Exact Solution and Bifurcation Curve for the Minkowski-Curvature Equation with Nonlinearity up+uq, p > 1

Cheng, Yan-Hsiou; Hung, Kuo-Chih

doi:10.3390/math13223680

Open AccessFeature PaperArticle

Exact Solution and Bifurcation Curve for the Minkowski-Curvature Equation with Nonlinearity u^p+u^q, p > 1

by

Yan-Hsiou Cheng

¹

and

Kuo-Chih Hung

^2,*

¹

Department of Mathematics and Information Education, National Taipei University of Education, Taipei City 106, Taiwan

²

Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411, Taiwan

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(22), 3680; https://doi.org/10.3390/math13223680

Submission received: 19 October 2025 / Revised: 12 November 2025 / Accepted: 13 November 2025 / Published: 17 November 2025

(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)

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Abstract

We study the bifurcation curve and exact multiplicity of positive solutions in the space

C^{2} ((- L, L)) \cap C ([- L, L])

for the Minkowski-curvature equation:

- {(\frac{u^{'} (x)}{\sqrt{1 - {(u^{'} (x))}^{2}}})}^{'} = λ (u^{p} + u^{q}), - L < x < L; u (- L) = u (L) = 0,

where

λ, L > 0

and

p > 1

. If

1 < p < q \leq 2 p + \sqrt{3 p^{2} - 3}

, we prove that the bifurcation curve is ⊂-shaped.

Keywords: positive solution; exact multiplicity; bifurcation curve; Minkowski-curvature equation

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MDPI and ACS Style

Cheng, Y.-H.; Hung, K.-C. Exact Solution and Bifurcation Curve for the Minkowski-Curvature Equation with Nonlinearity u^p+u^q, p > 1. Mathematics 2025, 13, 3680. https://doi.org/10.3390/math13223680

AMA Style

Cheng Y-H, Hung K-C. Exact Solution and Bifurcation Curve for the Minkowski-Curvature Equation with Nonlinearity u^p+u^q, p > 1. Mathematics. 2025; 13(22):3680. https://doi.org/10.3390/math13223680

Chicago/Turabian Style

Cheng, Yan-Hsiou, and Kuo-Chih Hung. 2025. "Exact Solution and Bifurcation Curve for the Minkowski-Curvature Equation with Nonlinearity u^p+u^q, p > 1" Mathematics 13, no. 22: 3680. https://doi.org/10.3390/math13223680

APA Style

Cheng, Y.-H., & Hung, K.-C. (2025). Exact Solution and Bifurcation Curve for the Minkowski-Curvature Equation with Nonlinearity u^p+u^q, p > 1. Mathematics, 13(22), 3680. https://doi.org/10.3390/math13223680

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Exact Solution and Bifurcation Curve for the Minkowski-Curvature Equation with Nonlinearity u^p+u^q, p > 1

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