[Paper Review] Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree

Alberto Boscaggin, Guglielmo Feltrin+1 more|arXiv (Cornell University)|Dec 22, 2015

Nonlinear Partial Differential Equations65 references20 citations

TL;DR

This paper establishes the existence of $3^m - 1$ positive $T$-periodic solutions for a super-sublinear indefinite second-order differential equation with a weight function $a(t)$ having $m$ positive humps separated by $m$ negative humps. Using coincidence degree theory on unbounded open sets, the authors prove high multiplicity of solutions for large positive parameters $\lambda$ and $\mu$, extending to Neumann and Dirichlet boundary conditions and establishing symbolic dynamics and subharmonic solutions.

ABSTRACT

We study the periodic boundary value problem associated with the second order nonlinear equation \\begin{equation*} u'' + ( \\lambda a^{+}(t) - \\mu a^{-}(t) ) g(u) = 0, \\end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $\\lambda, \\mu$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-periodic solutions when the weight function $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a $T$-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

Motivation & Objective

• To establish high multiplicity of positive periodic solutions for a class of second-order nonlinear ODEs with indefinite weights.
• To analyze the role of parameter scaling ($\lambda, \mu \to \infty$) in generating complex solution structures.
• To extend multiplicity results beyond Dirichlet conditions to Neumann and periodic settings using topological methods.
• To demonstrate the existence of abundant subharmonic solutions and symbolic dynamics in the solution set.

Proposed method

• Formulates the problem as $u'' + (\lambda a^+(t) - \mu a^-(t))g(u) = 0$, where $g$ is superlinear at zero and sublinear at infinity.
• Applies coincidence degree theory for locally compact operators on unbounded open sets to compute the degree of the associated operator.
• Uses a combinatorial lemma to compute the degree over disjoint subsets of the solution space indexed by $\mathcal{I}'$ and $\mathcal{J}'$, leveraging inclusion-exclusion and parity arguments.
• Establishes that the degree is $(-1)^{\#\mathcal{I}'}$, which implies the existence of at least one solution per nonempty subset of the $m$ positive humps.
• Applies the method to Neumann and Dirichlet boundary conditions by adapting the topological framework to the respective solution spaces.
• Derives symbolic dynamics and subharmonic solutions by analyzing the structure of the solution set under parameter variation.

Experimental results

Research questions

• RQ1 How many positive $T$-periodic solutions exist for a super-sublinear indefinite ODE when the weight function has $m$ positive and $m$ negative humps?
• RQ2 Can coincidence degree theory be extended to unbounded domains to prove multiplicity in indefinite problems with nonlinearities of mixed growth?
• RQ3 What role do the parameters $\lambda$ and $\mu$ play in generating high multiplicity of solutions when both are large?
• RQ4 Do the solution sets exhibit complex dynamics such as symbolic dynamics or subharmonic solutions under these conditions?
• RQ5 Can the results be extended from periodic to Neumann and Dirichlet boundary conditions?

Key findings

• For a weight function with $m$ positive and $m$ negative humps, the equation admits at least $3^m - 1$ positive $T$-periodic solutions when $\lambda$ and $\mu$ are sufficiently large.
• The solution multiplicity arises from the combinatorial structure of the positive humps and the topological degree computation over subsets of the solution space.
• The method applies to Neumann and Dirichlet boundary conditions, extending prior results limited to Dirichlet settings.
• The solution set contains infinitely many subharmonic solutions, indicating rich dynamical behavior.
• Symbolic dynamics is established in the solution set, implying chaotic-like behavior under parameter variation.
• The degree computation relies on a key combinatorial identity involving subsets of index sets, ensuring nontrivial degree and hence existence of solutions.