*Mathematical Methods in the Applied Sciences*Ground states for critical fractional
Schr\”{o}dinger-Poisson systems with vanishing
potentials

This paper deals with a class of fractional
Schr\”{o}dinger-Poisson system
\[\begin{cases}\displaystyle
(-\Delta )^{s}u+V(x)u-K(x)\phi
|u|^{2^*_s-3}u=a (x)f(u), &x
\in
\R^{3}\\
(-\Delta
)^{s}\phi=K(x)|u|^{2^*_s-1},
&x \in
\R^{3}\end{cases}
\]with a critical nonlocal term and multiple competing
potentials, which may decay and vanish at infinity, where $s
\in (\frac{3}{4},1)$, $ 2^*_s =
\frac{6}{3-2s}$ is the fractional critical exponent.
The problem is set on the whole space and compactness issues have to be
tackled. By employing the mountain pass theorem,
concentration-compactness principle and approximation method, the
existence of a positive ground state solution is obtained under
appropriate assumptions imposed on $V, K, a$ and $f$.

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