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 $\Delta_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply the results to nonlinear elliptic eigenvalue problem $- \Delta_{p} u = \lambda f(u)$, with Dirichlet boundary condition, where $f$ is a nondecreasing continuous differentiable function on $[0,\infty]$ 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 $\lambda_{p}^{*}$. In particular, we consider the nonlinearities $f(u) = e^u$ and $f(u) = (1+u)^{m}$ ($m > p-1$) and give explicit estimates on $\lambda_{p}^{*}$. 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 $N$.