Existence of a double S-shaped bifurcation curve with six positive solutions for a combustion problem (Q425317)

The authors study the bifurcation curve of positive solutions of the boundary value problem NEWLINE\[NEWLINE -u''(x) = \lambda \exp\left(\frac{\beta u}{\beta+u}\right), \quad 0<x<1, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0)=0, \quad \frac{u(1)}{u(1)+1} u'(1) + \left[1-\frac{u(1)}{u(1)+1}\right] u(1)=0, NEWLINE\]NEWLINE where \(\lambda,\beta>0\) are parameters. They prove that, for \(\beta>\beta_1\approx 6.459\) for some constant \(\beta_1\), the bifurcation curve is double S-shaped on the \((\lambda,\|u\|_\infty)\)-plane and the problem has at least six positive solutions for a certain range of positive \(\lambda\). They also provide proofs for some other bifurcation phenomena of the problem.