Bifurcation solution branches and their numerical approximations of a semi-linear elliptic problem with two parameters (Q1302261)

In the interval \((0,\pi)\), the authors consider the boundary value problem \[ u''+(\lambda+ s(\mu)) f(u)- \mu\sin x= 0,\quad u(0)= u(\pi)= 0. \] Here, \(\lambda\) and \(\mu\) are real parameters and \(f\), \(s\) smooth odd functions satisfying certain conditions. Due to \(f(0)= 0\), the problem has in particular the trivial solution \(u= 0\), \(\mu=0\), \(\lambda\in\mathbb{R}\). In the special case \(\mu= 0\), bifurcation from the trivial solution has been studied among others by \textit{E. I. Allgower}, \textit{K. Böhmer} and \textit{Z. Mei} [IMA J. Numer. Anal. 14, No. 2, 161-182 (1994; Zbl 0803.65103)]. Starting from known branching solutions of the special problem, the present authors apply continuation methods in order to find solutions for given values of the parameters \(\lambda\) and \(\mu\). Theoretical and numerical results are presented.