A high regularity result of solutions to modified pp-Stokes equations

This paper is concerned with a special quasilinear elliptic system, which can be seen as a perturbed pp-Laplacean, p∈(1,2)p∈(1,2), in the whole space RnRn. For its “shape”, it is close to the pp-Stokes system. However, our quasilinear second-order differential operator is given by means of ∇u∇u and not by its symmetric part, so that our system cannot be considered as a pp-Stokes system. Hence, it is called modified  pp-Stokes system. We look for the high regularity of the solutions (u,π)(u,π), in the sense of D2u,∇π∈Lq(Rn),q∈(n,∞)D2u,∇π∈Lq(Rn),q∈(n,∞). In particular, we get ∇u,π∈C0,α(Rn)∇u,π∈C0,α(Rn). As far as we know, such a result of high regularity is the first concerning the coupling of unknowns (u,π)(u,π). However, our result also holds for the pp-Laplacean, and it is the first high regularity result in an unbounded domain.