带有凹非线性项的平均曲率半正问题正解的确切个数

Abstract

Abstract:

In this paper, we study the exact multiplicity and bifurcation diagrams of positive solutions for the prescribed mean curvature problem in one-dimensional Minkowski space in the form of

$ \left\{\begin{array}{ll} -\left(\frac{u'}{\sqrt{1-u'^{2}}}\right)'=\lambda f(u), x\in(-L,L),\\ u(-L)=0=u(L), \end{array} \right. $

where $\lambda>0$ is a bifurcation parameter and $L>0$ is an evolution parameters, $f\in C^{2}([0,\infty), \mathbb{R})$ satisfies $f(0)<0$ and $f$ is concave for $0. In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of $\lambda$. The arguments are based upon a detailed analysis of the time map.

Key words: Minkowski space, Semipositone, Positive solution, Time map, Exact multiplicity

CLC Number:

O175.8

Li Xiaodong, Gao Hongliang, Xu Jing. Exact Multiplicity of Positive Solutions for a Semipositone Mean Curvature Problem with Concave Nonlinearity[J].Acta mathematica scientia,Series A, 2023, 43(5): 1341-1349.

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References 13

[1]	Bratu G. Sur les equation integrals non linéarires. Bull Soc Math France, 1914, 42: 113-142
[2]	Corsato C. Mathematical Analysis of Some Differential Models Involving the Euclidean or the Minkowski Mean Curvature Operator. Trieste: University of Trieste, 2015
[3]	Gao H L, Xu J. Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation. Boundary Value Problems, 2021, Article number 81
[4]	Gelfand I M. Some problems in the theory of quasilinear equations. American Mathematical Society Translations, 1963, 29(2): 295-381
[5]	Huang S Y. Global bifurcation diagrams for Liouville-Bratu-Gelfand problem with Minkowski-curvature operator. Journal of Dynamics and Differential Equations, 2021, 34: 2157-2172 doi: 10.1007/s10884-021-09982-4
[6]	Huang S Y. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure and Applied Analysis, 2018, 17(3): 1271-1294 doi: 10.3934/cpaa.2018061
[7]	Huang S Y. Classifification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications. Journal of Differential Equations, 2018, 264(9): 5977-6011 doi: 10.1016/j.jde.2018.01.021
[8]	Huang S Y. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure and Applied Analysis, 2019, 18(6): 3267-3284 doi: 10.3934/cpaa.2019147
[9]	Huang S Y. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete and Continuous Dynamical Systems-A, 2019, 39(6): 3443-3462
[10]	Huang S Y, Hwang M S. Bifurcation curves of positive solutions for the Minkowski-curvature problem with cubic nonlinearity. Electronic Journal of Qualitative Theory of Differential Equations, 2021, Article number 41
[11]	Hutten E H. Relativistic (non-linear) Oscillator. Nature, 1965, 205(4974): 892
[12]	Walter G. Classicai Mechanics-point Particles and Relativity. New York: Springer, 2004: 1-485
[13]	Zhang X M, Feng M Q. Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space. Communications in Contemporary Mathematics, 2019, 21(3): 111-137

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Exact Multiplicity of Positive Solutions for a Semipositone Mean Curvature Problem with Concave Nonlinearity

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