), 385. 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,, \emph{Comm. partial differential Equations}, 22 (1997), 1671. 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,, \emph{Arch. Rational Mech. Anal.}, 46 (1972), 81. Google Scholar

[4]

L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 3937. 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 $\mathbbR^n_+$,, \emph{J. Math. Anal. Appl.}, 389 (2012), 1365. 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 $\mathbbR^n_+$,, \emph{J. Funct. Anal.}, 256 (2013), 1522. 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,, \emph{Disc. Cont. Dyn. Sys.}, (2005), 164. 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,, \emph{Comm. Pure Appl. Anal.}, 4 (2005), 1. Google Scholar

[11]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, \emph{Disc. Cont. Dyn. Sys.}, 4 (2009), 1167. 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,, \emph{Disc. Cont. Dyn. Sys.}, 30 (2011), 1083. doi: 10.3934/dcds.2011.30.1083. Google Scholar

[13]

W. Chen and C. Li, A sup + inf inequality near $R = 0$,, \emph{Advances in Math}, 220 (2009), 219. 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,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[15]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, \emph{Disc. Cont. Dyn. Sys.}, 12 (2005), 347. Google Scholar

[16]

W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems,, \emph{J. Math. Anal. Appl.}, 377 (2011), 744. 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,, \emph{Rev. Mat. Iberoam.}, 20 (2004), 67. Google Scholar

[18]

W. F. Donoghue, Distributions and Fourier Transforms,, Vol. 32, (1969). Google Scholar

[19]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space,, \emph{Adv. Math.}, 229 (2012), 2835. 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,, \emph{Comm. Partial Differential Equations}, 34 (1981), 525. doi: 10.1080/03605308108820196. Google Scholar

[21]

L. Grafakos, Classical and Modern Fourier Analysis,, Pearson Education, (2004). Google Scholar

[22]

X. Han and G. Lu, Regularity of solutions to an integral equation associated with bessel potential,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 1111. doi: 10.3934/cpaa.2011.10.1111. Google Scholar

[23]

M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, \emph{Arch. Rational Mech. Anal.}, 105 (1989), 243. doi: 10.1007/BF00251502. Google Scholar

[24]

S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations,, \emph{Nonlinear Anal.}, 71 (2009), 4231. 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,, \emph{Comm. Pure Appl. Anal.}, 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453. Google Scholar

[26]

C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations,, \emph{Comm. Pure Appl. Anal.}, 6 (2009), 1925. doi: 10.3934/cpaa.2009.8.1925. Google Scholar

[27]

Y. Li and M. Zhu, Uniqueness theorems through the mothod of moving spheres,, \emph{Duke Math. J.}, 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8. Google Scholar

[28]

L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation,, \emph{J. Math. Anal. Appl.}, 342 (2008), 943. doi: 10.1016/j.jmaa.2007.12.064. Google Scholar

[29]

L. Ma and D. Chen, A Liouville type theorem for an integral system,, \emph{Commun. Pure Appl. Anal.}, 5 (2006), 855. 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,, \emph{Math. Comput. Modelling}, 49 (2009), 379. 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,, \emph{Adv. Math.}, 226 (2011), 2676. 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,, \emph{Adv. Math.}, 225 (2010), 3052. 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 $R^n$,, \emph{Arch. Rational Mech. Anal.}, 99 (1987), 115. 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,, \emph{Math. Z.}, 261 (2009), 805. doi: 10.1007/s00209-008-0352-3. Google Scholar

[35]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970). Google Scholar

[36]

Y. Zhao, Regularity and symmetry for solutions to a system of weighted integral equations,, \emph{J. Math. Anal. Appl.}, 391 (2012), 209. 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,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 977. 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, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

[1]

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

[2]

Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure & Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663

[3]

Lu Chen, Zhao Liu, Guozhen Lu. Qualitative properties of solutions to an integral system associated with the Bessel potential. Communications on Pure & Applied Analysis, 2016, 15 (3) : 893-906. doi: 10.3934/cpaa.2016.15.893

[4]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155

[5]

Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure & Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385

[6]

Xiaotao Huang, Lihe Wang. Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1121-1134. doi: 10.3934/cpaa.2017054

[7]

Roman Chapko. On a Hybrid method for shape reconstruction of a buried object in an elastostatic half plane. Inverse Problems & Imaging, 2009, 3 (2) : 199-210. doi: 10.3934/ipi.2009.3.199

[8]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[9]

Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475

[10]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957

[11]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925

[12]

Yutian Lei. Positive solutions of integral systems involving Bessel potentials. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2721-2737. doi: 10.3934/cpaa.2013.12.2721

[13]

Mingchun Wang, Jiankai Xu, Huoxiong Wu. On Positive solutions of integral equations with the weighted Bessel potentials. Communications on Pure & Applied Analysis, 2019, 18 (2) : 625-641. doi: 10.3934/cpaa.2019031

[14]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061

[15]

C. H. Arthur Cheng, John M. Hong, Ying-Chieh Lin, Jiahong Wu, Juan-Ming Yuan. Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 763-779. doi: 10.3934/dcdsb.2016.21.763

[16]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042

[17]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[18]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225

[19]

Lauri Harhanen, Nuutti Hyvönen. Convex source support in half-plane. Inverse Problems & Imaging, 2010, 4 (3) : 429-448. doi: 10.3934/ipi.2010.4.429

[20]

Paola Trebeschi. On the slightly compressible MHD system in the half-plane. Communications on Pure & Applied Analysis, 2004, 3 (1) : 97-113. doi: 10.3934/cpaa.2004.3.97

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]