Applications of Interpolation theory to the regularity of some quasilinear PDEs

We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -\div(|\nabla u|^{p-2}\nabla u)+V(x;u)=f, \end{equation*}where $\Omega$ is a bounded smooth domain of $\mathbb R^n$, $V$ is a nonlinear potential and $f$ belongs to non-standard spaces like Lorentz-Zygmund spaces. The results, which have been exposed by the third author in a talk presented in AGMA 2024, the International Scientific Online Conference «Algebraic and geometric methods of analysis», May 27-30, 2024, Ukraine, constitute only a part of results proved in detail in a paper coauthored with I. Ahmed, A. Fiorenza, A. Gogatishvili, A. El Hamidi and J.M. Rakotoson (\cite{AFFGER2023}). Moreover, we collect some well-known and new results of the identication of some interpolation spaces and we enrich some contents with details.