# American Institute of Mathematical Sciences

December  2016, 21(10): 3407-3428. doi: 10.3934/dcdsb.2016104

## Infinitely many solutions of the nonlinear fractional Schrödinger equations

 1 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China, China

Received  January 2015 Revised  March 2016 Published  November 2016

In this paper, we study the fractional Schrödinger equation \begin{equation*} (-\Delta)^{s} u+V(x)u=f(x,u), \quad x\in\mathbb{R}^{N}, \end{equation*} where $0< s <1$, $(-\Delta)^{s}$ denotes the fractional Laplacian of order $s$ and the nonlinearity $f$ is sublinear or superlinear at infinity. Under certain assumptions on $V$ and $f$, we prove that this equation has infinitely many solutions via variational methods, which unifies and sharply improves the recent results of Teng (2015) [33]. Moreover, we also consider the above equation with concave and critical nonlinearities, and obtain the existence of infinitely many solutions.
Citation: Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104
##### References:
 [1] S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion,, Physica A, 356 (2005), 403. doi: 10.1016/j.physa.2005.03.035. Google Scholar [2] B. Barrios, E. Colorado, A. de Pablo and U. Sáchez, On some critical problems for the fractional Laplacian operator,, J. Differential Equations, 252 (2012), 6133. doi: 10.1016/j.jde.2012.02.023. Google Scholar [3] T. Bartsch, A. Pankov and Z. Wang, Nonlinear Schröinger equations with steep potential well,, Commun. Contemp. Math., 3 (2001), 549. doi: 10.1142/S0219199701000494. Google Scholar [4] T. Bartsch, Z. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations,, in: Handbook of Differential Equations-Stationary Partial Differential Equations, 2 (2005), 1. doi: 10.1016/S1874-5733(05)80009-9. Google Scholar [5] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [6] J. Bertoin, Lévy Processes,, Cambridge Tracts in Mathematics, (1996). Google Scholar [7] Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions,, Nonlinearity, 28 (2015), 2247. doi: 10.1088/0951-7715/28/7/2247. Google Scholar [8] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar [10] G. Y. Chen and Y. Q. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 2359. doi: 10.3934/cpaa.2014.13.2359. Google Scholar [11] M. Cheng, Bound state for the fractional Schröinger equation with unbounded potential,, J. Math. Phys., 53 (2012). doi: 10.1063/1.3701574. Google Scholar [12] R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financ. Math. Ser., (2004). Google Scholar [13] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schröinger type problem involving the fractional laplacian,, Matematiche (Catania), 68 (2013), 201. Google Scholar [14] M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $(-\Delta)^su +u=u^p$ in $\mathbbR^N$ when $s$ is close to 1,, Comm. Math. Phys., 329 (2014), 383. doi: 10.1007/s00220-014-1919-y. Google Scholar [15] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schröinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237. doi: 10.1017/S0308210511000746. Google Scholar [16] M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,, Comm. Pure Appl. Math., 62 (2009), 198. doi: 10.1002/cpa.20253. Google Scholar [17] L. Jeanjean, On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer type problem set on $\mathbbR^N$,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787. doi: 10.1017/S0308210500013147. Google Scholar [18] N. Laskin, Fractional quantum mechanics and lévy path integrals,, Phys. Lett. A, 268 (2000), 298. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar [19] N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar [20] G. Molica Bisci, D. Repovš and R. Servadei, Nontrivial solutions for superlinear nonlocal problems,, preprint., (). Google Scholar [21] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [22] R. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Reg. Conf. Ser. in Math., (1986). doi: 10.1090/cbms/065. Google Scholar [23] P. H. Rabinowitz, On a class of nonlinear Schröinger equations,, Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [24] S. Secchi, Ground state solutions for nonlinear fractional Schröinger equations in $\mathbbR^N$,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4793990. Google Scholar [25] R. Servadei, A critical fractional Laplace equation in the resonant case,, Topol. Methods Nonlinear Anal., 43 (2014), 251. doi: 10.12775/TMNA.2014.015. Google Scholar [26] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete Contin. Dyn. Syst., 33 (2013), 2105. Google Scholar [27] R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension,, Commun. Pure Appl. Anal., 12 (2013), 2445. doi: 10.3934/cpaa.2013.12.2445. Google Scholar [28] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., 367 (2015), 67. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar [29] X. D. Shang and J. H. Zhang, Ground states for fractional Schrödinger equations with critical growth,, Nonlinearity, 27 (2014), 187. doi: 10.1088/0951-7715/27/2/187. Google Scholar [30] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 67. doi: 10.1002/cpa.20153. Google Scholar [31] W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar [32] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21. doi: 10.1007/s00526-010-0378-3. Google Scholar [33] K. M. Teng, Multiple solutions for a class of fractional Schröngdinger equations in $\mathbbR^N$,, Nonlinear Anal. Real World Appl., 71 (2015), 4927. doi: 10.1016/j.nonrwa.2014.06.008. Google Scholar [34] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial,, Order and Chaos, (2008). Google Scholar [35] H. Weitzner and 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. doi: 10.1016/S1007-5704(03)00049-2. Google Scholar [36] Willem, Minimax Theorems,, Birkhäser, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar [37] W. Zou, Variant fountain theorems and their applications,, Manuscripta Math., 104 (2001), 343. doi: 10.1007/s002290170032. Google Scholar

