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

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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

Lu Chen ^1, , Zhao Liu ^1, and Guozhen Lu ^2,

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

Abstract
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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.

Keywords: regularity, uniqueness., method of moving planes in integral forms, radial symmetry, Bessel potential.

Mathematics Subject Classification: Primary: 35J48; Secondary: 35B06, 45G1.

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

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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

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