| Summary: | 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.
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