Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation
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 ) , $$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-09-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-021-01558-x |
| _version_ | 1823938809453084672 |
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| author | Hongliang Gao Jing Xu |
| author_facet | Hongliang Gao Jing Xu |
| author_sort | Hongliang Gao |
| collection | DOAJ |
| description | Abstract In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation { − ( u ′ 1 − u ′ 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = 0 = u ( L ) , $$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime \,2}}} )'=\lambda f(u), &x\in (-L,L), \\ u(-L)=0=u(L), \end{cases} $$ where λ and L are positive parameters, f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) $f\in C[0,\infty ) \cap C^{2}(0,\infty )$ , and f ( u ) > 0 $f(u)>0$ for 0 < u < L $0< u< L$ . We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies f ″ ( u ) > 0 $f''(u)>0$ and u f ′ ( u ) ≥ f ( u ) + 1 2 u 2 f ″ ( u ) $uf'(u)\geq f(u)+\frac{1}{2}u^{2}f''(u)$ for 0 < u < L $0< u< L$ . In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map. |
| first_indexed | 2024-12-17T02:51:00Z |
| format | Article |
| id | doaj.art-f1a9b10b68714860a6f0368572f5b24a |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-17T02:51:00Z |
| publishDate | 2021-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-f1a9b10b68714860a6f0368572f5b24a2022-12-21T22:06:25ZengSpringerOpenBoundary Value Problems1687-27702021-09-012021111010.1186/s13661-021-01558-xBifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equationHongliang Gao0Jing Xu1Department of Mathematics, LanZhou Jiaotong UniversityDepartment of Mathematics, LanZhou Jiaotong UniversityAbstract In this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation { − ( u ′ 1 − u ′ 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = 0 = u ( L ) , $$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime \,2}}} )'=\lambda f(u), &x\in (-L,L), \\ u(-L)=0=u(L), \end{cases} $$ where λ and L are positive parameters, f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) $f\in C[0,\infty ) \cap C^{2}(0,\infty )$ , and f ( u ) > 0 $f(u)>0$ for 0 < u < L $0< u< L$ . 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 $f''(u)>0$ and u f ′ ( u ) ≥ f ( u ) + 1 2 u 2 f ″ ( u ) $uf'(u)\geq f(u)+\frac{1}{2}u^{2}f''(u)$ for 0 < u < L $0< u< L$ . 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.https://doi.org/10.1186/s13661-021-01558-xMinkowski-curvature equationExact multiplicityPositive solutionBifurcation curvesTime map |
| spellingShingle | Hongliang Gao Jing Xu Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation Boundary Value Problems Minkowski-curvature equation Exact multiplicity Positive solution Bifurcation curves Time map |
| title | Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation |
| title_full | Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation |
| title_fullStr | Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation |
| title_full_unstemmed | Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation |
| title_short | Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation |
| title_sort | bifurcation curves and exact multiplicity of positive solutions for dirichlet problems with the minkowski curvature equation |
| topic | Minkowski-curvature equation Exact multiplicity Positive solution Bifurcation curves Time map |
| url | https://doi.org/10.1186/s13661-021-01558-x |
| work_keys_str_mv | AT honglianggao bifurcationcurvesandexactmultiplicityofpositivesolutionsfordirichletproblemswiththeminkowskicurvatureequation AT jingxu bifurcationcurvesandexactmultiplicityofpositivesolutionsfordirichletproblemswiththeminkowskicurvatureequation |