2016, 15(6): 2059-2074. doi: 10.3934/cpaa.2016027

On the Hardy-Littlewood-Sobolev type systems

1. 

Department of Applied Mathematics, University of Colorado at Boulder, Colorado

2. 

Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China

3. 

Department of Applied Mathematics, University of Colorado at Boulder

Received  October 2015 Revised  June 2016 Published  September 2016

In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.
Citation: Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027
References:
[1]

F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems,, \emph{Disc. & Cont. Dynamics Sys.}, 34 (2014), 3317. doi: 10.3934/dcds.2014.34.3317.

[2]

H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semi-linear problems in $R^N$,, \emph{Indiana University Mathematics Journal}, 30 (1981), 141. doi: 10.1512/iumj.1981.30.30012.

[3]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 142 (1989), 615. doi: 10.1002/cpa.3160420304.

[4]

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.

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Commun. in Partial Differential Equations}, 30 (2005), 59. doi: 10.1081/PDE-200044445.

[6]

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.

[7]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().

[8]

Z. Cheng and C. Li, Shooting method with sign-changing nonlinearity,, \emph{Nonlinear Analysis: Theory, 114 (2015), 2. doi: 10.1016/j.na.2014.10.019.

[9]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Math. Anal. and Applications, 7A (1981), 369.

[10]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277. doi: 10.3934/dcds.2016.36.3277.

[11]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, \emph{Calc. Var. of Partial Differential Equations}, 45 (2012), 43. doi: 10.1007/s00526-011-0450-7.

[12]

C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, \emph{Comm. in Partial Differential Equation}, 41 (2016), 1029. doi: 10.1080/03605302.2016.1190376.

[13]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349. doi: 10.2307/2007032.

[14]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{Journal of Partial Differential Equations}, 19 (2006).

[15]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Quaderno Matematico}, (1982).

[16]

E. Mitidieri, A Rellich type identity and applications: Identity and applications,, \emph{Communications in partial differential equations}, 18 (1993), 125. doi: 10.1080/03605309308820923.

[17]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465.

[18]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$,, \emph{Soviet Math. Doklady}, 6 (1965), 1408.

[19]

P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.

[20]

P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. J. Math.}, 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036.

[21]

Pavol Quittner and Philippe Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States,, Springer, (2007).

[22]

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

[23]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635.

[24]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, \emph{Atti Semi. Mat. Fis. Univ. Modena}, 46 (1998), 369.

[25]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014.

[26]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, (1999), 207. doi: 10.1007/s002080050258.

show all references

References:
[1]

F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems,, \emph{Disc. & Cont. Dynamics Sys.}, 34 (2014), 3317. doi: 10.3934/dcds.2014.34.3317.

[2]

H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semi-linear problems in $R^N$,, \emph{Indiana University Mathematics Journal}, 30 (1981), 141. doi: 10.1512/iumj.1981.30.30012.

[3]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 142 (1989), 615. doi: 10.1002/cpa.3160420304.

[4]

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.

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Commun. in Partial Differential Equations}, 30 (2005), 59. doi: 10.1081/PDE-200044445.

[6]

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.

[7]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().

[8]

Z. Cheng and C. Li, Shooting method with sign-changing nonlinearity,, \emph{Nonlinear Analysis: Theory, 114 (2015), 2. doi: 10.1016/j.na.2014.10.019.

[9]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Math. Anal. and Applications, 7A (1981), 369.

[10]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277. doi: 10.3934/dcds.2016.36.3277.

[11]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, \emph{Calc. Var. of Partial Differential Equations}, 45 (2012), 43. doi: 10.1007/s00526-011-0450-7.

[12]

C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, \emph{Comm. in Partial Differential Equation}, 41 (2016), 1029. doi: 10.1080/03605302.2016.1190376.

[13]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349. doi: 10.2307/2007032.

[14]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{Journal of Partial Differential Equations}, 19 (2006).

[15]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Quaderno Matematico}, (1982).

[16]

E. Mitidieri, A Rellich type identity and applications: Identity and applications,, \emph{Communications in partial differential equations}, 18 (1993), 125. doi: 10.1080/03605309308820923.

[17]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465.

[18]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$,, \emph{Soviet Math. Doklady}, 6 (1965), 1408.

[19]

P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.

[20]

P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. J. Math.}, 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036.

[21]

Pavol Quittner and Philippe Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States,, Springer, (2007).

[22]

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

[23]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635.

[24]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, \emph{Atti Semi. Mat. Fis. Univ. Modena}, 46 (1998), 369.

[25]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014.

[26]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, (1999), 207. doi: 10.1007/s002080050258.

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