Advanced Search
Article Contents
Article Contents

On critical Choquard equation with potential well

  • * Minbo Yang is the corresponding author, he was partially supported by NSFC (11571317) and ZJNSF(LY15A010010)

    * Minbo Yang is the corresponding author, he was partially supported by NSFC (11571317) and ZJNSF(LY15A010010)
Zifei Shen and Fashun Gao were supported by NSFC (11671364).
Abstract Full Text(HTML) Related Papers Cited by
  • In this paper we are interested in the following nonlinear Choquard equation

    $-Δ u+(λ V(x)-β)u = \big(|x|^{-μ}* |u|^{2_{μ}^{*}}\big)|u|^{2_{μ}^{*}-2}u\;\;\;\;\;\;\;\;\;\;\mbox{in}\;\; \mathbb{R}^N,$

    where $λ, β∈\mathbb{R}^+$ , $0<μ<N, N≥4, 2_{μ}^{*} = (2N-μ)/(N-2)$ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function $V∈ \mathcal{C}(\mathbb{R}^N, \mathbb{R})$ such that $Ω : = \mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. If $β>0$ is a constant such that the operator $-Δ +λ V(x)-β$ is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$ for $λ$ large enough and also characterize the asymptotic behavior of the solutions as the parameter $λ$ goes to infinity. Furthermore, for any $0<β<β_{1}$ , we are able to prove the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $β_{1}$ is the first eigenvalue of $-Δ$ on $Ω$ with Dirichlet boundary condition.

    Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35A15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443.  doi: 10.1007/s00209-004-0663-y.
    [2] N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320.  doi: 10.1016/j.jfa.2005.11.010.
    [3] C. O. AlvesD. CassaniC. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger 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] C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55 (2016), Art. 48, 28 pp. doi: 10.1007/s00526-016-0984-9.
    [6] A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.
    [7] T. BartschA. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.  doi: 10.1142/S0219199701000494.
    [8] T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384.  doi: 10.1007/PL00001511.
    [9] V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93.  doi: 10.1007/BF00375686.
    [10] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.
    [11] B. BuffoniL. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc., 119 (1993), 179-186.  doi: 10.1090/S0002-9939-1993-1145940-X.
    [12] S. CingolaniM. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.  doi: 10.1007/s00033-011-0166-8.
    [13] M. Clapp and Y. Ding, Positive solutions of a Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys., 55 (2004), 592-605.  doi: 10.1007/s00033-004-1084-9.
    [14] M. Clapp and Y. Ding, Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential, Differential Integral Equations, 16 (2003), 981-992. 
    [15] M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.  doi: 10.1016/j.jmaa.2013.04.081.
    [16] Y. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034.  doi: 10.1016/j.jde.2010.03.022.
    [17] Y. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manus. Math., 112 (2003), 109-135.  doi: 10.1007/s00229-003-0397-x.
    [18] F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math.
    [19] F. Gao and M. Yang, On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl., 448 (2017), 1006-1041.  doi: 10.1016/j.jmaa.2016.11.015.
    [20] F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., https://doi.org/10.1142/S0219199717500377. doi: 10.1142/S0219199717500377.
    [21] M. GhimentiV. Moroz and J. Van Schaftingen, Least Action nodal solutions for ghe quadratic Choquard equation, Proc. Amer. Math. Soc., 145 (2017), 737-747.  doi: 10.1090/proc/13247.
    [22] M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019.
    [23] Y. Guo and Z. Tang, Multi-bump solutions for Schrödinger equation involving critical growth and potential wells, Discrete Contin. Dyn. Syst., 35 (2015), 3393-3415.  doi: 10.3934/dcds.2015.35.3393.
    [24] Y. Guo and Z. Tang, Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells, J. Differential Equations, 259 (2015), 6038-6071.  doi: 10.1016/j.jde.2015.07.015.
    [25] Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006.
    [26] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.
    [27] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105. 
    [28] E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, 1997. doi: 10.1090/gsm/014.
    [29] P.L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.
    [30] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.
    [31] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.
    [32] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equation: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12pp. doi: 10.1142/S0219199715500054.
    [33] S. Pekar, Untersuchungüber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.
    [34] R. Penrose, On gravity's role in quantum state reduction, Gen. Relativ. Gravitat., 28 (1996), 581-600.  doi: 10.1007/BF02105068.
    [35] G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299.  doi: 10.1016/j.jmaa.2009.10.061.
    [36] Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Commun. Pure Appl. Anal., 13 (2014), 237-248.  doi: 10.3934/cpaa.2014.13.237.
    [37] J. C. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169.
    [38] 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.
  • 加载中

Article Metrics

HTML views(2162) PDF downloads(422) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint