  We consider a strongly nonlinear differential equation of the following
general type $$(\Phi(a(t,x(t)) \, x'(t)))'= f(t,x(t),x'(t)), \quad \text{a.e.
on $[0,T]$}$$ where $f$ is a Carath\'edory function, $\Phi$ is a strictly
increasing homeomorphism (the $\Phi$-Laplacian operator) and the function $a$
is continuous and non-negative. We assume that $a(t,x)$ is bounded from below
by a non-negative function $h(t)$, independent of $x$ and such that $1/h \in
L^p(0,T)$ for some $p> 1$, and we require a weak growth condition of
Wintner-Nagumo type. Under these assumptions, we prove existence results for
the Dirichlet problem associated to the above equation, as well as for
different boundary conditions. Our approach combines fixed point techniques and
the upper/lower solutions method.
