  Under different assumptions on the potential functions $b$ and $c$, we study
the fractional equation $\left( I-\Delta \right)^{\alpha} u = \lambda b(x)
|u|^{p-2}u+c(x)|u|^{q-2}u$ in $\mathbb{R}^N$. Our existence results are based
on compact embedding properties for weighted spaces.
