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

Huang, Shao-Yuan ; Hsieh, Ping-Han

Czechoslovak Mathematical Journal, Tome 73 (2023) no. 4, pp. 1081-1098

Abstract (VO)

We study the exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems$$ \begin {cases} -[\phi (u^{\prime })]^{\prime }=\lambda u^{p} \Bigl (1-\dfrac {u}{N} \Bigr ) \text {in} \^^M( -L,L) , \\ u(-L)=u(L)=0,\end {cases} $$ 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.

Détail

DOI : 10.21136/CMJ.2023.0359-22

Classification : 34B15, 34B18, 34C23, 74G35
Keywords: positive solution; bifurcation curve; Minkowski-curvature problem, logistic problem

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