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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 and 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 $\mathbbR^n$, Commun. in PDE, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [2] 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. [3] L. Caffarelli and L. Silvestre, An extension problems related to the fractional Laplacian, Comm. PDE, 32 (2007), 1245C1260.2. doi: 10.1080/03605300600987306. [4] 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. [5] 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. [6] M. Cheng, Bound state for the fractional Schrödinger equation with undounded potential, J. Math. Phys., 53 (2012), 043507. doi: 10.1063/1.3701574. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104. doi: 10.1063/1.2235026. [17] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. [18] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108. [19] 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. [20] 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. [21] P. Rabinowitz, Minimax method in critical point theory with applications to differential equations, CBMS Amer. Math. Soc., 65, 1986. [22] P. Rabinowitz, On a class of nonlinear Schrödinguer equations, ZAMP, 43 (1992), 270-291. doi: 10.1007/BF00946631. [23] 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. [24] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. [25] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^n$, J. Math. Phys., 54 (2013), 031501. doi: 10.1063/1.4793990. [26] J. Zhang and W. Jiang, Existence and concentration of solutions for a fractional Schrödinger equations with sublinear nonlinearity,, \emph{arXiv:1502.02221v1}., ().

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##### References:
 [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. [2] 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. [3] L. Caffarelli and L. Silvestre, An extension problems related to the fractional Laplacian, Comm. PDE, 32 (2007), 1245C1260.2. doi: 10.1080/03605300600987306. [4] 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. [5] 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. [6] M. Cheng, Bound state for the fractional Schrödinger equation with undounded potential, J. Math. Phys., 53 (2012), 043507. doi: 10.1063/1.3701574. [7] 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. [8] 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. [9] 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. [10] 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. [11] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys., 47 (2006), 082104. doi: 10.1063/1.2235026. [17] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. [18] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108. [19] 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. [20] 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. [21] P. Rabinowitz, Minimax method in critical point theory with applications to differential equations, CBMS Amer. Math. Soc., 65, 1986. [22] P. Rabinowitz, On a class of nonlinear Schrödinguer equations, ZAMP, 43 (1992), 270-291. doi: 10.1007/BF00946631. [23] 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. [24] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. [25] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^n$, J. Math. Phys., 54 (2013), 031501. doi: 10.1063/1.4793990. [26] J. Zhang and W. Jiang, Existence and concentration of solutions for a fractional Schrödinger equations with sublinear nonlinearity,, \emph{arXiv:1502.02221v1}., ().
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