A strong comparison principle for the -Laplacian

We consider weak solutions of the differential inequality of p-Laplacian type

$- \Delta_p u - f(u) \le - \Delta_p v - f(v)$

such that $u\leq v$ on a smooth bounded domain in $\mathbb{R}^N$ and either $u$ or $v$ is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that $u<v$ on the boundary of the domain we prove that $u<v$, and assuming that $u\equiv v\equiv0$ on the boundary of the domain we prove $u < v$ unless $u \equiv v$. The novelty is that the nonlinearity $f$ is allowed to change sign. In particular, the result holds for the model nonlinearity $f(s) = s^q - \lambda s^{p-1}$ with $q >p-1$.