Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 102, pp. 1-16.

Title: Non-trivial solutions of fractional Schrodinger-Poisson systems
       with sum of  periodic and vanishing potentials

Authors: Mingzhu Yu (Central South Univ., Changsha, Hunan, China)
         Haibo Chen (Central South Univ., Changsha, Hunan, China)

Abstract:
 We consider the fractional Schrodinger-Poisson system
 $$\displaylines{
 (-\Delta)^{\alpha}u+V(x)u+K(x){\Phi}(x)u=f(x,u)-{\Gamma(x)}|u|^{q-2}u
 \quad\text{in }\mathbb{R}^3,\cr
 (-\Delta)^{\beta}{\Phi}=K(x)u^2\quad\text{in }\mathbb{R}^3,
 }$$
 where $\alpha,\beta\in(0,1]$, $4\alpha+2\beta>3$, $4\leq q<2_{\alpha}^{\ast}$,
 K(x), $\Gamma(x)$ and f(x,u) are periodic in x, V is coercive or
 $V=V_{\rm per}+V_{\rm loc}$ is a sum of a periodic potential $V_{\rm per}$
 and a localized potential $V_{\rm loc}$. If f has the subcritical growth,
 but higher than $\Gamma(x)|u|^{q-2}u$, we establish  the existence and
 nonexistence of ground state solutions are dependent on the sign of $V_{\rm loc}$.
 Moreover, we prove  that such a problem admits infinitely many pairs of
 geometrically distinct solutions provided that V is periodic and f
 is odd in u. Finally, we investigate the existence of ground state solutions
 in the case of coercive potential V.

Submitted March 22, 2018. Published September 04, 2019.
Math Subject Classifications: 35B38, 35G99.
Key Words: Fractional Schrodinger-Poisson system; coercive potential;
           periodic and localized potential; Nehari manifold; variational method.