# American Institute of Mathematical Sciences

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 and 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 $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst., 36 (2016), 577-600. [2] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [3] A. Chang and P. Yang, On uniqueness of solutions of nth order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102. doi: 10.4310/MRL.1997.v4.n1.a9. [4] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys., Vol. 4, 2010. [5] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [6] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. [7] 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. [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65. [9] 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. [10] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [11] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, vol. 7a of the book series Advances in Mathematics, Academic Press, New York, 1981. [12] X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potential, Commun. Pure Appl. Anal., 10 (2011), 1111-1119. doi: 10.3934/cpaa.2011.10.1111. [13] X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602. [14] C. Jin and C. Li, Symmetry of solution to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [15] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [16] N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., 32 (2012), 2187-2205. doi: 10.3934/dcds.2012.32.2187. [17] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453. [18] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. [19] 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. [20] W. Reichel, Characterization of balls by Riesz-potentials, Ann. Mat., 188 (2009), 235-245. doi: 10.1007/s10231-008-0073-6. [21] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318. [22] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York, Berlin, 1983. [23] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Ser. Appl. Math., vol. 32, Princeton Univ. Press, Princeton, NJ, 1970.

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##### References:
 [1] J. Bao, N. Lam and G. Lu, Polyharmonic equations with critical exponential growth in the whole space $\mathbb{R}^{N}$, Discrete Contin. Dyn. Syst., 36 (2016), 577-600. [2] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [3] A. Chang and P. Yang, On uniqueness of solutions of nth order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102. doi: 10.4310/MRL.1997.v4.n1.a9. [4] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys., Vol. 4, 2010. [5] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [6] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. [7] 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. [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65. [9] 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. [10] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. [11] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, vol. 7a of the book series Advances in Mathematics, Academic Press, New York, 1981. [12] X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potential, Commun. Pure Appl. Anal., 10 (2011), 1111-1119. doi: 10.3934/cpaa.2011.10.1111. [13] X. Han, G. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602. [14] C. Jin and C. Li, Symmetry of solution to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [15] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [16] N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete Contin. Dyn. Syst., 32 (2012), 2187-2205. doi: 10.3934/dcds.2012.32.2187. [17] C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Commun. Pure Appl. Anal., 6 (2007), 453-464. doi: 10.3934/cpaa.2007.6.453. [18] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 226 (2011), 2676-2699. [19] 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. [20] W. Reichel, Characterization of balls by Riesz-potentials, Ann. Mat., 188 (2009), 235-245. doi: 10.1007/s10231-008-0073-6. [21] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318. [22] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer-Verlag, New York, Berlin, 1983. [23] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Ser. Appl. Math., vol. 32, Princeton Univ. Press, Princeton, NJ, 1970.
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