A note on the global regularity results for strongly nonhomogeneous $p,q$-fractional problems and applications

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 $(-\mathrm{\Delta }{)}_p^{s_1}+(-\mathrm{\Delta }{)}_q^{s_2}$, for $s_2,s_1\in (0,1)$ and $1<p,q<\mathrm{\infty }$. 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}.