  In this paper, we consider the regularity of weak solutions (in an
appropriate space) to the elliptic partial differential equation
\begin{equation*} (-\Delta_{p})^{s} u + (-\Delta_{q})^{s} u = f(x) \quad
\text{in} \quad \mathbb{R}^{N}, \end{equation*} where $0<s<1$ and $ 2 \leq q
\leq p < N/s$. We prove that these solutions are locally in
$C^{0,\alpha}(\mathbb{R}^N)$, which seems to be optimal. Furthermore, we prove
the existence of solutions to the problem \begin{equation*} (-\Delta_{p})^{s} u
+ (-\Delta_{q})^{s} u = \vert u \vert^{p^{*}_{s}-2}u + \lambda g(x) \vert u
\vert^{r-2}u \,\,\, \text{in} \,\,\,\, \mathbb{R}^{N}, \end{equation*} where $1
< q\leq p < N/s$, $\lambda$ is a parameter and $g$ satisfies some conditions of
integrability. We also show that, if $g$ is bounded, then the solutions are
continuous and bounded.
