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1. A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” J. Funct. Anal. 14, 349381 (1973).
2. B. Barrios, E. Colorado, A. de Pablo, and U. Sánchez, “On some critical problems for the fractional Laplacian operator,” J. Differ. Equations 252, 61336162 (2012).
3. C. Brändle, E. Colorado, and A. de Pablo, “A concave-convex elliptic problem involving the fractional Laplacian,” Proc. - R. Soc. Edinburgh, Sect. A: Math. 143, 3971 (2013).
4. X. Cabré and Y. Sire, “Nonlinear equations for fractional Laplacians I: Regularity, maximum principle and Hamiltonian estimates,” Ann. Inst. Henri Poincare (C) Non Linear Anal. (published online).
5. L. Caffarelli and L. Silvestre, “An extension problem related to the fractional Laplacian,” Commun. Partial. Differ. Equ. 32, 12451260 (2007).
6. A. Capella, J. Dávila, L. Dupaigne, and Y. Sire, “Regularity of radial extremal solutions for some nonlocal semilinear equations,” Commun. Partial. Differ. Equ. 36, 13531384 (2011).
7. X. J. Chang, “Ground states of some fractional Schrödinger equations on ,” Proc. Edinb. Math. Soc. (to be published).
8. X. J. Chang and Z.-Q. Wang, “Ground state of scalar field equations involving fractional Laplacian with general nonlinearity,” Nonlinearity 26, 479494 (2013).
9. W. Chen, C. Li, and B. Ou, “Classification of solutions for an integral equation,” Commun. Pure Appl. Math. 59, 330343 (2006).
10. M. Cheng, “Bound state for the fractional Schrödinger equation with unbounded potential,” J. Math. Phys. 53, 043507 (2012).
11. V. Coti Zelati and M. Nolasco, “Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations,” Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend. Lincei, Mat. Appl. 22, 5172 (2011).
12. V. Coti Zelati and M. Nolasco, “Ground states for pseudo-relativistic Hartree equations of critical type,” Rev. Mat. Iberoam. (to be published).
13. S. Dipierro, G. Palatucci, and E. Valdinoci, “Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian,” Le Matematiche (Catania) 68, 201216 (2013).
14. E. B. Fabes, C. E. Kenig, and R. P. Serapioni, “The local regularity of solutions of degenerate elliptic equations,” Commun. Part. Differ. Equ. 7, 77116 (1982).
15. P. Felmer, A. Quaas, and J. G. Tan, “Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,” Proc. - R. Soc. Edinburgh, Sect. A: Math. 142, 12371262 (2012).
16. R. Frank and E. Lenzmann, “Uniqueness and nondegeneracy of ground states for (−Δ)sQ + QQα+1 = 0 in ,” e-print arXiv:1009.4042.
17. R. L. Frank, E. Lenzmann, and L. Silvestre, “Uniqueness of radial solutions for the fractional Laplacian,” e-print arXiv:1302.2652.
18. M. M. Fall and E. Valdinoci, “Uniqueness and nondegeneracy of positive solutions of (−Δ)su + u = up in when s is close to 1,” e-print arXiv:1301.4868.
19. L. Jeanjean and K. Tanaka, “A positive solution for an asymptotically linear elliptic problem on autonomous at infinity,” ESAIM Control Optim. Calc. Var. 7, 597614 (2002).
20. N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268, 298305 (2000).
21. N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
22. Y. Q. Li, Z.-Q. Wang, and J. Zeng, “Ground states of nonlinear Schrödinger equations with potentials,” Ann. Inst. Henry Poincaré, Anal. Non Linéaire 23, 829837 (2006).
23. P. L. Lions, “Symétrie et compacité dans les espaces de Sobolev,” J. Funct. Anal. 49, 315334 (1982).
24. G. Palatucci, O. Savin, and E. Valdinoci, “Local and global minimizers for a variational energy involving a fractional norm,” Ann. Mat. Pura Appl. 73(1–2), 3752 (2011).
25. P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics Vol. 65 (American Mathematical Society, Providence, RI, 1986).
26. P. H. Rabinowitz, “On a class of nonlinear Schrödinger equations,” Z. Angew. Math. Phys. 43, 270291 (1992).
27. M. Schechter, “A variation of the mountain pass lemma and applications,” J. London Math. Soc. 44, 491502 (1991).
28. S. Secchi, “Ground state solutions for nonlinear fractional Schröinger equations in ,” J. Math. Phys. 54, 031501 (2013).
29. R. Servadei and E. Valdinoci, “Mountain Pass solutions for non-local elliptic operators,” J. Math. Anal. Appl. 389, 887898 (2012).
30. R. Servadei and E. Valdinoci, “Variational methods for non-local operators of elliptic type,” Discrete Contin. Dyn. Syst. 33, 21052137 (2013).
31. R. Servadei and E. Valdinoci, “A Brezis-Nirenberg result for non-local critical equations in low dimension,” Commun. Pure Appl. Anal. (to be published).
32. C. A. Stuart, “Self-trapping of an electromagnetic 3eld and bifurcation from the essential spectrum,” Arch. Ration. Mech. Anal. 113, 6596 (1991).
33. C. A. Stuart, “Magnetic field wave equations for TM-modes in nonlinear optical waveguides,” in Reaction Diffusion Systems, edited by G. Caristi and E. Mitidieri (Marcel Dekker, New York, 1997).
34. C. A. Stuart and H. S. Zhou, “Applying the mountain pass theorem to an asymptotically linear elliptic equation on ,” Commun. Part. Differ. Equ. 24, 17311758 (1999).
35. A. Szulkin and T. Weth, “Ground state solutions for some indefinite variational problems,” J. Funct. Anal. 257, 38023822 (2009).

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This paper is devoted to a time-independent fractional Schrödinger equation of the form where ⩾ 2, ∈ (0, 1), (−Δ) stands for the fractional Laplacian. We apply the variational methods to obtain the existence of ground state solutions when (, ) is asymptotically linear with respect to at infinity.


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