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