Exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems

Shao-Yuan Huang; Ping-Han Hsieh

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 4, page 1081-1098
  • ISSN: 0011-4642

Abstract

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We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems - [ φ ( u ' ) ] ' = λ u p 1 - u N in ( - L , L ) , u ( - L ) = u ( L ) = 0 , where p > 1 , N > 0 , λ > 0 is a bifurcation parameter, L > 0 is an evolution parameter, and φ ( u ) is either φ ( u ) = u or φ ( u ) = u / 1 - u 2 . We prove that the corresponding bifurcation curve is ⊂ -shape. Thus, the exact multiplicity of positive solutions can be obtained.

How to cite

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Huang, Shao-Yuan, and Hsieh, Ping-Han. "Exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems." Czechoslovak Mathematical Journal 73.4 (2023): 1081-1098. <http://eudml.org/doc/299426>.

@article{Huang2023,
abstract = {We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems\[ \{\left\lbrace \begin\{array\}\{ll\} -[\phi (u^\{\prime \})]^\{\prime \}=\lambda u^\{p\} \Bigl (1-\dfrac\{u\}\{N\} \Bigr ) & \text\{in\} \ ( -L,L) , \\ u(-L)=u(L)=0,\end\{array\}\right.\} \] where $p>1$, $N>0$, $\lambda >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, and $\phi (u)$ is either $\phi (u)=u$ or $\phi (u)=u/\sqrt\{1-u^\{2\}\}$. We prove that the corresponding bifurcation curve is $\subset $-shape. Thus, the exact multiplicity of positive solutions can be obtained.},
author = {Huang, Shao-Yuan, Hsieh, Ping-Han},
journal = {Czechoslovak Mathematical Journal},
keywords = {positive solution; bifurcation curve; Minkowski-curvature problem; logistic problem},
language = {eng},
number = {4},
pages = {1081-1098},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems},
url = {http://eudml.org/doc/299426},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Huang, Shao-Yuan
AU - Hsieh, Ping-Han
TI - Exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 4
SP - 1081
EP - 1098
AB - We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems\[ {\left\lbrace \begin{array}{ll} -[\phi (u^{\prime })]^{\prime }=\lambda u^{p} \Bigl (1-\dfrac{u}{N} \Bigr ) & \text{in} \ ( -L,L) , \\ u(-L)=u(L)=0,\end{array}\right.} \] where $p>1$, $N>0$, $\lambda >0$ is a bifurcation parameter, $L>0$ is an evolution parameter, and $\phi (u)$ is either $\phi (u)=u$ or $\phi (u)=u/\sqrt{1-u^{2}}$. We prove that the corresponding bifurcation curve is $\subset $-shape. Thus, the exact multiplicity of positive solutions can be obtained.
LA - eng
KW - positive solution; bifurcation curve; Minkowski-curvature problem; logistic problem
UR - http://eudml.org/doc/299426
ER -

References

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