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We investigate qualitative properties of weak solutions of the Dirichletproblem for the equation $-\Delta_p u = \lambda m(x)|u|^{p-2}u + \etaa(x)|u|^{q-2}u + f(x)$ in a bounded domain $\Omega \subset \mathbb{R}^N$, where$q<p$. Under certain regularity and qualitative assumptions on the weights $m,a$ and the source function $f$, we identify ranges of parameters $\lambda$ and$\eta$ for which solutions satisfy maximum and antimaximum principles in weakand strong forms. Some of our results, especially on the validity of theantimaximum principle under low regularity assumptions, are new for theunperturbed problem with $\eta=0$, and among them there are results providingnew information even in the linear case $p=2$. In particular, we show that forany $p>1$ solutions of the unperturbed problem satisfy the antimaximumprinciple in a right neighborhood of the first eigenvalue of the $p$-Laplacianprovided $m,f \in L^\gamma(\Omega)$ with $\gamma>N$. For completeness, we alsoinvestigate the existence of solutions.

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