Remarks on bifurcations of \(u''+\mu u-u^k=0\quad (4\leq k\in\mathbb{Z}^+)\) (Q5942762)

The paper focuses on the study of bifurcation of solutions to the boundary value problem \[ F(u,\mu):= u''+\mu(u-u^k)=0, \qquad u(0)=u(\pi)=0 , \] from the trivial ones (\(k\geq 4\) is an integer and \(\mu\) is a real parameter). With \(k=2,3\) being studied previously by \textit{C. Li} [Acta Math. Sci. (Chin. Ed.) 20, No. 3, 336-341 (2000; Zbl 0962.34030)], of particular interest are values of \(k\geq 4\) (more difficult case). Using the singularity theory based on the Lyapunov-Schmidt reduction, bifurcation results are obtained. Results of this paper are closely related to results of \textit{C. Li} [Appl. Math. Mech., Engl. Ed. 21, No. 3, 265-274 (2000; Zbl 0982.34034) and J. Math. Res. Expo. 21, No. 1, 47-52 (2001; Zbl 0993.34039)]. Some consequences of this analysis to the evolution equation \(\partial u/\partial t=F(u, \mu)\) are outlined (stability of stationary solutions).