# American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131
##### References:
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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. 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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