show all references

##### References:
 [1] S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion,, Physica A, 356 (2005), 403. doi: 10.1016/j.physa.2005.03.035. Google Scholar [2] B. Barrios, E. Colorado, A. de Pablo and U. Sáchez, On some critical problems for the fractional Laplacian operator,, J. Differential Equations, 252 (2012), 6133. doi: 10.1016/j.jde.2012.02.023. Google Scholar [3] T. Bartsch, A. Pankov and Z. Wang, Nonlinear Schröinger equations with steep potential well,, Commun. Contemp. Math., 3 (2001), 549. doi: 10.1142/S0219199701000494. Google Scholar [4] T. Bartsch, Z. Wang and M. Willem, The Dirichlet problem for superlinear elliptic equations,, in: Handbook of Differential Equations-Stationary Partial Differential Equations, 2 (2005), 1. doi: 10.1016/S1874-5733(05)80009-9. Google Scholar [5] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [6] J. Bertoin, Lévy Processes,, Cambridge Tracts in Mathematics, (1996). Google Scholar [7] Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions,, Nonlinearity, 28 (2015), 2247. doi: 10.1088/0951-7715/28/7/2247. Google Scholar [8] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar [10] G. Y. Chen and Y. Q. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 2359. doi: 10.3934/cpaa.2014.13.2359. Google Scholar [11] M. Cheng, Bound state for the fractional Schröinger equation with unbounded potential,, J. Math. Phys., 53 (2012). doi: 10.1063/1.3701574. Google Scholar [12] R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financ. Math. Ser., (2004). Google Scholar [13] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schröinger type problem involving the fractional laplacian,, Matematiche (Catania), 68 (2013), 201. Google Scholar [14] M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $(-\Delta)^su +u=u^p$ in $\mathbbR^N$ when $s$ is close to 1,, Comm. Math. Phys., 329 (2014), 383. doi: 10.1007/s00220-014-1919-y. Google Scholar [15] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schröinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237. doi: 10.1017/S0308210511000746. Google Scholar [16] M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps,, Comm. Pure Appl. Math., 62 (2009), 198. doi: 10.1002/cpa.20253. Google Scholar [17] L. Jeanjean, On the existence of bounded Palais-Smale sequences and applications to a Landesman-Lazer type problem set on $\mathbbR^N$,, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787. doi: 10.1017/S0308210500013147. Google Scholar [18] N. Laskin, Fractional quantum mechanics and lévy path integrals,, Phys. Lett. A, 268 (2000), 298. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar [19] N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar [20] G. Molica Bisci, D. Repovš and R. Servadei, Nontrivial solutions for superlinear nonlocal problems,, preprint., (). Google Scholar [21] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [22] R. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Reg. Conf. Ser. in Math., (1986). doi: 10.1090/cbms/065. Google Scholar [23] P. H. Rabinowitz, On a class of nonlinear Schröinger equations,, Z. Angew. Math. Phys., 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar [24] S. Secchi, Ground state solutions for nonlinear fractional Schröinger equations in $\mathbbR^N$,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4793990. Google Scholar [25] R. Servadei, A critical fractional Laplace equation in the resonant case,, Topol. Methods Nonlinear Anal., 43 (2014), 251. doi: 10.12775/TMNA.2014.015. Google Scholar [26] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete Contin. Dyn. Syst., 33 (2013), 2105. Google Scholar [27] R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension,, Commun. Pure Appl. Anal., 12 (2013), 2445. doi: 10.3934/cpaa.2013.12.2445. Google Scholar [28] R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, Trans. Amer. Math. Soc., 367 (2015), 67. doi: 10.1090/S0002-9947-2014-05884-4. Google Scholar [29] X. D. Shang and J. H. Zhang, Ground states for fractional Schrödinger equations with critical growth,, Nonlinearity, 27 (2014), 187. doi: 10.1088/0951-7715/27/2/187. Google Scholar [30] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, Comm. Pure Appl. Math., 60 (2007), 67. doi: 10.1002/cpa.20153. Google Scholar [31] W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149. doi: 10.1007/BF01626517. Google Scholar [32] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21. doi: 10.1007/s00526-010-0378-3. Google Scholar [33] K. M. Teng, Multiple solutions for a class of fractional Schröngdinger equations in $\mathbbR^N$,, Nonlinear Anal. Real World Appl., 71 (2015), 4927. doi: 10.1016/j.nonrwa.2014.06.008. Google Scholar [34] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial,, Order and Chaos, (2008). Google Scholar [35] H. Weitzner and 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. doi: 10.1016/S1007-5704(03)00049-2. Google Scholar [36] Willem, Minimax Theorems,, Birkhäser, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar [37] W. Zou, Variant fountain theorems and their applications,, Manuscripta Math., 104 (2001), 343. doi: 10.1007/s002290170032. Google Scholar
 [1] João Marcos do Ó, Uberlandio Severo. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Communications on Pure & Applied Analysis, 2009, 8 (2) : 621-644. doi: 10.3934/cpaa.2009.8.621 [2] Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076 [3] D. Motreanu, Donal O'Regan, Nikolaos S. Papageorgiou. A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1791-1816. doi: 10.3934/cpaa.2011.10.1791 [4] Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991 [5] Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099 [6] Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427 [7] Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188 [8] Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008 [9] Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337 [10] Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555 [11] Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 [12] Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265 [13] Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa. On the orbital stability of fractional Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1267-1282. doi: 10.3934/cpaa.2014.13.1267 [14] Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359 [15] Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $l(s^2)$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056 [16] Wenxiong Chen, Congming Li, Jiuyi Zhu. Fractional equations with indefinite nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1257-1268. doi: 10.3934/dcds.2019054 [17] Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 [18] Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic & Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215 [19] Ryuji Kajikiya, Daisuke Naimen. Two sequences of solutions for indefinite superlinear-sublinear elliptic equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1593-1612. doi: 10.3934/cpaa.2014.13.1593 [20] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

2018 Impact Factor: 1.008