On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem. II Original Research Article

We study the evolution and qualitative behaviors of bifurcation curves of positive solutions for View the MathML source{−u″(x)=λfq,p(u)=λ(uq(1−sinu)+up),−10λ>0 is a bifurcation parameter, q<1q<1 is a positive bifurcation parameter, and p≥1p≥1 is an evolution parameter. We prove that, for given q<1q<1, there exist numbers p∗(q)>p∗(q)>1p∗(q)>p∗(q)>1 such that, on the (λ,‖u‖∞)(λ,‖u‖∞)-plane, the bifurcation curve has exactly one turning point where the curve turns to the left for p>p∗(q)p>p∗(q), it has at least three turning points for 1p∗(q)p>p∗(q). Our results extend some results of Wang [S.-H. Wang, On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem, Nonlinear Anal. 67 (2007) 1316–1328] from q=1q=1 to 0