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 ≥ 2) to the equation ( u' / sqrt{1-(u')^2} )' + λ a(t) g(u) = 0, where λ > 0 is a parameter, a(t) is a T-periodic sign-changing weight function and g:[0,+∞[→[0,+∞[ 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 ≤ q ≤ 1 < p, the equation has no positive T-periodic solutions for λ close to zero and two positive T-periodic solutions (a "small" one and a "large" one) for λ 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 Poincaré-Birkhoff fixed point theorem, after a careful asymptotic analysis of the T-periodic solutions for λ→+∞.

Positive periodic solutions to an indefinite Minkowski-curvature equation

Boscaggin, Alberto
;
Feltrin, Guglielmo
2020-01-01

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 ≥ 2) to the equation ( u' / sqrt{1-(u')^2} )' + λ a(t) g(u) = 0, where λ > 0 is a parameter, a(t) is a T-periodic sign-changing weight function and g:[0,+∞[→[0,+∞[ 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 ≤ q ≤ 1 < p, the equation has no positive T-periodic solutions for λ close to zero and two positive T-periodic solutions (a "small" one and a "large" one) for λ 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 Poincaré-Birkhoff fixed point theorem, after a careful asymptotic analysis of the T-periodic solutions for λ→+∞.
2020
269
5595
5645
https://arxiv.org/abs/1805.06659
Minkowski-curvature operator, indefinite weight, positive solutions, periodic solutions, subharmonic solutions, coincidence degree theory, Poincaré-Birkhoff theorem
Boscaggin, Alberto; Feltrin, Guglielmo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1668220
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