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

Abstract

We deal with the non-autonomous parameter-dependent second-order differential equation \begin{equation*} δ\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, $\delta >0$, $q\in L^{\infty }(\mathbb{R})$, $f:[0,1]\to \mathbb{R}$ is a continuous function with $f(0)=f(1)=0=f(\alpha )$ for some $\alpha \in ]0,1[$, $f(s)<0$ for all $s\in ]0,\alpha [$ and $f(s)>0$ for all $s\in ]\alpha ,1[$. 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 behaviour of such solutions both for $\delta \to 0^{+}$ and for $\delta \to +\infty$. Some numerical examples illustrate the stated results.

25 pages, 6 figures