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 problem $\Delta_p(u)=f$ with $p>2$. For all $0<\lambda\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+\lambda$ such that the (Besov-) smoothness of corresponding solutions $u$ is essentially limited to $1+\lambda / (p-1)$. The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savar\'e [J. Funct. Anal. 152:176-201, 1998], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130:298-329, 2016]. Keywords: Nonlinear and adaptive approximation, Besov space, regularity of solutions, $p$-Poisson problem.