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Abstract

In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation $\begin{matrix} {(- Δ)}^{α} u + V (x) u = A (ϵ x) f (u), x \in R^{N}, \end{matrix}$ where $α \in (0, 1)$ , ϵ is a positive parameter, $N > 2 α$ , ${(- Δ)}^{α}$ stands for the fractional Laplacian, f is a continuous function with subcritical growth, $V \in C (R^{N}, R)$ is a $Z^{N}$ -periodic function and $A \in C (R^{N}, R)$ satisfies some appropriate assumptions.

Keywords

Concentration of solutions ground state fractional Schrödinger equations periodic potential Nehari manifold

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References

C.O.

Alves and

G.F.

Germano, Existence and concentration of ground state solution for a class of indefinite variational problem, Commun. Pure Appl. Anal. 19 (2020), 2887–2906. doi:10.3934/cpaa.2020126.

C.O.

Alves and

O.H.

Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in

R^{N}

via penalization method, Calc. Var. Partial Differential Equations 55(47) (2016), 19.

Barrios,

Colorado,

de Pablo and

Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), 6133–6162. doi:10.1016/j.jde.2012.02.023.

Barrios,

Colorado,

de Pablo and

Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 39–71. doi:10.1017/S0308210511000175.

Bartsch,

Pankov and

Z.-Q.

Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math. 3 (2001), 1–21. doi:10.1142/S0219199701000275.

Bartsch and

Z.-Q.

Wang, Existence and multiplicity results for some superlinear elliptic problems on

R^{N}

, Comm. Partial Differential Equations 20 (1995), 1725–1741. doi:10.1080/03605309508821149.

Binhua,

Chen and

Liu, Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger–Choquard equation, Adv. Nonlinear Anal. 10(1) (2021), 311–330. doi:10.1515/anona-2020-0127.

Brezis, Functional Analysis, Sobolev Space and Partial Differential Equations, Springer, 2011.

Cabré and

Sire, Nnolinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23–53. doi:10.1016/j.anihpc.2013.02.001.

10.

X.J.

Chang and

Z.-Q.

Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity 26 (2013), 479–494. doi:10.1088/0951-7715/26/2/479.

11.

Coti Zelati and

P.H.

Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on

R^{N}

, Commun. Pure Appl. Math. 45 (1992), 1217–1269. doi:10.1002/cpa.3160451002.

12.

Fang and

Ji, On a fractional Schrödinger equation with periodic potential, Comput. Math. Appl. 78 (2019), 1517–1530. doi:10.1016/j.camwa.2019.03.044.

13.

Felmer,

Quaas and

J.G.

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

14.

Ji, Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger–Poisson system in

R^{3}

, Ann. Mat. Pura Appl. (4) 198(5) (2019), 1563–1579. doi:10.1007/s10231-019-00831-2.

15.

Ji, Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), 6071–6089.

16.

Ji and

Fang, Multiplicity of solutions for a perturbed fractional Schrödinger equation involving oscillatory terms, Electron. J. Differential Equations 126 (2018), 1–21.

17.

Kryszewski and

Szulkin, Generalized linking theorem with an alpplication to a semilinear Schrödinger equation, Adv. Differential Equations 3 (1988), 441–472.

18.

G.B.

Li and

Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math. 4 (2002), 763–776. doi:10.1142/S0219199702000853.

19.

S.B.

Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations 45 (2012), 1–9. doi:10.1007/s00526-011-0447-2.

20.

Z.L.

Liu and

Z.-Q.

Wang, On the Ambrosetti–Rabinowitz superlinear condition, Adv. Nonlinear Stud. 4 (2004), 561–572.

21.

Molica Bisci and

V.D.

Rădulescu, Ground state solutions of scalar field fractional for Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), 2985–3008. doi:10.1007/s00526-015-0891-5.

22.

Molica Bisci,

V.D.

Rădulescu and

Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2015.

23.

Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259–287. doi:10.1007/s00032-005-0047-8.

24.

Pu,

Liu and

C.L.

Tang, Existence of weak solutions for a class of fractional Schrödinger equations with periodic potential, Comput. Math. Appl. 73 (2017), 465–482. doi:10.1016/j.camwa.2016.12.004.

25.

P.H.

Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew Math. Phys. 43 (1992), 270–291. doi:10.1007/BF00946631.

26.

Secchi, On fractional Schrödinger equations in

R^{N}

without the Ambrosetti–Rabinowitz condition, Topol. Methods Nonlinear Anal. 47 (2016), 19–41.

27.

X.D.

Shang and

J.H.

Zhang, Ground states for fractional Schrödinger equations with critical growth, Nonlinearity 27 (2014), 187–207. doi:10.1088/0951-7715/27/2/187.

28.

Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67–112. doi:10.1002/cpa.20153.

29.

E.M.

Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

30.

Struwe, Variational Methods, Springer, 1996.

31.

Szulkin and

Weth, Ground state solutions for some indefinite problems, J. Funct. Anal. 257 (2009), 3802–3822. doi:10.1016/j.jfa.2009.09.013.

32.

X.F.

Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys. 53 (1993), 229–244. doi:10.1007/BF02096642.

33.

M.Q.

Xiang,

B.L.

Zhang and

V.D.

Rădulescu, Existence of solutions for perturbed fractional p-Laplacian equations, J. Differential Equations 260 (2016), 1392–1413. doi:10.1016/j.jde.2015.09.028.

34.

M.Q.

Xiang,

B.L.

Zhang and

V.D.

Rădulescu, Superlinear Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent, Adv. Nonlinear Anal. 9(1) (2020), 690–709. doi:10.1515/anona-2020-0021.

Existence and concentration of ground state solutions for a class of fractional Schrödinger equations

Abstract

Keywords

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