# American Institute of Mathematical Sciences

March 2019, 18(2): 625-641. doi: 10.3934/cpaa.2019031

## On Positive solutions of integral equations with the weighted Bessel potentials

 1 College of mathematical econometric, Hunan university, Hunan, Changsha, 410082, China 2 College of Sciences, Hunan Agriculture University, Hunan, Changsha, 410128, China 3 School of Mathematical Sciences, Xiamen University, Fujian, Xiamen 361005, China

* Corresponding author

Received  December 2017 Revised  June 2018 Published  October 2018

Fund Project: The first author is supported by the Scientific Research Fund of Hunan Provincial Education Department grant 16C0763 and 17C0754; The second author is supported by the NSF of China grant 11671086 and 11871208 and NSF of Hunan Province of China (No. 2018JJ2159); The third author is supported by the NSF of China grant 11771358

This paper is devoted to exploring the properties of positive solutions for a class of nonlinear integral equation(s) involving the Bessel potentials, which are equivalent to certain partial differential equations under appropriate integrability conditions. With the help of regularity lifting theorem, we obtain an integrability interval of positive solutions and then extend the integrability interval to the whole [1, ∞) by the properties of the Bessel kernels and some delicate analysis techniques. Meanwhile, the radial symmetry and the sharp exponential decay of positive solutions are also obtained. Furthermore, as an application, we establish the uniqueness theorem of the corresponding partial differential equations.

