Positive periodic solutions to an indefinite Minkowski-curvature equation

TL;DR

This paper studies positive periodic solutions of a Minkowski-curvature differential equation with sign-changing weights, establishing existence, multiplicity, and non-existence results depending on a parameter, using advanced topological and fixed point methods.

Contribution

It introduces new existence and multiplicity results for positive periodic solutions of a Minkowski-curvature equation with sign-changing weights, employing Mawhin's coincidence degree and Poincaré--Birkhoff theorems.

Findings

No positive T-periodic solutions for small lambda.

Existence of two positive T-periodic solutions for large lambda.

Presence of positive subharmonic solutions with arbitrarily large periods.

Abstract

We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., $T$-periodic) and subharmonic (i.e., $kT$-periodic for some integer $k \geq 2$) to the equation \begin{equation*} \Biggl{(} \dfrac{u'}{\sqrt{1-(u')^{2}}} \Biggr{)}' + \lambda a(t) g(u) = 0, \end{equation*} where $\lambda > 0$ is a parameter, $a(t)$ is a $T$-periodic sign-changing weight function and $g \colon \mathopen{[}0,+\infty\mathclose{[} \to \mathopen{[}0,+\infty\mathclose{[}$ is a continuous function having superlinear growth at zero. In particular, we prove that for both $g(u)=u^{p}$, with $p>1$, and $g(u)= u^{p}/(1+u^{p-q})$, with $0 \leq q \leq 1 < p$, the equation has no positive $T$-periodic solutions for $\lambda$ close to zero and two positive $T$-periodic solutions (a 'small' one and a 'large' one) for $\lambda$ large enough. Moreover, in both cases the 'small' $T$-periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period. The proof of the existence of $T$-periodic solutions relies on a recent extension of Mawhin's coincidence degree theory for locally compact operators in product of Banach spaces, while subharmonic solutions are found by an application of the Poincar\'e--Birkhoff fixed point theorem, after a careful asymptotic analysis of the $T$-periodic solutions for $\lambda \to +\infty$.