Positive periodic solutions to an indefinite Minkowski-curvature equation

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 equationwhere 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 → +∞.

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

“…f (x, u) ≡ 0) and existence/multiplicity of positive solutions of (1.1) is investigated. On this line of research, in the recent paper [6] we searched for positive periodic solutions (both harmonic and subharmonic) to the parameter-dependent equation…”

“…indefinite) weight function and g(u) is a nonlinear term satisfying g(0) = 0. Among other results, we proved therein that a two-solution theorem holds for the T -periodic boundary value problem associated with equation (1.2): more precisely, for weight functions a(x) satisfying the mean value condition T 0 a(x) dx < 0 and for a large class of nonlinear terms g(u) which are superlinear at zero (namely, g(u)/u → 0 for u → 0 + ), two positive T -periodic solutions of (1.2) exist, whenever the parameter λ is large enough (see [6,Theorem 3.1] for the precise statement of this result). We refer the reader to the introduction of [6] for several comments about this solvability pattern, arising as a result of a delicate interplay between the behaviors of the nonlinear differential operator driving equation (1.2) and the nonlinear term a(x)g(u) when u → +∞.…”

“…The technical tool used in [6] to prove the above mentioned result is topological degree theory in Banach spaces, along a line of research started in [14] and later developed and applied in several different situations (cf. [13]), always dealing with nonlinear BVPs for semilinear equations of the type u + q(x)g(u) = 0, (1.3) where q(x) is an indefinite weight function.…”

“…As far as the strongly nonlinear equation (1.2) is concerned, different strategies can be followed to achieve this goal. In [6], we chose to write (1.2) as the equivalent planar system…”

“…This approach, which looks very natural when dealing with the periodic problem, has the drawback of not being suited for other boundary conditions. In particular, in spite of the well-known strong analogies existing in this setting between the periodic and the Neumann boundary value problem (see, for instance, [9]), the possibility of proving the Neumann counterpart of the result in [6] is not discussed therein.…”