  We investigate qualitative properties of weak solutions of the Dirichlet
problem for the equation $-\Delta_p u = \lambda m(x)|u|^{p-2}u + \eta
a(x)|u|^{q-2}u + f(x)$ 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 weak
and strong forms. Some of our results, especially on the validity of the
antimaximum principle under low regularity assumptions, are new for the
unperturbed problem with $\eta=0$, and among them there are results providing
new information even in the linear case $p=2$. In particular, we show that for
any $p>1$ solutions of the unperturbed problem satisfy the antimaximum
principle in a right neighborhood of the first eigenvalue of the $p$-Laplacian
provided $m,f \in L^\gamma(\Omega)$ with $\gamma>N$. For completeness, we also
investigate the existence of solutions.
