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A proof of a conjecture on the multiplicity of positive solutions of an indefinite superlinear problem

Guglielmo Feltrin, Julián López-Gómez, Juan Carlos Sampedro

TL;DR

This work resolves a long-standing conjecture on the multiplicity of positive solutions for a one-dimensional indefinite superlinear Dirichlet problem by combining a Liouville-type a priori bound with a Leray–Schauder degree framework. The authors reformulate the boundary value problem as a fixed-point equation using the Green’s kernel, then partition the domain into $n$ positive-interval regions and construct corresponding degree-analytic sets. They prove that the Leray–Schauder degree is nonzero on the empty-index region and takes alternating signs on the index-containing regions, yielding at least $2^{n}-1$ nontrivial, strongly positive solutions for all $\\\lambda$ below a negative threshold $\\lambda_{c}$. This approach provides a rigorous multiplicity result for indefinite weights and superlinear nonlinearities, solving the GR Gómez-Reñasco–López-Gómez conjecture and extending degree-theoretic methods to this class of problems.

Abstract

This paper solves in a positive manner a conjecture stated in 2000 by R. Gómez-Reñasco and J. López-Gómez regarding the multiplicity of positive solutions of a paradigmatic class of superlinear indefinite boundary value problems.

A proof of a conjecture on the multiplicity of positive solutions of an indefinite superlinear problem

TL;DR

This work resolves a long-standing conjecture on the multiplicity of positive solutions for a one-dimensional indefinite superlinear Dirichlet problem by combining a Liouville-type a priori bound with a Leray–Schauder degree framework. The authors reformulate the boundary value problem as a fixed-point equation using the Green’s kernel, then partition the domain into positive-interval regions and construct corresponding degree-analytic sets. They prove that the Leray–Schauder degree is nonzero on the empty-index region and takes alternating signs on the index-containing regions, yielding at least nontrivial, strongly positive solutions for all below a negative threshold . This approach provides a rigorous multiplicity result for indefinite weights and superlinear nonlinearities, solving the GR Gómez-Reñasco–López-Gómez conjecture and extending degree-theoretic methods to this class of problems.

Abstract

This paper solves in a positive manner a conjecture stated in 2000 by R. Gómez-Reñasco and J. López-Gómez regarding the multiplicity of positive solutions of a paradigmatic class of superlinear indefinite boundary value problems.
Paper Structure (6 sections, 9 theorems, 135 equations)

This paper contains 6 sections, 9 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. A priori estimates
  3. Proof of Theorem \ref{['th1.1']}
  4. Topological degree setting
  5. The sets $\Omega^{\mathcal{I}}$ and the computation of the degree
  6. Completing the proof of Theorem \ref{['th1.1']}

Key Result

Theorem 1.1

Let $p>1$ and $a(x)$ satisfy hp-Ha and hp-G. Then, there exists $\lambda_{c}=\lambda_c(a)<0$ such that, for every $\lambda < \lambda_{c}$, the problem 1.1 possesses, at least, $2^{n}-1$ positive solutions.

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 6 more