Rocky Mountain Journal of Mathematics

On the existence of ground states of nonlinear fractional Schrödinger systems with close-to-periodic potentials

Gongming Wei and Xueliang Duan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We are concerned with the nonlinear fractional Schrödinger system \begin{equation} \begin{cases}(-\Delta )^{s} u+V_{1}(x)u=f(x,u)+\Gamma (x)|u|^{q-2}u|v|^{q} &\mbox {in } \mathbb {R}^{N},\\ (-\Delta )^{s} v+V_{2}(x)v=g(x,v)+\Gamma (x)|v|^{q-2}v|u|^{q} &\mbox {in } \mathbb {R}^{N},\\ u,v\in H^{s}(\mathbb {R}^{N}), \end{cases} \end{equation} where $(-\Delta )^{s}$ is the fractional Laplacian operator, $s\in (0,1)$, $N>2s$, $4\leq 2q\lt p\lt 2^{\ast }$, $2^{\ast }={2N}/({N-2s})$. $V_{i}(x)=V^{i}_{per }(x)+V^{i}_{loc }(x)$ is closed-to-periodic for $i=1,2$, $f$ and $g$ have subcritical growths and $\Gamma (x)\geq 0$ vanishes at infinity. Using the Nehari manifold minimization technique, we first obtain a bounded minimizing sequence, and then we adopt the approach of Jeanjean-Tanaka (2005) to obtain a decomposition of the bounded Palais-Smale sequence. Finally, we prove the existence of ground state solutions for the nonlinear fractional Schrödinger system.

Article information

Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1647-1683.

First available in Project Euclid: 19 October 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35A15: Variational methods 35J60: Nonlinear elliptic equations

Nonlinear fractional Schrödinger system Nehari manifold Fatou's lemma Lions' lemma Mountain Pass geometry Vitali convergence theorem


Wei, Gongming; Duan, Xueliang. On the existence of ground states of nonlinear fractional Schrödinger systems with close-to-periodic potentials. Rocky Mountain J. Math. 48 (2018), no. 5, 1647--1683. doi:10.1216/RMJ-2018-48-5-1647.

Export citation


  • A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
  • V. Benci, C.R. Grisanti and A.M. Micheletti, Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with $V(\infty)=0$, Topol. Meth. Nonlin. Anal. 26 (2005), 203–219.
  • B. Bieganowski, Solutions of the fractional Schrödinger equation with a sign-changing nonlinearity, J. Math. Anal. Appl. 450 (2017), 461–479.
  • B. Bieganowski and J. Mederski, Nonlinear Schrödinger equatins with sum of periodic and vanishing potentials and sign-changing nonlinearities, Comm. Pure Appl. Anal., to appear.
  • V. Coti-Zelati and P. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^{N}$, Comm. Pure Appl. Math. 45 (1992), 1217–1269.
  • W. Dörfler, A. Lechleiter, M. Plum, G. Schneider and C. Wieners, Photonic crystals: Mathematical analysis and numerical approximation, Springer, Basel, 2012.
  • G. Figueiredo and H.R. Quoirin, Ground states of elliptic problems involving non homogeneous operators, Indiana Univ. Math. J. 65 (2016), 779–795.
  • L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^{N}$, Indiana Univ. Math. J. 54 (2005), 443–464.
  • Y. Li, Z.-Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. Poincare Anal. 23 (2006), 829–837.
  • P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part I, Ann. Inst. Poincare Anal. 1 (1984), 109–145.
  • ––––, The concentration-compactness principle in the calculus of variations, The locally compact case, Part II, Ann. Inst. Poincare Anal. 1 (1984), 223–283.
  • S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Part. Diff. Eqs. 45 (2012), 1–9.
  • L.A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Eqs. 229 (2006), 743–767.
  • J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Part. Diff. Eqs. 41 (2016), 1426–1440.
  • ––––, Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum, Topol. Meth. Nonlin. Anal. 46 (2015), 755–771.
  • Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. \textbf 95 (1960), 101–123.
  • ––––, Characteristic values associated with a class of non-linear seond-order differential equations, Acta Math. \textbf 105 (1961), 141–175.
  • E.D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
  • A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259–287.
  • P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
  • R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102.
  • A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), 3802–3822.
  • ––––, The method of Nehari manifold, Handbook of nonconvex analysis and applications, International Press, Somerville, 2010.
  • M. Willem, Minimax theorems, Birkhäuser Verlag, Berlin, 1996.