May  2016, 15(3): 893-906. doi: 10.3934/cpaa.2016.15.893

Qualitative properties of solutions to an integral system associated with the Bessel potential

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China, China

2. 

Department of Mathematics, Wayne State University, Detroit, MI 48202

Received  August 2015 Revised  November 2015 Published  February 2016

In this paper, we study a differential system associated with the Bessel potential: \begin{eqnarray}\begin{cases} (I-\Delta)^{\frac{\alpha}{2}}u(x)=f_1(u(x),v(x)),\\ (I-\Delta)^{\frac{\alpha}{2}}v(x)=f_2(u(x),v(x)), \end{cases}\end{eqnarray} where $f_1(u(x),v(x))=\lambda_1u^{p_1}(x)+\mu_1v^{q_1}(x)+\gamma_1u^{\alpha_1}(x)v^{\beta_1}(x)$, $f_2(u(x),v(x))=\lambda_2u^{p_2}(x)+\mu_2v^{q_2}(x)+\gamma_2u^{\alpha_2}(x)v^{\beta_2}(x)$, $I$ is the identity operator and $\Delta=\sum_{j=1}^{n}\frac{\partial^2}{\partial x^2_j}$ is the Laplacian operator in $\mathbb{R}^n$. Under some appropriate conditions, this differential system is equivalent to an integral system of the Bessel potential type. By the regularity lifting method developed in [4] and [18], we obtain the regularity of solutions to the integral system. We then apply the moving planes method to obtain radial symmetry and monotonicity of positive solutions. We also establish the uniqueness theorem for radially symmetric solutions. Our nonlinear terms $f_1(u(x), v(x))$ and $f_2(u(x), v(x))$ are quite general and our results extend the earlier ones even in the case of single equation substantially.
Citation: 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
References:
[1]

J. Bao, N. Lam and G. Lu, Polyharmonic equations with critical exponential growth in the whole space $\mathbbR^n$,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 577.   Google Scholar

[2]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth,, \emph{Commun. Pure Appl. Math.}, 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

A. Chang and P. Yang, On uniqueness of solutions of nth order differential equations in conformal geometry,, \emph{Math. Res. Lett.}, 4 (1997), 91.  doi: 10.4310/MRL.1997.v4.n1.a9.  Google Scholar

[4]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series on Diff. Equa. and Dyn. Sys., (2010).   Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

W. Chen and C. Li, Regularity of solutions for a system of integral equations,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 1.   Google Scholar

[7]

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

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. Patial Differential Equations}, 30 (2005), 59.   Google Scholar

[9]

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

[10]

B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.   Google Scholar

[11]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$,, \emph{Mathematical Analysis and Applications}, (1981).   Google Scholar

[12]

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

[13]

X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation,, \emph{J. Differential Equations}, 252 (2012), 1589.   Google Scholar

[14]

C. Jin and C. Li, Symmetry of solution to some systems of integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar

[15]

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

[16]

N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 2187.  doi: 10.3934/dcds.2012.32.2187.  Google Scholar

[17]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, \emph{Commun. Pure Appl. Anal.}, 6 (2007), 453.  doi: 10.3934/cpaa.2007.6.453.  Google Scholar

[18]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Adv. Math.}, 226 (2011), 2676.   Google Scholar

[19]

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

[20]

W. Reichel, Characterization of balls by Riesz-potentials,, \emph{Ann. Mat.}, 188 (2009), 235.  doi: 10.1007/s10231-008-0073-6.  Google Scholar

[21]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 304.   Google Scholar

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Grundlehren der Mathematischen Wissenschaften, (1983).   Google Scholar

[23]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Ser. Appl. Math., (1970).   Google Scholar

show all references

References:
[1]

J. Bao, N. Lam and G. Lu, Polyharmonic equations with critical exponential growth in the whole space $\mathbbR^n$,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 577.   Google Scholar

[2]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth,, \emph{Commun. Pure Appl. Math.}, 42 (1989), 271.  doi: 10.1002/cpa.3160420304.  Google Scholar

[3]

A. Chang and P. Yang, On uniqueness of solutions of nth order differential equations in conformal geometry,, \emph{Math. Res. Lett.}, 4 (1997), 91.  doi: 10.4310/MRL.1997.v4.n1.a9.  Google Scholar

[4]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Book Series on Diff. Equa. and Dyn. Sys., (2010).   Google Scholar

[5]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[6]

W. Chen and C. Li, Regularity of solutions for a system of integral equations,, \emph{Commun. Pure Appl. Anal.}, 4 (2005), 1.   Google Scholar

[7]

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

[8]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. Patial Differential Equations}, 30 (2005), 59.   Google Scholar

[9]

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

[10]

B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.   Google Scholar

[11]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$,, \emph{Mathematical Analysis and Applications}, (1981).   Google Scholar

[12]

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

[13]

X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation,, \emph{J. Differential Equations}, 252 (2012), 1589.   Google Scholar

[14]

C. Jin and C. Li, Symmetry of solution to some systems of integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661.  doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar

[15]

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

[16]

N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 2187.  doi: 10.3934/dcds.2012.32.2187.  Google Scholar

[17]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, \emph{Commun. Pure Appl. Anal.}, 6 (2007), 453.  doi: 10.3934/cpaa.2007.6.453.  Google Scholar

[18]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Adv. Math.}, 226 (2011), 2676.   Google Scholar

[19]

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

[20]

W. Reichel, Characterization of balls by Riesz-potentials,, \emph{Ann. Mat.}, 188 (2009), 235.  doi: 10.1007/s10231-008-0073-6.  Google Scholar

[21]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Ration. Mech. Anal.}, 43 (1971), 304.   Google Scholar

[22]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Grundlehren der Mathematischen Wissenschaften, (1983).   Google Scholar

[23]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Ser. Appl. Math., (1970).   Google Scholar

[1]

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

[2]

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

[3]

Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015

[4]

Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235

[5]

Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082

[6]

Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102

[7]

Wenxiong Chen, Congming Li. Radial symmetry of solutions for some integral systems of Wolff type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1083-1093. doi: 10.3934/dcds.2011.30.1083

[8]

Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685

[9]

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

[10]

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

[11]

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

[12]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201

[13]

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

[14]

Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121

[15]

Martin Bauer, Thomas Fidler, Markus Grasmair. Local uniqueness of the circular integral invariant. Inverse Problems & Imaging, 2013, 7 (1) : 107-122. doi: 10.3934/ipi.2013.7.107

[16]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure & Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041

[17]

Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1

[18]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395

[19]

Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41

[20]

Dongbing Zha. Remarks on nonlinear elastic waves in the radial symmetry in 2-D. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4051-4062. doi: 10.3934/dcds.2016.36.4051

2018 Impact Factor: 0.925

Metrics

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

Other articles
by authors

[Back to Top]