On solvability in the small of higher order elliptic equations in grand-Sobolev spaces

This work deals with themth order elliptic equation with non-smooth coefficients in grand-Sobolev space generated by the norm of the grand-Lebesgue space L-q)(Omega), 1 < q < +infinity. These spaces are nonseparable, and therefore, to use classical methods for treating solvability problems in these spaces, you need to modify these methods. To this aim, we consider some subspace, where the infinitely differentiable functions are dense. Then we prove that this subspace is invariant with respect to the singular integral operator and with respect to the multiplication operator by a function from L-infinity. Finally, using classical method of parametrics, we prove the existence in the small of the solution to the considered equation in W-q)(m)(Omega).