Classification and evolution of bifurcation curves of semipositone problem with Minkowski-curvature operator and its applications

Abstract

In this paper, we study the classification and evolution of bifurcation curves of semipositone problem with Minkowski-curvature operator{−(u′/1−u′2)′=λf(u), in (−L,L),u(−L)=u(L)=0,where λ,L>0, f∈C2(0,∞), and the following conditions (H1)–(H2) hold:(H1)there exists β>0 such that (β−u)f(u)<0 for u>0 and u≠β.(H2)Let F(u)≡∫0uf(t)dt. Then F:[0,∞)⟶R is continuous and differentiable for u>0. And there exists η>0 such that (η−u)F(u)<0 for u>0 and u≠η. Notice that we allow f(0+)=−∞. In particular, we further obtain the exact shapes of the bifurcation curve as f is convex or concave. Finally, we apply these results in several problems.