The first and the third authors have proved the global existence
of holomorphic solutions to partial differential equations
with several Fuchsian variables in the sense of N.S. Madi,
under some assumptions on some coefficients and on the Fuchsian
characteristic polynomial. This article shows a similar global
existence of solutions which are holomorphic with respect
to Fuchsian variables and of projective Gevrey class with
respect to non-Fuchsian variables. The proof is based on
the concept of formal norms of Leray-Waelbroeck and the application
of Gevrey's operator of C. Wagschal.