We consider solutions to \(-\Delta_p u=f(x)\) in \(\Omega\), when \(p\) approaches the semilinear limiting case \(p = 2\) and we get third order estimates. As a consequence we deduce improved regularity properties of the stress field. The results presented in this seminar are contained in a recent paper with D. Baratta and B. Sciunzi. Finally, I present a \(L^\infty\)-estimate for the second derivative of the solution. This is a recent result in preparation with F. Iandoli. This talk is part of the activity of the research Project : PRIN 2022 PNRR project 2022AKNSE4 "Variational and Analytical aspects of Geometric PDEs" , founded by the European Union- Next Generation EU.