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: − u ″ ( t ) = f ( λ , u ( t ) ) , u ( t ) > 0 , t ∈ I : = ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter. We mainly treat the case where f ( u ) = λ u ± u p ( p > 1 ) 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 .