Positive periodic solutions to an indefinite Minkowski-curvature equation

[摘要] 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'/root 1 - (u')(2))' + lambda a(t)g(u) = 0, where lambda > 0 is a parameter, a(t) is a T-periodic sign-changing weight function and g: [0, +infinity[ -> [0, +infinity[ 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 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 Poincare-Birkhoff fixed point theorem, after a careful asymptotic analysis of the T-periodic solutions for lambda -> +infinity. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-07-17

[关键词] Minkowski-curvature operator;Indefinite weight;Positive periodic solutions;Subharmonic solutions;Coincidence degree theory;Poincare-Birkhoff theorem