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

We investigate qualitative properties of weak solutions the Dirichlet problem for equation $$-\Delta _p u = \lambda \,m(x)|u|^{p-2}u+ \eta \,a(x)|u|^{q-2}u+ f(x)$$ in a bounded domain $$\Omega \subset \mathbb {R}^N$$ , where $$q1$$ right neighborhood first eigenvalue p-Laplacian provided $$m,f \in L^\gamma (\Omega )$$ $$\gamma >N$$ For completeness, also existence solutions.