Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity

Abstract

We study the global structure of the oscillatory perturbed bifurcation problem which comes from the stationary logarithmic Schrödinger equation −u″(t)=λ(log(1+u(t))+sinu(t)),u(t)>0,t∈I≔(−1,1),u(±1)=0,-{u}^{^{\prime\prime} }\left(t)=\lambda (\log \left(1+u\left(t))+\sin u\left(t)),\hspace{1.0em}u\left(t)\gt 0,\hspace{1.0em}t\in I:= \left(-1,1),\hspace{1.0em}u\left(\pm 1)=0, where λ>0\lambda \gt 0 is a parameter. It is known that λ\lambda is a continuous function of α>0\alpha \gt 0, written as λ(α)\lambda \left(\alpha ), where α\alpha is the maximum norm α=‖uλ‖∞\alpha =\Vert {u}_{\lambda }{\Vert }_{\infty } of the solution uλ{u}_{\lambda } associated with λ\lambda . In the field of bifurcation theory, the study of global structures of bifurcation curves is one of the main subjects of research, and it is important to investigate the influence of the oscillatory term on the global structure of bifurcation curve. Due to the effect of sinu\sin u, it is reasonable to expect that an oscillatory term appears in the second term of the asymptotic formula for λ(α)\lambda \left(\alpha ) as α→∞\alpha \to \infty (cf. [1]). Contrary to expectation, we show that the asymptotic formula for λ(α)\sqrt{\lambda \left(\alpha )} as α→∞\alpha \to \infty does not contain oscillatory terms by the third term of λ(α)\sqrt{\lambda \left(\alpha )}. This result implies that the oscillatory term has almost no influence on the global structure of λ(α)\lambda \left(\alpha ). The result is, therefore, unexpected, new and novel, since such phenomenon as this is not known so far. For the proof, the involved time map method and stationary phase method are used.

Keywords

• oscillatory bifurcation
• logarithmic nonlinearity
• stationary phase method
• 34b18
• 34c23