Existence and multiplicity of solutions of quasilinear equations with convex or nonconvex reaction term

We give existence, nonexistence and multiplicity results of nonnegative solutions for Dirichlet problems of the form
$$ -\Delta_pv=\lambda f(x)(1+g(v))^{p-1}\quad\text{in}\quad\Omega,\qquad u=0\quad\text{on}\quad\partial\Omega, $$
where $\Delta_p$ is the $p$-Laplacian $(p>1)$, $g$ is nondecreasing, superlinear, and possibly convex, $\lambda>0$ and $f\in L^1(\Omega)$, $f\ge0$. New information on the extremal solutions is given. Equations with measure data are also considered.