Uniqueness, non-degeneracy, and exact multiplicity of positive solutions for superlinear elliptic problems

TL;DR

This work studies positive solutions of 1D indefinite, superlinear Dirichlet problems of the form $u''+(h^{+}(t)-\mu h^{-}(t))\,g(u)=0$ and proves that, for large $\mu$, there are exactly $2^{m}-1$ positive solutions, all nondegenerate. The authors combine shooting methods, Sturm comparison, and refined Kolodner–Coffman-type uniqueness results with a continuation-from-infinity strategy, leveraging limit profiles on positive subintervals to achieve an exact multiplicity count that matches and completes Feltrin–Zanolin results. They establish symmetry and BV-based regularity conditions on the weight $h^{+}$ to guarantee uniqueness and nondegeneracy, and provide a detailed numerical study illustrating bifurcations and symmetry-breaking phenomena as $\mu$ varies. The results offer a sharp, optimal multiplicity framework for indefinite superlinear problems and illuminate the role of limit profiles in the continuation analysis, with potential implications for related ODE and PDE systems.

Abstract

In this paper, we focus our attention on the positive solutions to second-order nonlinear ordinary differential equations of the form $u''+q(t)g(u)=0$, where $q$ is a sign-changing weight and $g$ is a superlinear function. We exploit the classical shooting approach and the comparison theorem to present non-degeneracy and exact multiplicity results for positive solutions. This completes the multiplicity results obtained by Feltrin and Zanolin. Numerical examples and some related open problems are also discussed.

Figures (11)

• Figure 1: Qualitative representation of $\tilde{u}$ and $\tilde{q}$ defined in \ref{['def-tilde-uq']}, where $\tilde{a}=2t^{*}-a$.
• Figure 2: The branches of positive solutions to \ref{['eq:Dmu']} with $h(t) = \sin(3\pi t)$ in $\mathopen[ 0,1\mathclose]$ and $g(u)=u^{3}$ (on the left) and graphs of solutions for some $\mu$ (on the right).
• Figure 3: Graphs of the unique positive solution to \ref{['eq:Dmu']} with $h(t) = \sin(3\pi t)$ in $\mathopen[ 0,1\mathclose]$ and $g(u)=u^{3}$ for various $\mu < 0$.
• Figure 4: The branches of positive solutions to \ref{['eq:Dmu']} with $h(t) = \sin(5\pi t)$ in $\mathopen[ 0,1\mathclose]$ and $g(u)=u^{3}$ (on the left) and graphs of solutions for some $\mu$ (on the right).
• Figure 5: Graphs of the unique positive solution to \ref{['eq:Dmu']} with $h(t) = \sin(5\pi t)$ in $\mathopen[ 0,1\mathclose]$ and $g(u)=u^{3}$ for various $\mu \le 1$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Theorem 2.3: Sturm's comparison Theorem
• Lemma 2.4
• proof
• Remark 2.5
• Proposition 2.6: Leibniz rule