On elliptic equations with subhomogeneous indefinite nonlinearity

We will discuss the existence and multiplicity, as well as some qualitative properties of nonnegative solutions of the zero Dirichlet problem for the quasilinear equation
$$-\Delta_p u - \lambda u^{p-1} = a(x) u^{q-1} $$
in a bounded domain, where $1<q<p$ and the function $a(x)$ is sign-changing. A distinctive feature of this problem is the fact that its nonnegative solutions do not necessarily satisfy the strong maximum principle. As a consequence, the set of solutions might have a rich structure. We will show, in particular, that for some $p \neq 2$ there are nontrivial effects which are impossible in the linear case $p=2$.