Pairs of Nodal Solutions for a Class of Nonlinear Problems with One-sided Growth Conditions

Abstract

Boundary value problems of Sturm-Liouville and periodic type for the second order nonlinear ODE u + λ f (t, u) = 0 are considered. Multiplicity results are obtained for λ positive and large under suitable growth restrictions on f (t, u) of superlinear type at u = 0 and of sublinear type at u = ∞. Only one-sided growth conditions are required. Applications are given to the equation u + λq(t)f(u) = 0, allowing also a weight function q(t) of nonconstant sign.Mathematics Subject Classification. 34C25, 34C28.

Subharmonic solutions of the prescribed curvature equation

“…We see that, for all (θ 0 , ρ 0 ) with θ 0 ∈ R and ρ 0 > 0, uniqueness and global continuability of solutions hold for any Cauchy problem associated with (21) and with (22). Let r − (· ; θ 0 , ρ 0 ), r + (· ; θ 0 , ρ 0 ) (in the sequel, sometimes denoted by r ∓ for simplicity) be the solutions of (21) and (22), respectively, satisfying r ∓ (θ 0 ) = ρ 0 .…”

“…As uniqueness of solutions holds for any Cauchy problem associated with (22), by a classical result on differential inequalities (see, e.g., [27, Section I.6, Theorem 6.1]), we conclude that (θ) ≤ γ(θ), for all θ ∈ [θ(t 0 ), θ(σ)], and hence ρ(t) ≤ γ(θ(t)), for all t ∈ [t 0 , σ]. In particular, we have…”

“…, m − 1, then it is referred to as a subharmonic solution of order m. As a general rule in this context, one tries to get as much information as possible about the minimality of the period. In particular, in [22] the existence of subharmonic solutions of (3) has been proved assuming that either f is superlinear at 0, i.e.,…”

“…uniformly in t. More precisely, it is shown in [22] that condition (4), even assumed only at 0 + , or at 0 − , implies the existence of two sequences of arbitrarily small subharmonic solutions having a prescribed number of zeroes and condition (5), even assumed only at +∞, or at −∞, implies the existence of two sequences of arbitrarily large subharmonic solutions having a prescribed number of zeroes. The proof is performed by a phase-plane analysis and relies on the Poincaré-Birkhoff fixed point theorem; the nodal properties of the solutions are obtained by using the rotation number which counts the number of turns of the solutions around the origin in the phase-plane.…”

“…Assuming that f is superlinear at 0, we prove in Theorem 3.4 the existence of small classical subharmonic solutions having suitable nodal properties; in this case the proof, which borrows some arguments from [22], is based on the use of the rotation number and on a version of the Poincaré-Birkhoff theorem given in [23,Theorem 8.2] and not requiring uniqueness of solutions for the Cauchy problems associated with (1). In particular, the following result holds.…”