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High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type

Pablo Cubillos, Julián López-Gómez, Andrea Tellini

TL;DR

This work analyzes a 1D degenerate superlinear boundary-value problem of Moore–Nehari type with piecewise-constant weights that vanish on κ symmetric intervals. It combines analytic results showing concentration of positive solutions on regions where the weight is nonzero with extensive numerical path-following to map global bifurcation diagrams as λ and ε vary. The main theoretical insight is that as $λ$ becomes very negative, the solution structure decomposes into $κ+1$ autonomous subproblems, supporting a conjectured multiplicity of $2^{κ+1}-1$ positive solutions, which is borne out by detailed computations for κ up to 3 and various $h$-values. The findings reveal how spatial heterogeneity can induce high multiplicity and complex bifurcation patterns, including isolas and recombination phenomena, and suggest multiplicity persists for a broad class of weights near the degenerate and autonomous limits.

Abstract

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $κ\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter $λ$ and on $κ$. Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as $λ\downarrow-\infty$, in the regions where the weight vanishes. Our result leads us to conjecture the existence of $2^{κ+1}-1$ solutions for sufficiently negative $λ$. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in $λ$ and the profiles of positive solutions. Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions.

High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type

TL;DR

This work analyzes a 1D degenerate superlinear boundary-value problem of Moore–Nehari type with piecewise-constant weights that vanish on κ symmetric intervals. It combines analytic results showing concentration of positive solutions on regions where the weight is nonzero with extensive numerical path-following to map global bifurcation diagrams as λ and ε vary. The main theoretical insight is that as becomes very negative, the solution structure decomposes into autonomous subproblems, supporting a conjectured multiplicity of positive solutions, which is borne out by detailed computations for κ up to 3 and various -values. The findings reveal how spatial heterogeneity can induce high multiplicity and complex bifurcation patterns, including isolas and recombination phenomena, and suggest multiplicity persists for a broad class of weights near the degenerate and autonomous limits.

Abstract

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter and on . Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as , in the regions where the weight vanishes. Our result leads us to conjecture the existence of solutions for sufficiently negative . On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in and the profiles of positive solutions. Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions.
Paper Structure (7 sections, 1 theorem, 49 equations, 30 figures, 3 tables)

This paper contains 7 sections, 1 theorem, 49 equations, 30 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Point-wise behavior of positive solutions as $\lambda \downarrow -\infty$
  3. The case $\kappa =1$ with $h\in (0,1)$
  4. The case $\kappa =2$ with $h\in (0,0.5)$
  5. The case $\kappa =3$ with $h=0.3$
  6. Effects of varying $\varepsilon\in [0,1]$ in case $\kappa=1$ with $h=0.1$
  7. Technical aspects related to the numerical experiments

Key Result

Theorem 2.1

Suppose that $a=a_{\kappa,0}$ (see 1.5) for some integer $\kappa\geq 1$, and, for $\lambda<\pi^2$, let $u_\lambda$ be a positive solution of 1.1. Then,

Figures (30)

  • Figure 1: Bifurcation diagram of positive solutions of \ref{['1.3']}.
  • Figure 2: Phase portrait of positive solutions of \ref{['2.3']}.
  • Figure 3: Profiles of the solutions of \ref{['1.1']} with $a=a_{2,0}$ for $\lambda=-3000$. Here $\alpha_1=0.175$ and $\beta_1=0.325$ (thus, $\alpha_2=0.675$ and $\beta_2=0.825$). These profiles show the phenomenology described in Theorem \ref{['th2.1']}.
  • Figure 4: An admissible weight function $a(x)$ with $\kappa=1$.
  • Figure 5: Bifurcation diagram relative to \ref{['1.1']} for $a(x)=a_{1,0}(x)$ with $h=0.05$ (left) and a zoom of it (right).
  • ...and 25 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • proof
  • Remark 2.2