A maximum principle for the $p$-Laplacian, an eigenvalue estimate and a stabilization phenomenon for the large-$p$ regime

We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First, we obtain a new lower bound for the first nontrivial eigenvalue of the $p$-Laplacian, which improves upon existing estimates in certain parameter regimes and for thin domains. Second, we prove an existence theorem for nonlinear boundary value problems of the form $$-{\mathrm{\Delta }}_{p}u=\lambda f(u)\text{in }\mathrm{\Omega },u=0\text{on }\mathrm{\partial }\mathrm{\Omega },$$ with $f$ nonnegative, continuous and nondecreasing. A striking consequence is the emergence of a \emph{stabilization phenomenon}: for every such nonlinearity there exists a threshold $p_0\text{\hspace{0.17em}⁣}:=p_0(f,\lambda ,\mathrm{\Omega })$ such that for all $p\ge p_0$ solutions exist. To our knowledge, this stabilization effect with respect to $p$, that apparently has not been observed before, suggests a connection to the $\mathrm{\infty }$-Laplacian.