  We study the existence, multiplicity, and certain qualitative properties of
solutions to the zero Dirichlet problem for the equation $-\Delta_p u = \lambda
|u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$,
where $1<q<p$, $\lambda\in\mathbb{R}$, and $a$ is a continuous sign-changing
weight function. Our primary interest concerns ground states and nonnegative
solutions which are positive in $\{x\in \Omega: a(x)>0\}$, when the parameter
$\lambda$ lies in a neighborhood of the critical value $\lambda^* =
\inf\left\{\int_\Omega |\nabla u|^p \, dx/\int_\Omega |u|^p \, dx: u\in
W_0^{1,p}(\Omega) \setminus \{0\},\ \int_\Omega a|u|^q\,dx \geq 0\,\right\}$.
Among main results, we show that if $p>2q$ and either $\int_\Omega
a\varphi_p^q\,dx=0$ or $\int_\Omega a\varphi_p^q\,dx>0$ is sufficiently small,
then such solutions do exist in a right neighborhood of $\lambda^*$. Here
$\varphi_p$ is the first eigenfunction of the Dirichlet $p$-Laplacian in
$\Omega$. This existence phenomenon is of a purely subhomogeneous and nonlinear
nature, since either in the superhomogeneous case $q>p$ or in the sublinear
case $q<p=2$ the nonexistence takes place for any $\lambda \geq \lambda^*$.
Moreover, we prove that if $p>2q$ and $\int_\Omega a\varphi_p^q\,dx>0$ is
sufficiently small, then there exist three nonzero nonnegative solutions in a
left neighborhood of $\lambda^*$, two of which are strictly positive in $\{x\in
\Omega: a(x)>0\}$.
