  In this article, we communicate with the glimpse of the proofs of global
regularity results for weak solutions to a class of problems involving
fractional $(p,q)$-Laplacian, denoted by
$(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and
$1<p,q<\infty$. We also obtain the boundary H\"older continuity results for the
weak solutions to the corresponding problems involving at most critical growth
nonlinearities. These results are almost optimal. Moreover, we establish Hopf
type maximum principle and strong comparison principle. As an application to
these new results, we prove the Sobolev versus H\"older minimizer type result,
which provides the multiplicity of solutions in the spirit of seminal work
\cite{Brezis-Nirenberg}.
