  This paper deals with the qualitative analysis of solutions
  to the following $(p,q)$-fractional equation: \begin{equation*}
\begin{array}{rllll} (-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x)
\big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\ba} \; \text{ in } \mb
R^N, \end{array} \end{equation*} \noi where $1< q< p$, $0<s_2\leq s_1<1$,
$ps_1=N$, $\ba\in[0,N)$, and $V,K:\mb R^N\to\mb R$, $f:\mb R\to \mb R$ are
continuous functions satisfying some natural hypotheses. We are concerned both
with the case when $f$ has a subcritical growth and with the critical framework
with respect to the exponential nonlinearity. By combining a Moser-Trudinger
type inequality for fractional Sobolev spaces with Schwarz symmetrization
techniques and related variational methods, we prove the existence of
nonnegative solutions.

Analysis of PDEs