Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications

Abstract

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for one-dimensional Minkowski-curvature problem − u / 1 − u 2 = λf (u), in (−L, L) , u(−L) = u(L) = 0, where λ > 0 is a bifurcation parameter, L > 0 is an evolution parameter, f ∈ C[0, ∞) ∩ C 2 (0, ∞) and there exists β > 0 such that (β − z) f (z) > 0 for z = β. In particular, we find that the bifurcation curve S L is monotone increasing for all L > 0 when f (u)/u is of Logistic type, and is either ⊂-shaped or S-shaped for large L > 0 when f (u)/u is of weak Allee effect type. Finally, we can apply these results to obtain the global bifurcation diagrams in some important applications including ecosystem model.