On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation −u′1+u′2′=λu1+up,−L<x<L,u(−L)=u(L)=0,\left\{\begin{array}{l}-{\left(\frac{{u}^{^{\prime} }}{\sqrt{1+{u}^{^{\prime} 2}}}\right)}^{^{\prime} }=\lambda {\left(...
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De Gruyter
2021
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oai:doaj.org-article:0c28dc2095504b5eb0296f9998f8d81a2021-12-05T14:10:53ZOn the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type2391-545510.1515/math-2021-0070https://doaj.org/article/0c28dc2095504b5eb0296f9998f8d81a2021-08-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0070https://doaj.org/toc/2391-5455We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation −u′1+u′2′=λu1+up,−L<x<L,u(−L)=u(L)=0,\left\{\begin{array}{l}-{\left(\frac{{u}^{^{\prime} }}{\sqrt{1+{u}^{^{\prime} 2}}}\right)}^{^{\prime} }=\lambda {\left(\frac{u}{1+u}\right)}^{p},\hspace{1.0em}-L\lt x\lt L,\\ u\left(-L)=u\left(L)=0,\end{array}\right. where λ\lambda is a bifurcation parameter, and L,p>0L,p\gt 0 are two evolution parameters. We prove that on the (λ,‖u‖∞)\left(\lambda ,\Vert u{\Vert }_{\infty })-plane, for 0<p≤240\lt p\le \frac{\sqrt{2}}{4}, the bifurcation curve is ⊃\supset -shaped bifurcation starting from (0,0)\left(0,0). And for p=1,f(u)=u1+up=1,f\left(u)=\frac{u}{1+u} is a logistic function, then the bifurcation curve is also ⊃\supset -shaped bifurcation starting from π24L2,0\left(\frac{{\pi }^{2}}{4{L}^{2}},0\right). While for p>1p\gt 1, the bifurcation curve is reversed ε\varepsilon -like shaped bifurcation if L>L∗L\gt {L}^{\ast }, and is exactly decreasing for λ>λ∗\lambda \gt {\lambda }^{\ast } if 0<L<L∗0\lt L\lt {L}_{\ast }.Zhang JiajiaQiao YuanhuaDuan LijuanMiao JunDe Gruyterarticlemean curvaturebifurcation curvelogistic function34b1834c2334b15MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 927-939 (2021) |
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EN |
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mean curvature bifurcation curve logistic function 34b18 34c23 34b15 Mathematics QA1-939 |
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mean curvature bifurcation curve logistic function 34b18 34c23 34b15 Mathematics QA1-939 Zhang Jiajia Qiao Yuanhua Duan Lijuan Miao Jun On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type |
| description |
We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation −u′1+u′2′=λu1+up,−L<x<L,u(−L)=u(L)=0,\left\{\begin{array}{l}-{\left(\frac{{u}^{^{\prime} }}{\sqrt{1+{u}^{^{\prime} 2}}}\right)}^{^{\prime} }=\lambda {\left(\frac{u}{1+u}\right)}^{p},\hspace{1.0em}-L\lt x\lt L,\\ u\left(-L)=u\left(L)=0,\end{array}\right. where λ\lambda is a bifurcation parameter, and L,p>0L,p\gt 0 are two evolution parameters. We prove that on the (λ,‖u‖∞)\left(\lambda ,\Vert u{\Vert }_{\infty })-plane, for 0<p≤240\lt p\le \frac{\sqrt{2}}{4}, the bifurcation curve is ⊃\supset -shaped bifurcation starting from (0,0)\left(0,0). And for p=1,f(u)=u1+up=1,f\left(u)=\frac{u}{1+u} is a logistic function, then the bifurcation curve is also ⊃\supset -shaped bifurcation starting from π24L2,0\left(\frac{{\pi }^{2}}{4{L}^{2}},0\right). While for p>1p\gt 1, the bifurcation curve is reversed ε\varepsilon -like shaped bifurcation if L>L∗L\gt {L}^{\ast }, and is exactly decreasing for λ>λ∗\lambda \gt {\lambda }^{\ast } if 0<L<L∗0\lt L\lt {L}_{\ast }. |
| format |
article |
| author |
Zhang Jiajia Qiao Yuanhua Duan Lijuan Miao Jun |
| author_facet |
Zhang Jiajia Qiao Yuanhua Duan Lijuan Miao Jun |
| author_sort |
Zhang Jiajia |
| title |
On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type |
| title_short |
On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type |
| title_full |
On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type |
| title_fullStr |
On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type |
| title_full_unstemmed |
On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type |
| title_sort |
on the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/0c28dc2095504b5eb0296f9998f8d81a |
| work_keys_str_mv |
AT zhangjiajia ontheevolutionarybifurcationcurvesfortheonedimensionalprescribedmeancurvatureequationwithlogistictype AT qiaoyuanhua ontheevolutionarybifurcationcurvesfortheonedimensionalprescribedmeancurvatureequationwithlogistictype AT duanlijuan ontheevolutionarybifurcationcurvesfortheonedimensionalprescribedmeancurvatureequationwithlogistictype AT miaojun ontheevolutionarybifurcationcurvesfortheonedimensionalprescribedmeancurvatureequationwithlogistictype |
| _version_ |
1718371582618894336 |