  We consider the forced problem $-\Delta_p u - V(x)|u|^{p-2} u = f(x)$, where
$\Delta_p$ is the $p$-Laplacian ($1<p<\infty$) in a domain $\Omega\subset
\mathbb{R}^N$, $V\ge 0$ and $Q_V (u) := \int_\Omega |\nabla u|^p\, dx -
\int_\Omega V|u|^p\,dx$ satisfies the condition (A) stated at the beginning of
the paper. We show that this problem has a solution for all $f$ in a suitable
space of distributions. Then we apply this result to some classes of functions
$V$ which in particular include the Hardy potential and the potential
$V(x)=\lambda_{1,p}(\Omega)$, where $\lambda_{1,p}(\Omega)$ is the Poincar\'e
constant on an infinite strip.
