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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| Autores principales: | , , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
De Gruyter
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/0c28dc2095504b5eb0296f9998f8d81a |
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| Sumario: | 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 }. |
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