Advanced Search
Article Contents
Article Contents

Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents

  • * Corresponding author

    * Corresponding author

The third author is supported by NSFC grant 11671364 and the Fourth author is supported by NSFC grant 11571317

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we are interested in the following critical coupled Hartree system

    $ \left\{\begin{array}{l} (-\Delta)^{s} \tilde{u}+\lambda_{1}\tilde{u} = \alpha_{1}\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{u}|^{2_{\mu}^{\ast}-2}\tilde{u}, \ \ &&\mbox{in} \ \Omega,\\ (-\Delta)^{s} \tilde{v}+\lambda_{2}\tilde{v} = \alpha_{2}\int_{\Omega}\frac{|\tilde{v}(z)|^{2_{\mu}^{\ast}} }{|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}\\ \quad +\beta\int_{\Omega}\frac{|\tilde{u}(z)|^{2_{\mu}^{\ast}}} {|x-z|^{\mu}}dz|\tilde{v}|^{2_{\mu}^{\ast}-2}\tilde{v}, \ \ &&\mbox{in} \ \Omega,\\ \tilde{u} = \tilde{v} = 0, \ \ && \mbox{on} \ \partial \Omega, \end{array} \right. $

    where $ 0<s<1 $, $ \alpha_{1}, \alpha_{2}>0 $, $ \beta\neq0 $, $ 4s<\mu<N $, $ 2_{\mu}^{\ast} = (2N-\mu)/(N-2s) $, $ \Omega\subset\mathbb{R}^N(N\geq3) $ is a smooth bounded domain, $ -\lambda_{1}(\Omega)<\lambda_{1}, \lambda_{2}<0 $ with $ \lambda_{1}(\Omega) $ the first eigenvalue of $ (-\Delta)^{s} $ under the Dirichlet boundary condition. Assume that the nonlinearity and the coupling terms are both of the upper critical growth due to the Hardy–Littlewood–Sobolev inequality, by applying the Dirichlet-to-Neumann map, we are able to obtain the existence of the ground state solution of the critical coupled Hartree system.

