Existence of Infinitely Many Non-Trivial Bifurcation Points

Abstract

We consider the non-linear two point boundary value problem where λ > 0,f ∈ C², f′ ≥ 0, f(0) < 0 and lim_{u → ∞} f(u) > 0. By considering the non-negative as well as all sign changing solutions, we establish the existence of infinitely many non-trivial bifurcation points. Further, when f is superlinear, we prove that there exists a constant λ* > 0, such that for each λ ∈ (0, λ*) there are exactly two solutions with m interior zeros for every m = 1,2, …We apply our results to the case when f(u) = u ³ - k; k > 0, and also discuss the evolution of the bifurcation diagram as k → 0.