A strong comparison principle for the p-laplacian

We consider weak solutions of the differential inequality of p-Laplacian type -(p)u - f(u) <= - Delta(p)v - f(v) such that u <= v on a smooth bounded domain in RN 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 equivalent to v equivalent to 0 on the boundary of the domain we prove u < v unless u equivalent to 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.