Positive periodic solutions to an indefinite Minkowski-curvature equation

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 $k\ge 2$) 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 $g(u)=u^p\mathrm{/}(1+u^{p-q})$, with $0\le q\le 1<p$, 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 $\lambda \to +\mathrm{\infty }$.