Positive subharmonic solutions to nonlinear ODEs with indefinite weight

Abstract

We prove that the superlinear indefinite equationwhere p > 1 and a(t) is a T -periodic sign-changing function satisfying the (sharp) mean value condition T 0 a(t) dt < 0, has positive subharmonic solutions of order k for any large integer k, thus providing a further contribution to a problem raised by G. J. Butler in its pioneering paper [16]. The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincaré-Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).

Positive subharmonic solutions to superlinear ODEs with indefinite weight

“…In this section we present a symplectic approach, based on the Poincaré-Birkhoff fixed point theorem, which allows to obtain infinitely many subharmonics to (E ) u + q(t)g(u) = 0, as stated in Theorem 1.1. We refer to [11] for the missing details.…”

“…This latter aspect places in the investigation on indefinite equations of the form −∆u = q(x)g(u), u ∈ Ω ⊆ R N , that arise in many models concerning population dynamics, differential geometry and mathematical physics, and for which only non-negative solutions make sense. Concerning indefinite problems, we mention the contributions [2,3,6,31] and we refer to the introductions in [1,9,22,25,47] for a more complete discussion and bibliography on the subject.…”

Positive periodic solutions to an indefinite Minkowski-curvature equation

“…As already mentioned in the introduction, the proof of Theorem 5.1 relies on the Poincaré-Birkhoff fixed point theorem, on the lines of [14,20]. Before giving the proof, we need the following crucial lemma.…”

“…Then, apart from some technical difficulties arising from the presence of the nonlinear differential operator, the key point of the argument consists in proving that the principal eigenvalue µ 0 of the T -periodic Sturm-Liouville problem ϕ (u s,λ (t))w + µ + λa(t)g (u s,λ (t)) w = 0 is strictly negative when λ is large enough. This in turn can be proved (via an algebraic trick inspired by the one introduced in [23] and already used in [14]) using in an essential way the asymptotic analysis for the solution u s,λ (t) as λ → +∞ developed in Section 4.…”

“…Even if we do not exclude that this approach could be successful in our setting, we find more convenient to adopt a completely different method, still relying on the variational structure of the equation but using a celebrated result of Hamiltonian dynamical systems theory: the Poincaré-Birkhoff fixed point theorem. More precisely, we follow the strategy introduced in [20] (to which we also refer for a general bibliography on the subject of subharmonic solutions, as well as for a description of the Poincaré-Birkhoff theorem and its applications in the theory of ODEs) and later refined in [14,18], dealing with second order equations like u + h(t, u) = 0 and finding subharmonic solutions once a T -periodic solution u * (t) is available. Roughly, the solution u * (t) is required to have a non-trivial Morse index and, under suitable conditions on the behaviour of h(t, u) at infinity, for k large enough many kT -periodic solutions u k (t) are found, with a precise number of zeros for u k (t) − u * (t).…”