November  2019, 24(11): 6071-6089. doi: 10.3934/dcdsb.2019131

Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term

1. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

2. 

Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

* Corresponding author: Chao Ji

Received  October 2017 Revised  August 2018 Published  November 2019 Early access  May 2019

Fund Project: C. Ji was supported by Shanghai Natural Science Foundation(18ZR1409100), NSFC (grant No. 11301181, 11771324) and China Postdoctoral Science Foundation funded project.

In this paper we are concerned with the fractional Schrödinger equation $ (-\Delta)^{\alpha} u+V(x)u = f(x, u) $, $ x\in {{\mathbb{R}}^{N}} $, where $ f $ is superlinear, subcritical growth and $ u\mapsto\frac{f(x, u)}{\vert u\vert} $ is nondecreasing. When $ V $ and $ f $ are periodic in $ x_{1},\ldots, x_{N} $, we show the existence of ground states and the infinitely many solutions if $ f $ is odd in $ u $. When $ V $ is coercive or $ V $ has a bounded potential well and $ f(x, u) = f(u) $, the ground states are obtained. When $ V $ and $ f $ are asymptotically periodic in $ x $, we also obtain the ground states solutions. In the previous research, $ u\mapsto\frac{f(x, u)}{\vert u\vert} $ was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.

Citation: Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131
References:
[1]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${{\mathbb{R}}^{N}}$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[3]

J. Chabrowski, Variational Methods for Potential Operator Equations, de Gruyter, Berlin, 1997. doi: 10.1515/9783110809374.

[4]

K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0.

[5]

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.

[6]

R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman Hall/CRC Financial Mathematics Series, Boca Raton, 2004.

[7]

E. Di NezzaG. Palatucci and E. Valdinaci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[8]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[9]

N. Laskin, Fractional Schrödinger equations, Phys. Rev., 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.

[10]

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.

[11]

R. Metzler and J. Klafter, The random walls guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[12]

G. Molica Bisci and V. 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.

[13] G. Molica BisciV. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.
[14]

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

[15]

F. O. de PaviaW. Kryszewski and A. Szulkin, Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term, Proc. Amer. Math. Soc., 145 (2017), 4783-4794.  doi: 10.1090/proc/13609.

[16]

P. PucciM. Q. Xia and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.

[17]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$, J. Math. Phys., 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990.

[18]

S. Secchi, On fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$ without the Ambrosetti-Rabinowitz condition, Topol. Methods Nonlinear Anal., 47 (2016), 19-41. 

[19]

L. 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.

[20]

M. Struwe, Variational Methods, 2$^{nd}$ edition, 34. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03212-1.

[21]

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

[22]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[23]

H. Zhang, J. X. Xu and F. B. Zhao, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$, J. Math. Phys., 56 (2015), 091502, 13pp. doi: 10.1063/1.4929660.

[24]

X. Zhong and W. Zou, Ground state and multiple solutions via generalized Nehari mandifold, Nonlinear Anal., 102 (2014), 251-263.  doi: 10.1016/j.na.2014.02.018.

show all references

References:
[1]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${{\mathbb{R}}^{N}}$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[3]

J. Chabrowski, Variational Methods for Potential Operator Equations, de Gruyter, Berlin, 1997. doi: 10.1515/9783110809374.

[4]

K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0.

[5]

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.

[6]

R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman Hall/CRC Financial Mathematics Series, Boca Raton, 2004.

[7]

E. Di NezzaG. Palatucci and E. Valdinaci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[8]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[9]

N. Laskin, Fractional Schrödinger equations, Phys. Rev., 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.

[10]

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.

[11]

R. Metzler and J. Klafter, The random walls guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[12]

G. Molica Bisci and V. 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.

[13] G. Molica BisciV. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.
[14]

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

[15]

F. O. de PaviaW. Kryszewski and A. Szulkin, Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term, Proc. Amer. Math. Soc., 145 (2017), 4783-4794.  doi: 10.1090/proc/13609.

[16]

P. PucciM. Q. Xia and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.

[17]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$, J. Math. Phys., 54 (2013), 031501, 17pp. doi: 10.1063/1.4793990.

[18]

S. Secchi, On fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$ without the Ambrosetti-Rabinowitz condition, Topol. Methods Nonlinear Anal., 47 (2016), 19-41. 

[19]

L. 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.

[20]

M. Struwe, Variational Methods, 2$^{nd}$ edition, 34. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03212-1.

[21]

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

[22]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[23]

H. Zhang, J. X. Xu and F. B. Zhao, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in ${{\mathbb{R}}^{N}}$, J. Math. Phys., 56 (2015), 091502, 13pp. doi: 10.1063/1.4929660.

[24]

X. Zhong and W. Zou, Ground state and multiple solutions via generalized Nehari mandifold, Nonlinear Anal., 102 (2014), 251-263.  doi: 10.1016/j.na.2014.02.018.

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