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

We deal with a non-autonomous parameter-dependent second-order differential equation driven by the Minkowski-curvature operator, with a positively weighted bistable nonlinearity. Under suitable assumptions on the weight, we prove the existence of strictly increasing heteroclinic solutions and homoclinic solutions with a unique change of monotonicity, based on a careful phase-plane analysis. Then, we analyze the asymptotic behavior of such solutions for small/large values of the parameter. Some numerical examples illustrate the stated results.