Global Bifurcation Diagrams for Liouville–Bratu–Gelfand Problem with Minkowski-Curvature Operator

• Published in: Journal of dynamics and differential equations 2022-09, Vol.34 (3), p.2157-2172
• Main Author: Huang, Shao-Yuan
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Bifurcations; Curvature; Mathematical analysis; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Parameters; Partial Differential Equations
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-021-09982-4

In this paper, we study global bifurcation diagrams and the exact multiplicity of positive solutions for Minkowski-curvature problem - u ′ / 1 - u ′ 2 ′ = λ exp u , in - L , L , u ( - L ) = u ( L ) = 0 , where λ > 0 is a bifurcation parameter and L > 0 is an evolution parameter. It can be viewed as a variant of the one-dimensional Liouville–Bratu–Gelfand problem. We prove that there exists L 0 > 0 such that the bifurcation curve S L is S-shaped for L > L 0 and is monotone increasing for 0 < L ≤ L 0 . We also study, in the L , λ , u ∞ -space, the shape and structure of the bifurcation surface. Finally, we will make a list which shows the different properties of bifurcation curves for Minkowski-curvature problem, corresponding semilinear problem and corresponding prescribed curvature problem.