# American Institute of Mathematical Sciences

November  2019, 12(7): 2115-2125. doi: 10.3934/dcdss.2019136

## Ground states of nonlinear Schrödinger equations with fractional Laplacians

 1 Center for Applied Mathematics, Guangzhou University, Guangzhou 510405, China 2 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Qinqin Zhang

Received  December 2017 Revised  April 2018 Published  December 2018

Fund Project: This work was supported by National Natural Science Foundation (11701114, 11471085) and the program for Changjiang scholars and Innovative Recearch Team in univesity (Grant No.IRT1226)

Inspired by Schaftingen [15], we develop a symmetric variational principle for the field equation involving a fractional Laplacians
 \begin{equation*} \left\{ \begin{aligned} (-\Delta)^\alpha u+u& = f(u), x\in\mathbb{R}^N,\\ u(x)&\geq 0. \end{aligned} \right. \end{equation*}
As an application, we prove the existence of symmetric ground states in the fractional Sobolev space
 $H^\alpha (\mathbb{R}^N)$
. These results improve some known ones in the literature. An example is also given to illustrate our results.
Citation: Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136
##### References:
 [1] S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407. Google Scholar [2] F. Almgren and E. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc, 2 (1989), 683-773. doi: 10.1090/S0894-0347-1989-1002633-4. Google Scholar [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [4] D. Applebaum, Lévy processes-from probability to finance and quantum groups, Not. Am. Math. Soc, 51 (2004), 1336-1347. Google Scholar [5] X. 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. Google Scholar [6] E. Di Nezza, G. Palatucci 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. Google Scholar [7] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, LXVIII, 68 (2013), 201-216. Google Scholar [8] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrodinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar [9] P. Felmer, A. Quaas, M. Tang and J. Yu, Monotonicity properties for ground states of the scalar field equation, Ann.I.Poincare-AN, 25 (2008), 105-119. doi: 10.1016/j.anihpc.2006.12.003. Google Scholar [10] M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253. Google Scholar [11] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^N$, Proc. Amer. Math. Soc, 131 (2002), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1. Google Scholar [12] L. Jeanjean and K. Tanaka, A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^N$ autonomous at infinity, ESAIM Control Optim. Calc. Var, 7 (2002), 597-614. doi: 10.1051/cocv:2002068. Google Scholar [13] E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar [14] S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9. doi: 10.1007/s00526-011-0447-2. Google Scholar [15] J. Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math., 7 (2005), 463-481. doi: 10.1142/S0219199705001817. Google Scholar [16] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013. Google Scholar [17] X. Tang, Non-Nehari manifold method for asymptoticallyperiodic Schrödinger equations,, Sci. China Math., 58 (2015), 715-728. doi: 10.1007/s11425-014-4957-1. Google Scholar [18] X. Tang, Non-Nehari-manifold method for asymptoticallylinear Schrödinger equation,, J. Aust. Math. Soc., 98 (2015), 104-116. doi: 10.1017/S144678871400041X. Google Scholar [19] X. Tang, X. Lin and J. Yu, Nontrivial solutions for Schrödinger equation with local supper-quadratic conditions, J. Dyn. Diff. Equat., accepted for publication. doi: 10.1007/s10884-018-9662-2. Google Scholar [20] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial, Order and Chaos, Patras University Press, 2008. Google Scholar [21] Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124. doi: 10.1007/s00526-013-0706-5. Google Scholar [22] 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), 273-281. doi: 10.1016/S1007-5704(03)00049-2. Google Scholar [23] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. 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), 403-407. Google Scholar [2] F. Almgren and E. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc, 2 (1989), 683-773. doi: 10.1090/S0894-0347-1989-1002633-4. Google Scholar [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [4] D. Applebaum, Lévy processes-from probability to finance and quantum groups, Not. Am. Math. Soc, 51 (2004), 1336-1347. Google Scholar [5] X. 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. Google Scholar [6] E. Di Nezza, G. Palatucci 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. Google Scholar [7] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, LXVIII, 68 (2013), 201-216. Google Scholar [8] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrodinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar [9] P. Felmer, A. Quaas, M. Tang and J. Yu, Monotonicity properties for ground states of the scalar field equation, Ann.I.Poincare-AN, 25 (2008), 105-119. doi: 10.1016/j.anihpc.2006.12.003. Google Scholar [10] M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253. Google Scholar [11] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbb{R}^N$, Proc. Amer. Math. Soc, 131 (2002), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1. Google Scholar [12] L. Jeanjean and K. Tanaka, A positive solution for an asymptotically linear elliptic problem on $\mathbb{R}^N$ autonomous at infinity, ESAIM Control Optim. Calc. Var, 7 (2002), 597-614. doi: 10.1051/cocv:2002068. Google Scholar [13] E. Lieb and M. Loss, Analysis, Grad. Stud. Math., vol. 14, Amer. Math. Soc, Providence, RI, 2001. doi: 10.1090/gsm/014. Google Scholar [14] S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9. doi: 10.1007/s00526-011-0447-2. Google Scholar [15] J. Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math., 7 (2005), 463-481. doi: 10.1142/S0219199705001817. Google Scholar [16] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013. Google Scholar [17] X. Tang, Non-Nehari manifold method for asymptoticallyperiodic Schrödinger equations,, Sci. China Math., 58 (2015), 715-728. doi: 10.1007/s11425-014-4957-1. Google Scholar [18] X. Tang, Non-Nehari-manifold method for asymptoticallylinear Schrödinger equation,, J. Aust. Math. Soc., 98 (2015), 104-116. doi: 10.1017/S144678871400041X. Google Scholar [19] X. Tang, X. Lin and J. Yu, Nontrivial solutions for Schrödinger equation with local supper-quadratic conditions, J. Dyn. Diff. Equat., accepted for publication. doi: 10.1007/s10884-018-9662-2. Google Scholar [20] L. Vlahos, H. Isliker, Y. Kominis and K. Hizonidis, Normal and Anomalous Diffusion: A Tutorial, Order and Chaos, Patras University Press, 2008. Google Scholar [21] Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124. doi: 10.1007/s00526-013-0706-5. Google Scholar [22] 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), 273-281. doi: 10.1016/S1007-5704(03)00049-2. Google Scholar [23] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl., vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar
 [1] Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091 [2] 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 [3] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [4] Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131 [5] Daniele Garrisi, Vladimir Georgiev. Orbital stability and uniqueness of the ground state for the non-linear Schrödinger equation in dimension one. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4309-4328. doi: 10.3934/dcds.2017184 [6] Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026 [7] Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074 [8] Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $\mathbb{R} ^{3}$. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079 [9] Marco A. S. Souto, Sérgio H. M. Soares. Ground state solutions for quasilinear stationary Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 99-116. doi: 10.3934/cpaa.2013.12.99 [10] Zhanping Liang, Yuanmin Song, Fuyi Li. Positive ground state solutions of a quadratically coupled schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (3) : 999-1012. doi: 10.3934/cpaa.2017048 [11] Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120 [12] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813 [13] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71 [14] Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168 [15] Maoding Zhen, Jinchun He, Haoyuan Xu, Meihua Yang. Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6523-6539. doi: 10.3934/dcds.2019283 [16] Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009 [17] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [18] Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025 [19] Zhitao Zhang, Haijun Luo. Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction. Communications on Pure & Applied Analysis, 2018, 17 (3) : 787-806. doi: 10.3934/cpaa.2018040 [20] Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214

2018 Impact Factor: 0.545