Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super–sublinear case

Abstract

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equationwhere g : [0, +∞[ → [0, +∞[ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when T 0 a(t) dt < 0 and λ > 0 is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations. * Work performed under the auspices of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Positive subharmonic solutions to nonlinear ODEs with indefinite weight

“…In such a way, we can complement -in the direction of proving the existence of positive subharmonics -recent results dealing with positive harmonic solutions in the asymptotically linear case σ = γ − 1 (see [22,Corollary 3.7]) and in the sublinear one σ > γ − 1 (see [9,11]). It is worth noticing that, according to [9,Theorem 4.3], in this latter case a further positive T -periodic solution (having maximum greater than ρ) to (3.4) appears. This second solution is expected to have typically zero Morse index, and no positive subharmonic solutions oscillating around it.…”

Positive periodic solutions to an indefinite Minkowski-curvature equation

“…For more details and proofs, we refer the reader to [15,28,35,48,49] and the references therein. In our applications, given an open set O ⊆ X and an L-completely continuous operator N , in order to prove that D L (L − N , O) is well-defined or equivalently that the set x ∈ O ∩ dom L : Lx = N x is compact in X, we proceed in this manner.…”

“…[29,Theorem 3.2]), while two positive T -periodic solutions to (1.6) exist, but only for λ sufficiently large (cf. [15,Theorem 4.4 and Theorem 4.6]). Roughly speaking, we can say that the superlinearity at zero, when paired with (a # ) (and further suitable technical assumptions, see (g 0 ) and (g 0 ) in Section 1.1), still provides, in the indefinite periodic setting, the desired geometry near zero, thus implying that the topological degree is equal to 1 on small balls (it is impressive to interpret (a # ) as a non-resonance condition pushing the term λa(t)g(u)/u below the principal eigenvalue for u → 0 + ).…”

“…The proof of Theorem 3.1 relies on topological degree theory, according to the aforementioned strategy of showing that a suitable topological degree associated with (1.1) is equal to 1 on small and large balls centered at the origin and is equal to 0 (for large values of the parameter λ) on a ball of intermediate radius. While the needed technical estimates are inspired by the ones in [15,29] for the semilinear equation (1.4), the most delicate point in our strongly nonlinear setting is actually the definition of the degree. In [44,51], some continuations theorems have been proposed and used for equations driven by nonlinear operators: they all rely on the formulation of (1.1) as a fixed point problem on a Banach space of T -periodic functions and on a direct use of the Leray-Schauder degree theory.…”

Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefinite weight

“…we write it as a fixed point equation in the Banach space C([0, R]), by exploiting a strategy comparable to the one used, for instance, in [2,16] (see also Remark 2.2). At this point, Leray-Schauder degree theory can be applied: the computation of the degree on three different balls of the space C([0, R]) leads to the result, similarly as in [6,7]. The paper is organized as follows.…”