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 { (x Φ((−∞ a(t, ) x = (t ν )) 1, x0(x t)) (+ ) ∞0 = ) f = (t, ν2 x(t), x0(t)) in R where ν1, ν2 ∈ R, Φ: R → R is a strictly increasing homeomorphism extending the classical p-Laplacian, a is a nonnegative continuous function on R×R which can vanish on a set having zero Lebesgue measure and f is a Carathéodory function on R × R2