Exact multiplicity and bifurcation curves of positive solutions of generalized logistic problems
Shao-Yuan Huang, Ping-Han Hsieh · Czechoslovak Mathematical Journal · 2023
summary: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.