On the solvability of singular boundary value problems on the real line in the critical growth case

Combining fixed point techniques with the method of lower-upper solutions we prove the existence of at least one weak solution for the following boundary value problem \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} \left( \, \Phi(a(t, x(t)) \, x'(t) )\, \right)' = f(t, x(t), x'(t)) &\mbox{ in } \mathbb{R}\\ x(-\infty) = \nu_{1}, \quad x(+\infty) = \nu_{2} \end{array} \right. \end{equation*} $\end{document} where \begin{document}$ \nu_{1}, \nu_{2}\in \mathbb{R} $\end{document} , \begin{document}$ \Phi: \mathbb{R} \rightarrow \mathbb{R} $\end{document} is a strictly increasing homeomorphism extending the classical \begin{document}$ p $\end{document} -Laplacian, \begin{document}$ a $\end{document} is a nonnegative continuous function on \begin{document}$ \mathbb{R} \times \mathbb{R} $\end{document} which can vanish on a set having zero Lebesgue measure and \begin{document}$ f $\end{document} is a Caratheodory function on \begin{document}$ \mathbb{R} \times \mathbb{R}^{2} $\end{document} .