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

We deal with the non-autonomous parameter-dependent second-order differential equation delta(v 'root 1-(v ')2)'+q(t)f(v) = 0, t is an element of R, driven by a Minkowski-curvature operator. Here,delta >0,q is an element of L infinity(R),f: [0,1]-> Ris a continuous function withf(0) =f(1) = 0 =f(alpha)for some alpha is an element of]0,1[,f(s)<0for alls is an element of]0,alpha[ andf(s)>0for alls is an element of]alpha,1[. Based on a careful phase-plane analysis, under suitable assumptions onqwe prove the existence of strictlyincreasing heteroclinic solutions and of homoclinic solutions with a unique changeof monotonicity. Then, we analyze the asymptotic behavior of such solutions bothfor delta -> 0+and for delta ->+infinity. Some numerical examples illustrate the stated results.(c) 2023 The Author(s). Published by Elsevier Ltd. This is an open access article underthe CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

This article deals with a non-autonomous parameter-dependent second-order differential equation driven by a Minkowski-curvature operator. It proves the existence of strictly increasing heteroclinic solutions and homoclinic solutions with a unique change of monotonicity under suitable assumptions, and analyzes the asymptotic behavior of these solutions.