Heteroclinic solutions of boundary value problems on the real line involving singular Φ-Laplacian operators

We discuss the solvability of the following strongly nonlinear BVP: { ( a ( x ( t ) ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , t ∈ R , x ( − ∞ ) = α , x ( + ∞ ) = β where α < β , Φ : ( − r , r ) → R is a general increasing homeomorphism with bounded domain (singular Φ-Laplacian), a is a positive, continuous function and f is a Carathéodory nonlinear function. We give conditions for the existence and non-existence of heteroclinic solutions in terms of the behavior of y ↦ f ( t , x , y ) and y ↦ Φ ( y ) as y → 0 , and of t ↦ f ( t , x , y ) as | t | → + ∞ . Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions.