Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions

We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand equation with mixed boundary conditions given by{u″(x)+λexp⁡(aua+u)=0,00c>0, the bifurcation curve is strictly increasing on the (λ,‖u‖∞)(λ,‖u‖∞)-plane, and there exists a positive λ0λ0 such that the problem has no positive solution for 0<λ<λ00<λ<λ0 and exactly one positive solution for λ≥λ0λ≥λ0. While for a≥a1(≈4.107), there exists c1(=c1(a))>1.057 such that, on the (λ,‖u‖∞)(λ,‖u‖∞)-plane, (i) when 0