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Global Bifurcation Diagrams for Liouville–Bratu–Gelfand Problem with Minkowski-Curvature Operator
Journal of Dynamics and Differential Equations ( IF 1.4 ) Pub Date : 2021-03-20 , DOI: 10.1007/s10884-021-09982-4
Shao-Yuan Huang

In this paper, we study global bifurcation diagrams and the exact multiplicity of positive solutions for Minkowski-curvature problem

$$\begin{aligned} \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime }=\lambda \exp u,\text { in }\left( -L,L\right) , \\ u(-L)=u(L)=0, \end{array} \right. \end{aligned}$$

where \(\lambda >0\) is a bifurcation parameter and \(L>0\) is an evolution parameter. It can be viewed as a variant of the one-dimensional Liouville–Bratu–Gelfand problem. We prove that there exists \(L_{0}>0\) such that the bifurcation curve \(S_{L}\) is S-shaped for \(L>L_{0}\) and is monotone increasing for \(0<L\le L_{0}\). We also study, in the \(\left( L,\lambda ,\left\| u\right\| _{\infty }\right) \)-space, the shape and structure of the bifurcation surface. Finally, we will make a list which shows the different properties of bifurcation curves for Minkowski-curvature problem, corresponding semilinear problem and corresponding prescribed curvature problem.



中文翻译:

具有Minkowski曲率算子的Liouville–Bratu–Gelfand问题的全局分叉图

在本文中,我们研究了全局分叉图和Minkowski曲率问题的正解的精确多重性

$$ \ begin {aligned} \ left \ {\ begin {array} {l}-\ left(u ^ {\ prime} / \ sqrt {1- {u ^ {\ prime}} ^ {2}} \ right )^ {\ prime} = \ lambda \ exp u,\ text {in} \ left(-L,L \ right),\\ u(-L)= u(L)= 0,\ end {array} \对。\ end {aligned} $$

其中\(\ lambda> 0 \)是分叉参数,\(L> 0 \)是演化参数。可以将其视为一维Liouville–Bratu–Gelfand问题的变体。我们证明了存在\(L_ {0}> 0 \) ,使得分支曲线\(S_ {L} \)是S形为\(L> L_ {0} \),并且单调增加\( 0 <L \ le L_ {0} \)。我们还在\(\ left(L,\ lambda,\ left \ | u \ right \ | _ {\ infty} \ right)\)中研究空间,分叉表面的形状和结构。最后,我们将给出一个列表,显示针对Minkowski曲率问题,相应的半线性问题和相应的规定曲率问题的分岔曲线的不同性质。

更新日期:2021-03-21
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