Classification of bifurcation curves for the Minkowski-curvature problem involving general sign-changing nonlinearity
Abstract In this paper, we study the classification of bifurcation curves of positive solutions of the Minkowski-curvature problem { − ( u ′ / 1 − u ′ 2 ) ′ = λ f ( u ) , in ( − L , L ) , u ( − L ) = u ( L ) = 0 , $$ \left \{ \textstyle\begin{array}{l} -\left ( u^{\prime }/\sqrt{1-{u^{\prime }}^{2...
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2025-08-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-025-02110-x |
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| Summary: | Abstract In this paper, we study the classification of bifurcation curves of positive solutions of the Minkowski-curvature problem { − ( u ′ / 1 − u ′ 2 ) ′ = λ f ( u ) , in ( − L , L ) , u ( − L ) = u ( L ) = 0 , $$ \left \{ \textstyle\begin{array}{l} -\left ( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right ) ^{\prime }= \lambda f(u),\text{ in }\left ( -L,L\right ) , \\ u(-L)=u(L)=0,\end{array}\displaystyle \right . $$ where λ , L > 0 $\lambda ,L>0$ , and f ∈ C 2 ( 0 , ∞ ) $f\in C^{2}(0,\infty )$ is sign-changing. We identify and correct a significant error in (He et al. in AIMS Math. 7:17001–17018, 2022), and further refine their results. In contrast to (He et al. in AIMS Math. 7:17001–17018, 2022), which considers only the case f ( 0 + ) ≥ 0 $f(0^{+})\geq 0$ , we extend the analysis to function f satisfying the case − ∞ ≤ f ( 0 + ) < 0 $-\infty \leq f(0^{+})<0$ . Finally, we establish sufficient conditions for determining the exact shape of the bifurcation curve. |
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| ISSN: | 1687-2770 |