On the lack of interior regularity of the p ‐Poisson problem with p > 2

In this note we are concerned with interior regularity properties of the p ‐Poisson problemΔ p ( u ) = f with p > 2 . For all 0 < λ ≤ 1 we constuct right‐hand sides f of differentiability − 1 + λ such that the (Besov‐) smoothness of corresponding solutions u is essentially limited to 1 + λ / ( p − 1 ) . The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savaré [J. Funct. Anal. 152 (1998), 176–201], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130 (2016), 298–329].