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Liouville type theorems for stable solutions of the weighted fractional Lane-Emden system

  • *Corresponding author: Hatem Hajlaoui

    *Corresponding author: Hatem Hajlaoui
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  • In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system

    $ \begin{align*} (-\Delta)^s u = h(x)v^p,\quad (-\Delta)^s v = h(x)u^q, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N, \end{align*} $

    where $ 1<q\leq p $ and $ h $ is a positive continuous function in $ \mathbb{R}^N $ satisfying $ {\liminf_{|x|\to \infty}}\frac{h(x)}{|x|^\ell} > 0 $ with $ \ell > 0. $ Our results generalize the results established in [23] for the Laplacian case (correspond to $ s = 1 $) and improve the previous work [12]. As a consequence, we prove classification result for stable solutions to the weighted fractional Lane-Emden equation $ (-\Delta)^s u = h(x)u^p $ in $ \mathbb{R}^N $.

    Mathematics Subject Classification: Primary: 35J47, 35J15; Secondary: 35B53, 35B35.


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  • [1] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.
    [2] W. ChenL. Dupaigne and M. Ghergu, A new critical curve for the Lane-Emden system, Discrete Contin. Dyn. Syst., 34 (2014), 2469-2479.  doi: 10.3934/dcds.2014.34.2469.
    [3] C. Chen and H. Wang, Liouville theorems for the weighted Lane-Emden equation with finite Morse indices, Math. Methods Appl. Sci., 40 (2017), 4674-4682.  doi: 10.1002/mma.4333.
    [4] C. Cowan, Liouville theorems for stable Lane-Emden systems and biharmonic problems, Nonlinearity, 26 (2013), 2357-2371.  doi: 10.1088/0951-7715/26/8/2357.
    [5] C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.  doi: 10.1090/S0002-9939-2011-11351-0.
    [6] C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Calc. Var. PDE., 49 (2014), 291-305.  doi: 10.1007/s00526-012-0582-4.
    [7] E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differ. Equ., 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.
    [8] J. DávilaL. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.  doi: 10.1090/tran/6872.
    [9] S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^n$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp. doi: 10.1007/978-88-7642-601-8.
    [10] Y. Du and Z. Guo, Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations, Adv. Differ. Equ., 18 (2013), 737-768. 
    [11] A. T. Duong, A Liouville type theorem for non-linear elliptic systems involving advection terms, Complex Var. Elliptic Equ., 63 (2018), 1704-1720.  doi: 10.1080/17476933.2017.1403427.
    [12] A. T. Duong and V. H. Nguyen, Liouville type theorems for some fractional elliptic problems, Nonlinear Anal., 210 (2021), 112383.  doi: 10.1016/j.na.2021.112383.
    [13] A. T. Duong and D. H. Pham, Liouville-type theorem for fractional Kirchhoff equations with weights, Bulletin of the Iranian Mathematical Society, 47 (2021), 1585-1597.  doi: 10.1007/s41980-020-00460-z.
    [14] A. T. Duong and Q. H. Phan, Liouville type theorem for nonlinear elliptic system involving Grushin operator, J. Math. Anal. Appl., 454 (2017), 785-801.  doi: 10.1016/j.jmaa.2017.05.029.
    [15] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u = f(u)$ in $\Bbb R^N$, J. Eur. Math. Soc. (JEMS), 12 (2010), 855-882.  doi: 10.4171/JEMS/217.
    [16] L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solutions for the Liouville system, in: Geometric Partial Differential Equations, Publications of the Scuola Normale Superiore/CRM Series, 15 (2013), 139-144. doi: 10.1007/978-88-7642-473-1_7.
    [17] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^N$, J. Math. Pures Appl., 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001.
    [18] A. Farina and S. Hasegawa, Liouville-type theorems and existence results for stable solutions to weighted Lane-Emden equations, Proceedings of the Royal Society of Edinburgh Section A Mathematics, 150 (2020), 1567-1579.  doi: 10.1017/prm.2018.160.
    [19] M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533.  doi: 10.3934/dcds.2014.34.2513.
    [20] M. Fazly and J. Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math., 18 (2016), 1650005, 24 pp. doi: 10.1142/S021919971650005X.
    [21] M. Fazly and J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, Amer. J. Math., 139 (2017), 433-460.  doi: 10.1353/ajm.2017.0011.
    [22] C. GuiW.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in ${\bf{R}}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.
    [23] H. HajlaouiA. Harrabi and F. Mtiri, Liouville theorems for stable solutions of the weighted Lane-Emden system, Discrete Contin. Dyn. Syst., 37 (2017), 265-279.  doi: 10.3934/dcds.2017011.
    [24] A. Harrabi, Interpolation inequality and some applications, preprint, 2022, arXiv: 2105.04058v5.
    [25] L.-G. Hu, Liouville type results for semi-stable solutions of the weighted Lane-Emden system, J. Math. Anal. Appl., 432 (2015), 429-440.  doi: 10.1016/j.jmaa.2015.06.032.
    [26] L.-G. Hu, Liouville type theorems for stable solutions of the weighted elliptic system with the advection term: $p \ge \vartheta >1$, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 7, 30 pp. doi: 10.1007/s00030-018-0498-6.
    [27] L.-G. Hu and J. Zeng, Liouville type theorems for stable solutions of the weighted elliptic system, J. Math. Anal. Appl., 437 (2016), 882-901.  doi: 10.1016/j.jmaa.2016.01.032.
    [28] W. Jeong and Y. Lee, Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential, Nonlinear Anal., 87 (2013), 126-145.  doi: 10.1016/j.na.2013.04.007.
    [29] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508.
    [30] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${\mathbb R}^ N$, Differential Integral Equations, 9 (1996), 465-479. 
    [31] E. Mitidieri and S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384. 
    [32] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416. doi: 10.1112/S0024609305004248.
    [33] F. Mtiri and D. Ye, Liouville theorems for stable at infinity solutions of Lane-Emden system, Nonlinearity, 32 (2019), 910-926.  doi: 10.1088/1361-6544/aaf078.
    [34] P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.  doi: 10.1137/11085428X.
    [35] B. Rahal and C. Zaidi, On the classification of stable solutions of the fractional equation, Potential Anal., 50 (2019), 565-579.  doi: 10.1007/s11118-018-9694-6.
    [36] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9 (1996), 635-653. 
    [37] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.
    [38] C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), 1705-1727.  doi: 10.1016/j.jfa.2011.11.017.
    [39] J. Wei and D. Ye, Liouville theorems for stable solutions of biharmonic problem, Math. Ann., 356 (2013), 1599-1612.  doi: 10.1007/s00208-012-0894-x.
    [40] H. Yang and W. Zou, Symmetry of components and Liouville-type theorems for semilinear elliptic systems involving the fractional Laplacian, Nonlinear Anal., 180 (2019), 208-224.  doi: 10.1016/j.na.2018.10.006.
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