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

[摘要] We consider the set of positive solutions (lambda, u) of the semilinear Sturm-Liouville boundary value problem -u= lambdau + f(u) in (0, pi), u(0) = u(pi) = 0 where f: [0, proportional to) --> R is Lipschitz continuous and lambda is a real parameter. We suppose that f(s) oscillates, as s --> infinity, in such a manner that the problem is not linearizable at u = infinity but does, nevertheless, have a continuum C of positive solutions bifurcating from infinity. We investigate the relationship between the oscillations of f and those of C in the lambda-\u\(0) plane at large \u\(0) In particular, we discuss whether C oscillates infinitely often over a single point lambda, or over an interval I (of positive length) of lambda 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 lambda is an element of I. (C) 2000 Academic Press.

[发布日期] 2000-12-15