Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the $p$-Laplacian and application

We derive a priori bounds for positive supersolutions of $ - \Delta_{p} u = \rho(x) f(u) $,where$p>1$and$\Delta_{p}$isthe$p$-Laplace operator, in a smooth bounded domain of $R^N$ with zero Dirichlet boundary conditions. We apply the results to nonlinear elliptic eigenvalue problem $ - \Delta_{p} u = \lambda f(u) $,withDirichletboundarycondition,where$ f $ is a nondecreasing continuous differentiable function on $[0,\mathrm{\infty }]$ such that $f(0)>0$, $f(t{)}^{\frac{1}{p-1}}$ is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter ${\lambda }_p^{\ast }$. In particular, we consider the nonlinearities $f(u)=e^u$ and $ f(u) = (1+u)^{m} $($ m > p-1 $)andgiveexplicitestimateson$ \lambda_{p}^{*} $. As a by-product of our results, we obtain a lower bound for the principal eigenvalue of the $p$-Laplacian that improves obtained results in the recent literature for some range of $p$ and $N$.