Citation: 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
##### References:
 [1] W. Chen and C. Li, Methods on Nonlinear Elliptic equations, AIMS Ser. Differ. Equ. Dyn. Syst., Vol. 4, 2010. [2] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equation, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [3] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 54 (2006), 330-343. doi: 10.1002/cpa.20116. [4] C. Coffman, Uniqueness of the ground state solution for $\triangle u-u+u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. doi: 10.1007/BF00250684. [5] D. J. Frantzeskakis, Frantzeskakis, Dark solitons in atomic Bose-Einstein condesates: from theory to experiments, J. Physic A, 43 (2010), 213001. doi: 10.1088/1751-8113/43/21/213001. [6] Y. Guo, B. Li and J. Wei, Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents, J. Differential Equations, 10 (2014), 3463-3495. doi: 10.1016/j.jde.2014.02.007. [7] Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^n$, Comm. Partial Differential Equations, 33 (2008), 263-284. doi: 10.1080/03605300701257476. [8] 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. [9] C. Jin and C. Li, Qualitative analysis of some systems of equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5. [10] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [11] M. Kwong, Uniqueness of positive solutions of $\triangle u -u+ u^p = 0 \ {\rm in} \ \mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [12] Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799. doi: 10.1016/j.jde.2012.11.008. [13] Y. Lei, Radial symmetry and asymptotic estimates for positive solutions to a singular integral equation, Taiwanese J. Math., 20 (2016), 473-489. doi: 10.11650/tjm.20.2016.6150. [14] Y. Lei, Positive solutions of integral system involving Bessel potential, Comm. Pure Appl. Anal., 12 (2011), 2721-2737. doi: 10.3934/cpaa.2013.12.2721. [15] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7. [16] 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. [17] C. Li and L. Ma, Uniqueness of positive bound states to schrödinger systems with critical expoents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [18] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 114 (1983), 349-374. doi: 10.2307/2007032. [19] E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, Vol.14, Amer. Math. Soc., Providence, R.I., 2001. doi: 10.1090/gsm/014. [20] T. Lin and J. Wei, Spikes in two coupled nonlinear schrodinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [21] G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7. [22] 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. [23] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 3 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [24] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\triangle u + f(u) = 0 \ {\rm in} \ \mathbb{R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145. doi: 10.1007/BF00275874. [25] J. Xu, S. Jiang and H. Wu, Some properties of positive solutions for an integral system with the double weighted Riesz potentials, Comm. Pure Appl. Anal., 15 (2016), 2117-2134. doi: 10.3934/cpaa.2016030. [26] J. Xu, H. Wu and Z. Tan, The non-existence results for a class of integral equation, J. Differential Equations, 256 (2014), 1873-1902. doi: 10.1016/j.jde.2013.12.009. [27] J. Xu, H. Wu and Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations, J. Math. Anal. Appl., 427 (2015), 307-319. doi: 10.1016/j.jmaa.2015.02.043. [28] Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, Nonlinear Anal., 75 (2012), 1989-1999. doi: 10.1016/j.na.2011.09.051. [29] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math. Vol. 120, Springverlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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##### References:
 [1] W. Chen and C. Li, Methods on Nonlinear Elliptic equations, AIMS Ser. Differ. Equ. Dyn. Syst., Vol. 4, 2010. [2] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equation, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [3] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 54 (2006), 330-343. doi: 10.1002/cpa.20116. [4] C. Coffman, Uniqueness of the ground state solution for $\triangle u-u+u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. doi: 10.1007/BF00250684. [5] D. J. Frantzeskakis, Frantzeskakis, Dark solitons in atomic Bose-Einstein condesates: from theory to experiments, J. Physic A, 43 (2010), 213001. doi: 10.1088/1751-8113/43/21/213001. [6] Y. Guo, B. Li and J. Wei, Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents, J. Differential Equations, 10 (2014), 3463-3495. doi: 10.1016/j.jde.2014.02.007. [7] Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^n$, Comm. Partial Differential Equations, 33 (2008), 263-284. doi: 10.1080/03605300701257476. [8] 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. [9] C. Jin and C. Li, Qualitative analysis of some systems of equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457. doi: 10.1007/s00526-006-0013-5. [10] C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670. doi: 10.1090/S0002-9939-05-08411-X. [11] M. Kwong, Uniqueness of positive solutions of $\triangle u -u+ u^p = 0 \ {\rm in} \ \mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. [12] Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799. doi: 10.1016/j.jde.2012.11.008. [13] Y. Lei, Radial symmetry and asymptotic estimates for positive solutions to a singular integral equation, Taiwanese J. Math., 20 (2016), 473-489. doi: 10.11650/tjm.20.2016.6150. [14] Y. Lei, Positive solutions of integral system involving Bessel potential, Comm. Pure Appl. Anal., 12 (2011), 2721-2737. doi: 10.3934/cpaa.2013.12.2721. [15] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7. [16] 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. [17] C. Li and L. Ma, Uniqueness of positive bound states to schrödinger systems with critical expoents, SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301. [18] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 114 (1983), 349-374. doi: 10.2307/2007032. [19] E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, Vol.14, Amer. Math. Soc., Providence, R.I., 2001. doi: 10.1090/gsm/014. [20] T. Lin and J. Wei, Spikes in two coupled nonlinear schrodinger equations, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 403-439. doi: 10.1016/j.anihpc.2004.03.004. [21] G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7. [22] 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. [23] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., 3 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020. [24] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\triangle u + f(u) = 0 \ {\rm in} \ \mathbb{R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145. doi: 10.1007/BF00275874. [25] J. Xu, S. Jiang and H. Wu, Some properties of positive solutions for an integral system with the double weighted Riesz potentials, Comm. Pure Appl. Anal., 15 (2016), 2117-2134. doi: 10.3934/cpaa.2016030. [26] J. Xu, H. Wu and Z. Tan, The non-existence results for a class of integral equation, J. Differential Equations, 256 (2014), 1873-1902. doi: 10.1016/j.jde.2013.12.009. [27] J. Xu, H. Wu and Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations, J. Math. Anal. Appl., 427 (2015), 307-319. doi: 10.1016/j.jmaa.2015.02.043. [28] Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system, Nonlinear Anal., 75 (2012), 1989-1999. doi: 10.1016/j.na.2011.09.051. [29] W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Math. Vol. 120, Springverlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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