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On the Hardy-Littlewood-Sobolev type systems

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  • 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.
    Mathematics Subject Classification: Primary: 35J91.

    Citation:

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  • [1]

    F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems, Disc. & Cont. Dynamics Sys., 34 (2014), 3317-3339.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 $R^N$, Indiana University Mathematics Journal, 30 (1981), 141-157.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, Comm. Pure Appl. Math., 142 (1989), 615-622.doi: 10.1002/cpa.3160420304.

    [4]

    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.

    [5]

    W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. in Partial Differential Equations, 30 (2005), 59-65.doi: 10.1081/PDE-200044445.

    [6]

    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.

    [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, Nonlinear Analysis: Theory, Methods and Applications, 114 (2015), 2-12.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 $R^n$, Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies, 7A (1981), 369-402.

    [10]

    Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.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, Calc. Var. of Partial Differential Equations, 45 (2012), 43-61.doi: 10.1007/s00526-011-0450-7.

    [12]

    C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth, Comm. in Partial Differential Equation, 41 (2016), 1029-1039.doi: 10.1080/03605302.2016.1190376.

    [13]

    E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.doi: 10.2307/2007032.

    [14]

    J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, Journal of Partial Differential Equations, 19 (2006), 256.

    [15]

    E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Quaderno Matematico, (1982), 285.

    [16]

    E. Mitidieri, A Rellich type identity and applications: Identity and applications, Communications in partial differential equations, 18 (1993), 125-151.doi: 10.1080/03605309308820923.

    [17]

    E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differ. Integral Equations, 9 (1996), 465-479.

    [18]

    S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$, Soviet Math. Doklady, 6 (1965), 1408-1411.

    [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, Duke Math. J., 139 (2007), 555-579.doi: 10.1215/S0012-7094-07-13935-8.

    [20]

    P. Pucci and J. Serrin, A general variational identity, Indiana Univ. J. Math., 35 (1986), 681-703.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, Arch. Rat. Mech. Anal., 43 (1971), 304-318.

    [23]

    J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equations, 9 (1996), 635-654.

    [24]

    J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semi. Mat. Fis. Univ. Modena, 46 (1998), 369-380.

    [25]

    P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Advances in Mathematics, 221 (2009), 1409-1427.doi: 10.1016/j.aim.2009.02.014.

    [26]

    J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., (1999), 207-228.doi: 10.1007/s002080050258.

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