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

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 $\mathbb{R}^N$ with zero Dirichlet boundary conditions. We apply our results to the nonlinear elliptic eigenvalue problem $-\Delta_p u = \lambda f(u)$, with Dirichlet boundary condition, where $f$ is a nondecreasing continuous differentiable function on such that f(0)>0, $f(t) ^{1/(p-1)}$ 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