Exact multiplicity of boundary blow-up solutions for a bistable problem (Q929117)

The paper concerns the exact multiplicity of positive solutions \(u\leq 1/n\) of the problem \[ u''(x)+\lambda f(u(x))=0 \quad (-1<x<1), \qquad u(-1)=u(1)=1/n\in (1,\infty], \] where \(\lambda>0\), \(0\leq n<1\), and \(f(u)=u(u-\sigma)(1-u)\) \((\sigma\in (1/2,1))\). By using the time-mapping method (quadrature method), the authors prove that there exists \(\lambda^\ast>0\) such that the problem has exactly three solutions for \(\lambda>\lambda^\ast\), exactly two solutions for \(\lambda=\lambda^\ast\), and exactly one solution for \(\lambda<\lambda^\ast\).