    Mathematics Subject Classification: 35J25, 35J60, 35A15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] B. AbdellaouiV. Felli and I. Peral, Some remarks on systems of elliptic equations doubly critical in the whole $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 34 (2009), 97-137.  doi: 10.1007/s00526-008-0177-2.
    [2] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 26-61. 
    [3] C. O. AlvesD. CassaniC. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrodinger equation in $\mathbb{R}^{2}$, J. Differential Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021.
    [4] C. O. AlvesF. GaoM. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988.  doi: 10.1016/j.jde.2017.05.009.
    [5] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020.
    [6] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, A concave–convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, in press. doi: 10.1017/S0308210511000175.
    [7] B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 263 (2017), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.
    [8] T. BartschZ. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6.
    [9] L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas., 7 (2000), 210-230. 
    [10] S. Bhattarai, On fractional Schrödinger systems of Choquard type, J. Differential Equations, 263 (2017), 3197-3229.  doi: 10.1016/j.jde.2017.04.034.
    [11] H. Brézis and T. Kato, Remarks on the Schrödinger operator with regular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. 
    [12] L. A. Caffarelli and L. Silvestre., An extension problem related to the fractional Laplacian, Commun. PDEs., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.
    [13] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.
    [14] 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.
    [15] Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8.
    [16] Z. ChenC. Lin and W. Zou, Sign-changing solutions and phase separation for an elliptic system with critical exponent, Comm. Partial Differential Equations, 39 (2014), 1827-1859.  doi: 10.1080/03605302.2014.908391.
    [17] Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.  doi: 10.1007/s00526-014-0717-x.
    [18] F. DalfovoS. GiorginiL. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. 
    [19] E. Dancer and J. Wei, Spike Solutions in coupled nonlinear Schrödinger equations with Attractive Interaction, Tran. Amer. Math. Soc., 361 (2009), 1189-1208.  doi: 10.1090/S0002-9947-08-04735-1.
    [20] A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.
    [21] F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.
    [22] F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun.Contemp. Math., 20 (2018), no.4, 1750037, 22 pp. doi: 10.1142/S0219199717500377.
    [23] Z. GuoS. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706.  doi: 10.1016/j.jmaa.2016.08.069.
    [24] N. Ikoma and K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 40 (2011), 449-480.  doi: 10.1007/s00526-010-0347-x.
    [25] N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 180, Springer, 1972.
    [26] E. Lenzmann, Uniqueness of ground states for Hardy–Littlewood–Sobolev Hartree equations, Anal. PDE., 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.
    [27] M. Lewin and E. Lenzmann, On singularity formation for the $L^{2}$–critical boson star equation, Nonlinearity, 24 (2011), 3515-3540.  doi: 10.1088/0951-7715/24/12/009.
    [28] E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, AMS, Providence, Rhode Island, 2001.
    [29] T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéare., 221 (2005), 403-439.  doi: 10.1016/j.anihpc.2004.03.004.
    [30] T. Lin and J. Wei, Ground state of N-coupled nonlinear Schrödinger equations in $\mathbb{R}^n$, $n\leq3$, Commun. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x.
    [31] T. Lin and J. Wei, Spikes in two-component systems of Nonlinear Schrödinger equations with trapping potentials, J. Differential Equations, 299 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011.
    [32] T. Lin and J. Wei, Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations, Nonlinearty, 19 (2006), 2755-2773.  doi: 10.1088/0951-7715/19/12/002.
    [33] A. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232. 
    [34] Z. Liu and Z. Wang, Multiple bound states of nonlinear Schrödinger systems, Commun. Math. Phys., 282 (2008), 721–731. doi: 10.1007/s00220-008-0546-x.
    [35] H. Luo, Ground state solutions of Pohožaev type and Nehari type for a class of nonlinear Choquard equations, J. Math. Anal. Appl., 467 (2018), 842–862. doi: 10.1016/j.jmaa.2018.07.055.
    [36] L. MaiaE. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002.
    [37] C. Menyuk, Nonlinear pulse propagation in birefringence optical fiber, IEEE J. Quantum Electron, 23 (1987), 174-176. 
    [38] C. Menyuk, Pulse propagation in an elliptically birefringent Kerr medium, IEEE J. Quantum Electron, 25 (1989), 2674-2682. 
    [39] E. MontefuscoB. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71.  doi: 10.4171/JEMS/103.
    [40] V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equation, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2.
    [41] D. Mugnai, Pseudo–relativistic Hartree equation with general nonlinearity: existence, non-existence and variational identities, Adv. Nonlinear Stud., 13 (2013), 799-823.  doi: 10.1515/ans-2013-0403.
    [42] S. Peng, Y. Peng and Z. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations, 55 (2016), Art. 142, 30 pp. doi: 10.1007/s00526-016-1091-7.
    [43] S. PengW. Shuai and Q. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differential Equations, 263 (2017), 709-731.  doi: 10.1016/j.jde.2017.02.053.
    [44] A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002.
    [45] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional laplacian,, Trans. Amer. Math. Soc., 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.
    [46] Z. ShenF. Gao and M. Yang, Groundstates for nonlinear fractional Choquard equations with general nonlinearities, Math. Meth. Appl. Sci., 14 (2016), 4082-4098.  doi: 10.1002/mma.3849.
    [47] B. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbb{R}^{N}$, Commun. Math. Phys., 271 (2007), 199-221.  doi: 10.1007/s00220-006-0179-x.
    [48] E. M. SteinSingular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30, Princeton University Press, 1970. 
    [49] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.  doi: 10.1007/s00526-010-0378-3.
    [50] J. Wang and J. Shi, Standing waves for a coupled nonlinear Hartree equations with nonlocal interaction, Calc. Var. Partial Differential Equations, 56 (2017), Art. 168, 36 pp. doi: 10.1007/s00526-017-1268-8.
    [51] J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schröinger equations, Arch. Ration. Mech. Anal., 190 (2008), 83-106.  doi: 10.1007/s00205-008-0121-9.
    [52] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.
    [53] M. YangY. Wei and Y. Ding, Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities, Z. Angew. Math. Phys., 65 (2014), 41-68.  doi: 10.1007/s00033-013-0317-1.
    [54] Y. Zheng, Z. Shen and M. Yang, On critical pseudo-relativistic Hartree equation with potential well, Preprint.
  • 加载中

Article Metrics

HTML views(507) PDF downloads(425) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint