Heteroclinic solutions for a class of boundary value problems associated with singular equations

Abstract We obtain existence results for strongly nonlinear BVPs of type (P) ( Φ ( k ( t ) x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) a.e. on  [ 0 , ∞ ) , x ( 0 ) = ν 1 , x ( ∞ ) = ν 2 where Φ : R → R is a strictly increasing homeomorphism such that Φ ( 0 ) = 0 (the Φ -Laplacian operator ), k : [ 0 , ∞ ) → R is a non-negative continuous function which may vanish on a subset of [ 0 , ∞ ) of measure zero, f is a Caratheodory function and ν 1 , ν 2 ∈ R are fixed. Under mild assumptions, including a weak form of a Nagumo–Wintner growth condition, we prove the existence of heteroclinic solutions of (P) in the Sobolev space W loc 1 , p ( [ 0 , ∞ ) ) . Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions.