On heteroclinic solutions for BVPs involving ϕ ‐Laplacian operators without asymptotic or growth assumptions

In this paper we consider the second order discontinuous equation in the real line,( ϕ ( a ( t ) u ′ ( t ) ) ) ′=f ( t , u ( t ) , u ′ ( t ) ) , a.e. t ∈ R ,u ( − ∞ )=A ,u ( + ∞ ) = B ,with ϕ an increasing homeomorphism such that ϕ ( 0 ) = 0 and ϕ ( R ) = R , a ∈ C ( R ) with a ( t ) > 0 , for t ∈ R , f : R 3 → R a L 1 ‐Carathéodory function and A , B ∈ R verifying an adequate relation. We remark that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities ϕ and f . Moreover, as far as we know, our main result is even new when ϕ ( y ) = y , that is, for the equation( a ( t ) u ′ ( t ) ) ′ = f ( t , u ( t ) , u ′ ( t ) ) , a.e. t ∈ R .