Asymptotic Oscillations of Continua of Positive Solutions of a Semilinear Sturm Liouville Problem

We consider the set of positive solutionsŽ , u.of the semilinear Sturm Liouville boundary value problem u u fŽu. in Ž0, ., uŽ0. uŽ . 0, where f: 0, . is Lipschitz continuous and is a real parameter. We suppose that fŽs. oscillates, as s , in such a manner that the problem is not linearizable at u 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 u 0plane at large u 0. In particular, we discuss whether C oscillates infinitely often over a single point , or over an interval I Žof positive length. of 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 I.