Local structure of bifurcation curves for nonlinear Sturm–Liouville problems

Abstract

We consider the nonlinear bifurcation problem arising in population dynamics and nonlinear Schrödinger equation:and establish the precise asymptotic expansion formulas for the bifurcation curve near the bifurcation point λ = π 2 in L q -framework. Together with the result of the global behavior of the bifurcation curve, we understand completely the structure of the bifurcation curve.We also consider the nodal solution u n,λ of the equation −u (t) = λ(u(t) + |u(t)| p−1 u(t)) with u(0) = u(π ) = 0 and establish an asymptotic expansion formula for λ with respect to the gradient norm of the solution associated with λ as λ → n 2 .

Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems

“…Our methods to prove Theorems 2-4 are based on the precise calculation of the time map. We prove Theorems 5 and 6 by the method developed in [13]. By using Theorems 2, 4, and 5, we prove Theorem 7.…”