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

Abstract

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 ${\lambda }_p^{\ast }$. In particular, we consider the nonlinearities $f(u)=e^u$ and $f(u)=(1+u{)}^m$ ($m>p-1$) and give explicit estimates on ${\lambda }_p^{\ast }$. 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 $N$.