## Taiwanese Journal of Mathematics

### Multiple Solutions of Nonlinear Schrödinger Equations with the Fractional $p$-Laplacian

#### Abstract

We use two variant fountain theorems to prove the existence of infinitely many weak solutions for the following fractional $p$-Laplace equation$(-\Delta)^\alpha_p u + V(x) |u|^{p-2}u = f(x,u), \quad x \in \mathbb{R}^N,$where $N \geq 2$, $p \geq 2$, $\alpha \in (0,1)$, $(-\Delta)^\alpha_p$ is the fractional $p$-Laplacian and $f$ is either asymptotically linear or subcritical $p$-superlinear growth. Under appropriate assumptions on $V$ and $f$, we prove the existence of infinitely many nontrivial high or small energy solutions. Our results generalize and extend some existing results.

#### Article information

Source
Taiwanese J. Math., Volume 21, Number 5 (2017), 1017-1035.

Dates
Revised: 20 December 2016
Accepted: 4 January 2017
First available in Project Euclid: 1 August 2017

https://projecteuclid.org/euclid.twjm/1501599181

Digital Object Identifier
doi:10.11650/tjm/7947

Mathematical Reviews number (MathSciNet)
MR3707882

Zentralblatt MATH identifier
06871357

#### Citation

Luo, Huxiao; Tang, Xianhua; Li, Shengjun. Multiple Solutions of Nonlinear Schrödinger Equations with the Fractional $p$-Laplacian. Taiwanese J. Math. 21 (2017), no. 5, 1017--1035. doi:10.11650/tjm/7947. https://projecteuclid.org/euclid.twjm/1501599181

#### References

• L. Ambrosio, G. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math. 134 (2011), no. 3-4, 377–403.
• D. Applebaum, Lévy processes–-from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), no. 11, 1336–1347.
• B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), no. 11, 6133–6162.
• L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), no. 9, 1111–1144.
• L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 8, 1245–1260.
• P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 6, 1237–1262.
• B. Ge, Multiple solutions of nonlinear Schrödinger equation with the fractional Laplacian, Nonlinear Anal. Real World Appl. 30 (2016), 236–247.
• A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl. 414 (2014), no. 1, 372–385.
• ––––, Weyl-type laws for fractional $p$-eigenvalue problems, Asymptot. Anal. 88 (2014), no. 4, 233-245.
• N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), no. 4-6, 298–305.
• R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), no. 31, R161–R208.
• E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.
• X. H. Tang, Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl. 401 (2013), no. 1, 407-415.
• K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. Real World Appl. 14 (2013), no. 1, 867–874.
• ––––, Multiple solutions for a class of fractional Schrödinger equations in $\mb{R}^N$, Nonlinear Anal. Real World Appl. 21 (2015), 76–86.
• J. Xu, Z. Wei and W. Dong, Existence of weak solutions for a fractional Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), no. 1-3, 1215–1222.
• J. Zhang, X. Tang and W. Zhang, Semiclassical solutions for a class of Schrödinger system with magnetic potentials, J. Math. Anal. Appl. 414 (2014), no. 1, 357–371.
• ––––, Existence of multiple solutions of Kirchhoff type equation with sign-changing potential, Appl. Math. Comput. 242 (2014), 491–499.
• ––––, Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, J. Math. Anal. Appl. 420 (2014), no. 2, 1762–1775.
• ––––, Existence of infinitely many solutions for a quasilinear elliptic equation, Appl. Math. Lett. 37 (2014), 131–135.
• ––––, Existence and multiplicity of stationary solutions for a class of Maxwell-Dirac system, Nonlinear Anal. 127 (2015), 298–311.
• ––––, Ground state solutions for a class of nonlinear Maxwell-Dirac system, Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 785–798.
• ––––, Ground states for diffusion system with periodic and asymptotically periodic nonlinearity, Comput. Math. Appl. 71 (2016), no. 2, 633–641.
• J. Zhang, W. Zhang and X. Xie, Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Commun. Pure Appl. Anal. 15 (2016), no. 2, 599–622.
• W. Zhang, X. Tang and J. Zhang, Infinitely many solutions for fourth-order elliptic equations with sign-changing potential, Taiwanese J. Math. 18 (2014), no. 2, 645–659.
• W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001), no. 3, 343–358.