On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem

We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for {−u″(x)=λ(u(1−sinu)+up),−10 is a bifurcation parameter and p≥1 is an evolution parameter. On the (λ,‖u‖∞)-plane, we prove that the bifurcation curve has exactly one turning point where the curve turns to the left for p>2, it is a monotone curve for p=2, it has at least two turning points for 1