Existence of solutions for fractional Schrödinger equation with asymptotically periodic terms
-
2325
Downloads
-
4695
Views
Authors
Da-Bin Wang
- Department of Applied Mathematics, Lanzhou University of Technology, 730050 Lanzhou, People’s Republic of China.
Man Guo
- Department of Applied Mathematics, Lanzhou University of Technology, 730050 Lanzhou, People’s Republic of China.
Wen Guan
- Department of Applied Mathematics, Lanzhou University of Technology, 730050 Lanzhou, People’s Republic of China.
Abstract
In this paper, we investigate the following nonlinear fractional Schr¨odinger equation
\[(-\Delta)^su + V(x)u = f(x, u), x \in \mathbb{R}^N,\]
where \(s \in (0, 1),N > 2\) and \((-\Delta)^s\) is fractional Laplacian operator. We prove that the problem has a non-trivial solution under
asymptotically periodic case of \(V\) and \(f\) at infinity. Moreover, the nonlinear term \(f\) does not satisfy any monotone condition and
Ambrosetti-Rabinowitz condition.
Share and Cite
ISRP Style
Da-Bin Wang, Man Guo, Wen Guan, Existence of solutions for fractional Schrödinger equation with asymptotically periodic terms, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 2, 625--636
AMA Style
Wang Da-Bin, Guo Man, Guan Wen, Existence of solutions for fractional Schrödinger equation with asymptotically periodic terms. J. Nonlinear Sci. Appl. (2017); 10(2):625--636
Chicago/Turabian Style
Wang, Da-Bin, Guo, Man, Guan, Wen. "Existence of solutions for fractional Schrödinger equation with asymptotically periodic terms." Journal of Nonlinear Sciences and Applications, 10, no. 2 (2017): 625--636
Keywords
- asymptotically periodic
- Fractional Schrödinger equation
- variational method.
MSC
References
-
[1]
C. O. Alves, M. A. S. Souto, S. H. M. Soares, Schrödinger -Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 377 (2011), 584–592.
-
[2]
A. Ambrosetti, M. Badiale, S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285–300.
-
[3]
B. Barrios, E. Colorado, A. de Pablo, U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2011), 6133–6162.
-
[4]
H. Berestycki, P.-L. Lions, Nonlinear scalar field equations, II, Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347–375.
-
[5]
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (1996)
-
[6]
G. M. Bisci, V. D. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985–3008.
-
[7]
G. M. Bisci, V. D. Rădulescu, R. Servadei, Variational methods for nonlocal fractional problems, With a foreword by Jean Mawhin,/ Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (2016)
-
[8]
X. Cabré, J.-G. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052–2093.
-
[9]
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245–1260.
-
[10]
X. Chang, Z.-Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity , Nonlinearity, 26 (2013), 479–494.
-
[11]
M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 54 (2012), 7 pages.
-
[12]
R. de Marchi, Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh Sect. A , 145 (2015), 745–757.
-
[13]
M. del Pino, P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1–32.
-
[14]
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573.
-
[15]
Y.-H. Ding, F.-H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, 30 (2007), 231–249.
-
[16]
S. Dipierro, G. Palatucci, E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania), 68 (2013), 201–216.
-
[17]
P. Felmer, A. Quaas, J.-G. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237–1262.
-
[18]
J. Giacomoni, P. K. Mishra, K. Sreenadh, Fractional elliptic equations with critical exponential nonlinearity, Adv. Nonlinear Anal., 5 (2016), 57–74.
-
[19]
L. Jeanjean, K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, 21 (2004), 287–318.
-
[20]
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298–305.
-
[21]
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 7 pages.
-
[22]
L. Li, V. Rădulescu, D. Repovš, Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness, Int. J. Nonlinear Sci. Numer. Simul., 17 (2016), 325–333.
-
[23]
G.-B. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763–776.
-
[24]
H. F. Lins, E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890–2905.
-
[25]
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society , Providence, RI (1986)
-
[26]
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270–291.
-
[27]
M. Schechter, A variation of the mountain pass lemma and applications, J. London Math. Soc., 44 (1991), 491–502.
-
[28]
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^N\), J. Math. Phys., 54 (2013), 17 pages.
-
[29]
S. Secchi, On fractional Schrödinger equations in \(\mathbb{R}^N\)without the Ambrosetti-Rabinowitz condition, Topol. Methods Nonlinear Anal., 47 (2016), 19–41.
-
[30]
X.-D. Shang, J.-H. Zhang , Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187–207.
-
[31]
E. A. B. Silva, G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth , Calc. Var. Partial Differential Equations, 39 (2010), 1–33.
-
[32]
E. A. B. Silva, G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935–2949.
-
[33]
A. Szulkin, T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, Int. Press, Somerville, MA, (2010), 597–632.
-
[34]
J.-G. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21–41 .
-
[35]
K.-M. Teng, Multiple solutions for a class of fractional Schrödinger equations in \(\mathbb{R}^N\), Nonlinear Anal. Real World Appl., 21 (2015), 76–86.
-
[36]
H. Weitzner, G. M. Zaslavsky, Some applications of fractional equations , Chaotic transport and complexity in classical and quantum dynamics, Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 273–281.
-
[37]
M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. , Boston, MA (1996)
-
[38]
H. Zhang, J.-X. Xu, F.-B. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in \(\mathbb{R}^N\), J. Math. Phys., 56 (2015), 13 pages.
-
[39]
X. Zhang, B.-L. Zhang, D. Repovš, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal., 142 (2016), 48–68.
-
[40]
X. Zhang, B.-L. Zhang, M.-Q. Xiang, Ground states for fractional Schrödinger equations involving a critical nonlinearity, Adv. Nonlinear Anal., 5 (2016), 293–314.