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

TL;DR

This paper investigates the existence and behavior of homoclinic and heteroclinic solutions for a non-autonomous Minkowski-curvature differential equation, providing phase-plane analysis, asymptotic results, and numerical illustrations.

Contribution

It introduces new existence results for specific solutions of Minkowski-curvature equations with variable coefficients, using phase-plane analysis and asymptotic analysis.

Findings

Existence of strictly increasing heteroclinic solutions under certain conditions.

Existence of homoclinic solutions with a single monotonicity change.

Asymptotic behavior of solutions as elta o 0^+ and elta o +ty.

Abstract

We deal with the non-autonomous parameter-dependent second-order differential equation \begin{equation*} \delta \left( \dfrac{v'}{\sqrt{1-(v')^{2}}} \right)' + q(t) f(v)= 0, \quad t\in\mathbb{R}, \end{equation*} driven by a Minkowski-curvature operator. Here, $\delta>0$, $q\in L^{\infty}(\mathbb{R})$, $f\colon\mathopen{[}0,1\mathclose{]}\to\mathbb{R}$ is a continuous function with $f(0)=f(1)=0=f(\alpha)$ for some $\alpha \in \mathopen{]}0,1\mathclose{[}$, $f(s)<0$ for all $s\in\mathopen{]}0,\alpha\mathclose{[}$ and $f(s)>0$ for all $s\in\mathopen{]}\alpha,1\mathclose{[}$. 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 $\delta \to 0^{+}$ and for $\delta\to+\infty$. Some numerical examples illustrate the stated results.