Fractional $p\mathrm{\&}q$ Laplacian problems in ${\mathbb{R}}^N$ with critical growth

We deal with the following nonlinear problem involving fractional $p\mathrm{\&}q$ Laplacians: \begin{equation*} (-\Delta)^{s}{p}u+(-\Delta)^{s}{q}u+|u|^{p-2}u+|u|^{q-2}u=\lambda h(x) f(u)+|u|^{q^{}_{s}-2}u \mbox{ in } $\mathbb{R}$^{N}, \end{equation} where $s\in (0,1)$,$1<p<q<\frac{N}{s}$,$q^{*}_{s}=\frac{Nq}{N-sq}$,$\lambda>0$ is a parameter, $h$ is a nontrivial bounded perturbation and $f$ is a superlinear continuous function with subcritical growth. Using suitable variational arguments and concentration-compactness lemma, we prove the existence of a nontrivial non-negative solution for $\lambda$ sufficiently large.