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

Feliz Minhós

Mathematische Nachrichten, 2019, vol. 292, issue 4, 850-858

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

Date: 2019
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