American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data
  • CPAA Home
  • This Issue
  • Next Article
    Logarithmically improved criteria for Euler and Navier-Stokes equations
November  2013, 12(6): 2721-2737. doi: 10.3934/cpaa.2013.12.2721

Positive solutions of integral systems involving Bessel potentials

Yutian Lei 1,

1. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097

Received  October 2012 Revised  December 2012 Published  May 2013

This paper is concerned with integral systems involving the Bessel potentials. Such integral systems are helpful to understand the corresponding PDE systems, such as some static Shrödinger systems with the critical and the supercritical exponents. We use the lifting lemma on regularity to obtain an integrability interval of solutions. Since the Bessel kernel does not have singularity at infinity, we extend the integrability interval to the whole $[1,\infty]$. Next, we use the method of moving planes to prove the radial symmetry for the positive solution of the system. Based on these results, by an iteration we obtain the estimate of the exponential decay of those solutions near infinity. Finally, we discuss the uniqueness of the positive solution of PDE system under some assumption.
Keywords: method of moving planes, integrability intervals, Integral equations, Bessel potential, decay rate., Hardy-Littlewood-Sobolev inequality, radial symmetry.
Mathematics Subject Classification: Primary: 45E10, 45G0.
Citation: 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
References:
[1]

J. Bourgain, Global solutions of nonlinear Schrödinger equations,, in, 46 (1999).   Google Scholar

[2]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.   Google Scholar

[3]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547.   Google Scholar

[4]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[7]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in, (1981).   Google Scholar

[8]

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

[9]

F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality,, Math. Res. Lett., 14 (2007), 373.   Google Scholar

[10]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[11]

T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations,, coherent solitons in coupled nonlinear Schr\, 86 (2001), 5043.   Google Scholar

[12]

Y. Lei, On the regularity of positive solutions of a class of Choquard type equations,, Math. Z., 273 (2013), 883.  doi: 10.1007/s00209-012-1036-6.  Google Scholar

[13]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system,, Calc. Var. Partial Differential Equations, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[14]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.   Google Scholar

[15]

C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar

[16]

Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153.   Google Scholar

[17]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.   Google Scholar

[18]

T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 403.  doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[19]

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

[20]

L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type system,, Math. Comput. Modelling, 49 (2009), 379.  doi: 10.1016/j.mcm.2008.06.010.  Google Scholar

[21]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rational Mech. Anal., 195 (2010), 455.  doi: 10.1007/s00205-008-0208-3.  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 Function,", Princetion Math. Series, (1970).   Google Scholar

[24]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207.   Google Scholar

[25]

W. Ziemer, "Weakly Differentiable Functions,", Graduate Texts in Math. Vol. 120, (1989).   Google Scholar

show all references

References:
[1]

J. Bourgain, Global solutions of nonlinear Schrödinger equations,, in, 46 (1999).   Google Scholar

[2]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.   Google Scholar

[3]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations,, Ann. of Math., 145 (1997), 547.   Google Scholar

[4]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[7]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, in, (1981).   Google Scholar

[8]

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

[9]

F. Hang, On the integral systems related to Hardy-Littlewood-sobolev inequality,, Math. Res. Lett., 14 (2007), 373.   Google Scholar

[10]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations,, Calc. Var. Partial Differential Equations, 26 (2006), 447.  doi: 10.1007/s00526-006-0013-5.  Google Scholar

[11]

T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations,, coherent solitons in coupled nonlinear Schr\, 86 (2001), 5043.   Google Scholar

[12]

Y. Lei, On the regularity of positive solutions of a class of Choquard type equations,, Math. Z., 273 (2013), 883.  doi: 10.1007/s00209-012-1036-6.  Google Scholar

[13]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system,, Calc. Var. Partial Differential Equations, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[14]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, Invent. Math., 123 (1996), 221.   Google Scholar

[15]

C. Li and L. Ma, Uniqueness of positive bound states to Schrödinger systems with critical exponents,, SIAM J. Math. Anal., 40 (2008), 1049.  doi: 10.1137/080712301.  Google Scholar

[16]

Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres,, J. Eur. Math. Soc., 6 (2004), 153.   Google Scholar

[17]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349.   Google Scholar

[18]

T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, Ann. Inst. H. Poincare Anal. Non Lineaire, 22 (2005), 403.  doi: 10.1016/j.anihpc.2004.03.004.  Google Scholar

[19]

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

[20]

L. Ma and D. Chen, Radial symmetry and uniqueness for positive solutions of a Schrödinger type system,, Math. Comput. Modelling, 49 (2009), 379.  doi: 10.1016/j.mcm.2008.06.010.  Google Scholar

[21]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, Arch. Rational Mech. Anal., 195 (2010), 455.  doi: 10.1007/s00205-008-0208-3.  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 Function,", Princetion Math. Series, (1970).   Google Scholar

[24]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207.   Google Scholar

[25]

W. Ziemer, "Weakly Differentiable Functions,", Graduate Texts in Math. Vol. 120, (1989).   Google Scholar

[1]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462
[2]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445
[3]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258
[4]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250
[5]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273
[6]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[7]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[8]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260
[9]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076
[10]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[11]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
[12]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[13]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101
[14]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[15]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030
[16]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265
[17]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
[18]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106
[19]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[20]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (48)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences