A STRONG COMPARISON PRINCIPLE FOR THE p-LAPLACIAN

We consider weak solutions of the differential inequality of pLaplacian type −∆pu− f(u) ≤ −∆pv − 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 ≡ v ≡ 0 on the boundary of the domain we prove u < v unless u ≡ v. The novelty is that the nonlinearity f is allowed to change sign. In particular, the result holds for the model nonlinearity f(s) = sq − λsp−1 with q > p− 1.