On the antimaximum principle for the p-Laplacian and its sublinear perturbations

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $$-\Delta _p u = \lambda \,m(x)|u|^{p-2}u+ \eta \,a(x)|u|^{q-2}u+ f(x)$$ - Δ p u = λ m ( x ) | u | p - 2 u + η a ( x ) | u | q - 2 u + f ( x ) in a bounded domain $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N , where $$q 1$$ p > 1 solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the p-Laplacian provided $$m,f \in L^\gamma (\Omega )$$ m , f ∈ L γ ( Ω ) with $$\gamma >N$$ γ > N . For completeness, we also investigate the existence of solutions.