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

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Abstract

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

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

Dates
First available in Project Euclid: 19 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1539936040

Digital Object Identifier
doi:10.1216/RMJ-2018-48-5-1647

Mathematical Reviews number (MathSciNet)
MR3866563

Zentralblatt MATH identifier
06958796

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

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

Citation

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. https://projecteuclid.org/euclid.rmjm/1539936040


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