2023/09/23 by Guglielmo Feltrin, Feltrin, Guglielmo, Maurizio Garrione +1
Mathematics · #34C37 #35J93 #37J46 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2309.13286
openalex publication_date 2023/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We deal with the non-autonomous parameter-dependent second-order differential equation δ( \dfracv'√1-(v')2 )' + q(t) f(v)= 0, t∈ℝ, driven by a Minkowski-curvature operator. Here, δ>0, q∈ L∞(ℝ), f\colon\mathopen[0,1\mathclose]→ℝ is a continuous function with f(0)=f(1)=0=f(α) for some α∈ \mathopen]0,1\mathclose[, f(s)<0 for all s∈\mathopen]0,α\mathclose[ and f(s)>0 for all s∈\mathopen]α,1\mathclose[. 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 behaviour of such solutions both for δ→ 0+ and for δ→+∞. Some numerical examples illustrate the stated results.