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.
Citation
Gongming Wei. Xueliang Duan. "On the existence of ground states of nonlinear fractional Schrödinger systems with close-to-periodic potentials." Rocky Mountain J. Math. 48 (5) 1647 - 1683, 2018. https://doi.org/10.1216/RMJ-2018-48-5-1647
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