Heteroclinic solutions for classical and singular φ-Laplacian non-autonomous differential equations

Abstract

In this paper, we consider the second order discontinuous differential equation in the real line, (a(t,u)ϕ(u′))′=f(t,u,u′),a.e.t∈R,u(−∞)=ν−,u(+∞)=ν+, with ϕ an increasing homeomorphism such that ϕ(0)=0 and ϕ(R)=R, a∈C(R2,R) with a(t,x)>0 for (t,x)∈R2, f:R3→R a L1-Carathéodory function and ν−,ν+∈R such that ν−<ν+. The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities ϕ and f. To the best of our knowledge, this result is even new when ϕ(y)=y, that is for equation (a(t,u(t))u′(t))′=f(t,u(t),u′(t)),a.e.t∈R. Moreover, these results can be applied to classical and singular ϕ-Laplacian equations and to the mean curvature operator.