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Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well

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  • In this paper we study the non-linear fractional Schrödinger equation with steep potential well \begin{eqnarray} (-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u)\ in\ R^{n}, \ u\in H^{\alpha}(R^n), \end{eqnarray} where $(-\Delta)^\alpha$ ($\alpha \in (0,1)$) denotes the fractional Laplacian, $\lambda$ is a parameter, $V\in C(\mathbb{R}^n)$ and $V^{-1}(0)$ has nonempty interior. Under some suitable conditions, the existence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is also explored.
    Mathematics Subject Classification: Primary: 35J35; Secondary: 35J60.


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  • [1]

    T. Bartsch and Z. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^n$, Commun. in PDE, 20 (1995), 1725-1741.doi: 10.1080/03605309508821149.


    T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential, Discrete Contin. Dyn. Syst., 33 (2013), 7-26.


    L. Caffarelli and L. Silvestre, An extension problems related to the fractional Laplacian, Comm. PDE, 32 (2007), 1245C1260.2.doi: 10.1080/03605300600987306.


    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.


    A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilnear equations, Comm. PDE, 36 (2011), 1353C1384.doi: 10.1080/03605302.2011.562954.


    M. Cheng, Bound state for the fractional Schrödinger equation with undounded potential, J. Math. Phys., 53 (2012), 043507.doi: 10.1063/1.3701574.


    G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Comm. Pure Appl. Anal., 13 (2014), 2359-2376.doi: 10.3934/cpaa.2014.13.2359.


    J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.doi: 10.1016/j.jde.2013.10.006.


    J. Dávila, M. Del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235.doi: 10.2140/apde.2015.8.1165.


    E. Di Nezza, G. Patalluci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math., 136 (2012), 521-573.doi: 10.1016/j.bulsci.2011.12.004.


    J. Dong and M.Xu, Some solutions to the space fractional Schrödinger equation using momentum representation method, J. Math. Phys., 48 (2007), 072105.doi: 10.1063/1.2749172.


    M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.doi: 10.1088/0951-7715/28/6/1937.


    P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A., 142 (2012), 1237-1262.doi: 10.1017/S0308210511000746.


    P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var., 54 (2015), 75-98.doi: 10.1007/s00526-014-0778-x.


    P. Felmer and C. Torres, Radial symmetry of ground state for a fractional nonlinear Schrödinger equation, Comm. Pure and Applied Ana., 13 (2014), 2395-2406.doi: 10.3934/cpaa.2014.13.2395.


    X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104.doi: 10.1063/1.2235026.


    N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.doi: 10.1016/S0375-9601(00)00201-2.


    N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108.doi: 10.1103/PhysRevE.66.056108.


    J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer, Berlin, 1989.doi: 10.1007/978-1-4757-2061-7.


    E. de Oliveira, F. Costa and J. Vaz, The fractional Schrödinger equation for delta potentials, J. Math. Phys., 51 (2012), 123517.doi: 10.1063/1.3525976.


    P. Rabinowitz, Minimax method in critical point theory with applications to differential equations, CBMS Amer. Math. Soc., 65, 1986.


    P. Rabinowitz, On a class of nonlinear Schrödinguer equations, ZAMP, 43 (1992), 270-291.doi: 10.1007/BF00946631.


    R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.doi: 10.1016/j.jmaa.2011.12.032.


    R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.


    S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^n$, J. Math. Phys., 54 (2013), 031501.doi: 10.1063/1.4793990.


    J. Zhang and W. Jiang, Existence and concentration of solutions for a fractional Schrödinger equations with sublinear nonlinearity, arXiv:1502.02221v1.

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