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

Chao Ji

doi:10.3934/dcdsb.2019131

Article Contents

2019, Volume 24, Issue 11: 6071-6089. Doi: 10.3934/dcdsb.2019131

This issue Previous Article Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria Next Article A Max-Cut approximation using a graph based MBO scheme

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
^* Corresponding author: Chao Ji

Received: October 2017

Revised: August 2018

Early access: May 2019

Published: August 2019

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

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Abstract

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.

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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 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.
[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 Bisci, V. 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 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.
[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.
[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.