Beyond the strong maximum principle for quasilinear equations

We study an extension of the strong maximum principle for quasilinear Dirichlet problems with sign-changing right-hand sides. For equations driven by the p-Laplacian, we establish sufficient conditions ensuring that supersolutions remain strictly positive in the domain even when the datum is indefinite. We also show that, under additional assumptions, positive solutions may fail to satisfy the Hopf-Oleinik boundary point lemma and instead exhibit flat boundary behavior, characterized by a vanishing weighted normal derivative. These results extend the previous linear theory to the quasilinear setting and are applied to sub-homogeneous indefinite equations and to a sign-indefinite version of Vazquez's strong maximum principle.