Homoclinic and heteroclinic solutions for non-autonomous Minkowski-curvature equations

Abstract

We deal with the non-autonomous parameter-dependent second-order differential equation δ(v′1−(v′)2)′+q(t)f(v)=0,t∈R,driven by a Minkowski-curvature operator. Here, δ>0, q∈L∞(R), f:[0,1]→R is a continuous function with f(0)=f(1)=0=f(α) for some α∈]0,1[, f(s)<0 for all s∈]0,α[ and f(s)>0 for all s∈]α,1[. Based on a careful phase-plane analysis, under suitable assumptions on q we prove the existence of strictly increasing heteroclinic solutions and of homoclinic solutions with a unique change of monotonicity. Then, we analyze the asymptotic behavior of such solutions both for δ→0+ and for δ→+∞. Some numerical examples illustrate the stated results.