Local structure of bifurcation curves for nonlinear Sturm–Liouville problems

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>0λ>0 is a parameter. We mainly treat the case where f(u)=λu±upf(u)=λu±up (p>1p>1) and establish the precise asymptotic expansion formulas for the bifurcation curve near the bifurcation point λ=π2λ=π2 in LqLq-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 un,λun,λ of the equation −u″(t)=λ(u(t)+|u(t)|p−1u(t))−u″(t)=λ(u(t)+|u(t)|p−1u(t)) with u(0)=u(π)=0u(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→n2λ→n2.