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March  2016, 15(2): 535-547. doi: 10.3934/cpaa.2016.15.535

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

1. 

Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Trujillo, Perú, Chile

Received  July 2015 Revised  December 2015 Published  January 2016

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.
Citation: César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535
References:
[1]

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

[2]

T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 7. Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problems related to the fractional Laplacian,, \emph{Comm. PDE}, 32 (2007). doi: 10.1080/03605300600987306. Google Scholar

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X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

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

[6]

M. Cheng, Bound state for the fractional Schrödinger equation with undounded potential,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.3701574. Google Scholar

[7]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 2359. doi: 10.3934/cpaa.2014.13.2359. Google Scholar

[8]

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

[9]

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

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

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

[12]

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

[13]

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

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

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

[16]

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation,, \emph{J. Math. Phys.}, 47 (2006). doi: 10.1063/1.2235026. Google Scholar

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

[18]

N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar

[19]

J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems,, Applied Mathematical Sciences \textbf{74}, 74 (1989). doi: 10.1007/978-1-4757-2061-7. Google Scholar

[20]

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

[21]

P. Rabinowitz, Minimax method in critical point theory with applications to differential equations,, \emph{CBMS Amer. Math. Soc.}, 65 (1986). Google Scholar

[22]

P. Rabinowitz, On a class of nonlinear Schrödinguer equations,, \emph{ZAMP}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar

[23]

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

[24]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 2105. Google Scholar

[25]

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

[26]

J. Zhang and W. Jiang, Existence and concentration of solutions for a fractional Schrödinger equations with sublinear nonlinearity,, \emph{arXiv:1502.02221v1}., (). Google Scholar

show all references

References:
[1]

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

[2]

T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 7. Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problems related to the fractional Laplacian,, \emph{Comm. PDE}, 32 (2007). doi: 10.1080/03605300600987306. Google Scholar

[4]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[5]

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

[6]

M. Cheng, Bound state for the fractional Schrödinger equation with undounded potential,, \emph{J. Math. Phys.}, 53 (2012). doi: 10.1063/1.3701574. Google Scholar

[7]

G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 2359. doi: 10.3934/cpaa.2014.13.2359. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation,, \emph{J. Math. Phys.}, 47 (2006). doi: 10.1063/1.2235026. Google Scholar

[17]

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

[18]

N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar

[19]

J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems,, Applied Mathematical Sciences \textbf{74}, 74 (1989). doi: 10.1007/978-1-4757-2061-7. Google Scholar

[20]

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

[21]

P. Rabinowitz, Minimax method in critical point theory with applications to differential equations,, \emph{CBMS Amer. Math. Soc.}, 65 (1986). Google Scholar

[22]

P. Rabinowitz, On a class of nonlinear Schrödinguer equations,, \emph{ZAMP}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar

[23]

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

[24]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 2105. Google Scholar

[25]

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

[26]

J. Zhang and W. Jiang, Existence and concentration of solutions for a fractional Schrödinger equations with sublinear nonlinearity,, \emph{arXiv:1502.02221v1}., (). Google Scholar

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