American Institute of Mathematical Sciences

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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

Chao Ji 1,2,,

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  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.

Keywords: Fractional logarithmic Schrödinger equation, periodic potential, coercive potential, bounded potential, nonsmooth critical point theory.
Mathematics Subject Classification: Primary: 35J60, 35R11; Secondary: 47J30.
Citation: 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
References:
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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. Google Scholar

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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. Google Scholar

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

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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. Google Scholar

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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. Google Scholar

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

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E. Di Nezza, G. 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. Google Scholar

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

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

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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F. O. de Pavia, W. 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. Google Scholar

[16]

P. Pucci, M. 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. Google Scholar

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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. Google Scholar

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

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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. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

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

[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. Google Scholar

[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. Google Scholar

[6]

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

[7]

E. Di Nezza, G. 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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[15]

F. O. de Pavia, W. 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. Google Scholar

[16]

P. Pucci, M. 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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

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