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We consider a strongly nonlinear differential equation of the followinggeneral 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 strictlyincreasing homeomorphism (the $\Phi$-Laplacian operator) and the function $a$is continuous and non-negative. We assume that $a(t,x)$ is bounded from belowby a non-negative function $h(t)$, independent of $x$ and such that $1/h \inL^p(0,T)$ for some $p> 1$, and we require a weak growth condition ofWintner-Nagumo type. Under these assumptions, we prove existence results forthe Dirichlet problem associated to the above equation, as well as fordifferent boundary conditions. Our approach combines fixed point techniques andthe upper/lower solutions method.

Existence results for boundary value problems associated with singular strongly nonlinear equations