Symmetry and monotonicity results for solutions of vectorial 𝑝-Stokes systems

Abstract

In this paper we shall study qualitative properties of a p p -Stokes type system, namely − Δ p u = − d i v ⁡ ( | D u | p − 2 D u ) = f ( x , u )  in  Ω , \begin{equation*} -{\boldsymbol \Delta }_p{\boldsymbol u}=-\operatorname {\mathbf {div}}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol u}) \text { in $\Omega $}, \end{equation*} where Δ p {\boldsymbol \Delta }_p is the p p -Laplacian vectorial operator. More precisely, under suitable assumptions on the domain Ω \Omega and the function f \boldsymbol { f} , it is deduced that system solutions are symmetric and monotone. Our main results are derived from a vectorial version of the weak and strong comparison principles, which enable to proceed with the moving-planes technique for systems. As far as we know, these are the first qualitative kind results involving vectorial operators.