Global multiplicity results in a Moore-Nehari type problem with a spectral parameter

TL;DR

The paper analyzes the global bifurcation structure of positive solutions to a Moore–Nehari type problem with a piecewise-constant weight $a_h(x)$, treating the spectral parameter $\lambda$ as the main bifurcation parameter and $h$ as a deformation between autonomous and linear limits. It proves the existence and uniqueness of the symmetric positive solution for all $\lambda<\pi^2$ and describes a continuous symmetric branch $\mathscr{C}_{h,s}^+$ with endpoint behavior and a priori bounds, while also establishing multiple solution phenomena: for $\lambda<0$, at least two asymmetric positive solutions exist for $\lambda$ below a negative threshold; for $\lambda\in(\pi^2/4,\pi^2)$ and $h$ near 1, two asymmetric positive solutions emerge as well, both arising from detailed phase-plane and Mountain Climbing analyses. As $h\uparrow1$, all positive solutions blow up pointwise on $(0,1)$ for $\lambda<\pi^2$, yet a shadow linear limit $-u''=\pi^2u$ emerges, and metasolution-type convergence to scaled sine modes occurs along appropriate sequences. These results extend the understanding of global bifurcation in heterogeneous media and singular perturbation effects in nonlinear eigenvalue problems."

Abstract

This paper analyzes the structure of the set of positive solutions of a Moore-Nehari type problem, where $a\equiv a_h$ is a piece-wise constant function defined for some $h\in (0,1)$. In our analysis, $λ$ is regarded as a bifurcation parameter, whereas $h$ is viewed as a deformation parameter between the autonomous case when $a=1$ and the linear case when $a=0$. In this paper, besides establishing some of the multiplicity results suggested by previous numerical experiments (see Cubillos, López-Gómez and Tellini, 2024), we have analyzed the asymptotic behavior of the positive solutions of the problem as $h\uparrow 1$, when the shadow system of the problem is the linear equation $-u''=π^2 u$. This is the first paper where such a problem has been addressed. Numerics is of no help in analyzing this singular perturbation problem because the positive solutions blow-up point-wise in $(0,1)$ as $h\uparrow 1$ if $λ<π^2$.

Figures (8)

• Figure 1: Phase plane diagrams of ($\mathscr{N}$) (left) and ($\mathscr{L}$) (right) when $\lambda<0$. Integral curves with positive energy are plotted in red, while those with zero energy are plotted in black and those with negative energy are colored in blue.
• Figure 2: The phase plane route of a symmetric positive solution of \ref{['1.1']} in the case $\lambda<0$: nonlinear interval $[0,\tfrac{1-h}{2}]$ (left), linear interval $(\tfrac{1-h}{2},\tfrac{1+h}{2})$ (center) and nonlinear interval $[\tfrac{1+h}{2},1]$ (right).
• Figure 3: A symmetric positive solution of \ref{['1.1']} in the case $\lambda\in(0,\pi^2)$.
• Figure 4: The component $\mathscr{C}_{h,s}^+$ of \ref{['1.1']} positive symmetric solutions (left) and a possible component $\mathscr{C}_h^+$ of \ref{['1.1']} positive solutions (right).
• Figure 5: The sets $\Gamma_0$ in blue light and $\Gamma_1$ in green (left), as well as the sets $\Gamma_{0,\frac{1-h}{2}}^+$ in blue light and $\Gamma_{1,\frac{1+h}{2}}^-$ in green (right).

Theorems & Definitions (18)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• proof