On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type

Abstract

Abstract We study the bifurcation diagrams and exact multiplicity of positive solutions for the one-dimensional prescribed mean curvature equation − u ′ 1 + u ′ 2 ′ = λ u 1 + u p , − 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 > 0 L,p\gt 0 are two evolution parameters. We prove that on the ( λ , ‖ u ‖ ∞ ) \left(\lambda ,\Vert u{\Vert }_{\infty }) -plane, for 0 < p ≤ 2 4 0\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 ) = u 1 + u p=1,f\left(u)=\frac{u}{1+u} is a logistic function, then the bifurcation curve is also ⊃ \supset -shaped bifurcation starting from π 2 4 L 2 , 0 \left(\frac{{\pi }^{2}}{4{L}^{2}},0\right) . While for p > 1 p\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 } .