Asymptotic oscillations of continua of positive solutions of a semilinear Sturm–Liouville problem

TY - JOUR

T1 - Asymptotic oscillations of continua of positive solutions of a semilinear Sturm–Liouville problem

AU - Davidson, F. A.

AU - Rynne, B. P.

PY - 2000/12/15

Y1 - 2000/12/15

N2 - We consider the set of positive solutions (λ, u) of the semilinear Sturm–Liouville boundary value problemwhere f: [0, ∞) → is Lipschitz continuous and λ is a real parameter. We suppose that f(s) oscillates, as s → ∞, in such a manner that the problem is not linearizable at u = ∞ but does, nevertheless, have a continuum of positive solutions bifurcating from infinity. We investigate the relationship between the oscillations of f and those of in the λ–|u|0 plane at large |u|0. In particular, we discuss whether oscillates infinitely often over a single point λ, or over an interval I (of positive length) of λ values. An immediate consequence of such oscillations over I is the existence of infinitely many solutions, of arbitrarily large norm |u|0, of the problem for all values of λ ∈ I.

AB - We consider the set of positive solutions (λ, u) of the semilinear Sturm–Liouville boundary value problemwhere f: [0, ∞) → is Lipschitz continuous and λ is a real parameter. We suppose that f(s) oscillates, as s → ∞, in such a manner that the problem is not linearizable at u = ∞ but does, nevertheless, have a continuum of positive solutions bifurcating from infinity. We investigate the relationship between the oscillations of f and those of in the λ–|u|0 plane at large |u|0. In particular, we discuss whether oscillates infinitely often over a single point λ, or over an interval I (of positive length) of λ values. An immediate consequence of such oscillations over I is the existence of infinitely many solutions, of arbitrarily large norm |u|0, of the problem for all values of λ ∈ I.

U2 - 10.1006/jmaa.2000.7009

DO - 10.1006/jmaa.2000.7009

M3 - Article

SN - 0022-247X

VL - 252

SP - 617

EP - 630

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -