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<βα<β, Φ:(−r,r)→RΦ:(−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)y↦f(t,x,y) and y↦Φ(y)y↦Φ(y) as y→0y→0, and of t↦f(t,x,y)t↦f(t,x,y) as |t|→+∞|t|→+∞. Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions.