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- Title
Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation.
- Authors
Gao, Hongliang; Xu, Jing
- Abstract
In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation { − (u ′ 1 − u ′ 2 ) ′ = λ f (u) , x ∈ (− L , L) , u (− L) = 0 = u (L) , where λ and L are positive parameters, f ∈ C [ 0 , ∞) ∩ C 2 (0 , ∞) , and f (u) > 0 for 0 < u < L . We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies f ″ (u) > 0 and u f ′ (u) ≥ f (u) + 1 2 u 2 f ″ (u) for 0 < u < L . In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.
- Subjects
DIRICHLET problem; EQUATIONS; MULTIPLICITY (Mathematics); CURVES
- Publication
Boundary Value Problems, 2021, Vol 2021, Issue 1, p1
- ISSN
1687-2762
- Publication type
Article
- DOI
10.1186/s13661-021-01558-x