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

Abstract

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:R³→R a L¹-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 equation

(a(t)u′(t))′=f(t,u(t),u′(t)), a.e.t∈R.