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Three solutions with precise sign properties for Gierer-Meinhardt type system

Abdelkrim Moussaoui

TL;DR

This work analyzes a sign-coupled Gierer–Meinhardt elliptic system with Neumann boundary conditions on a bounded smooth domain. It proves the existence of three distinct solutions: two with opposite constant signs and a third nodal pair when $\beta_1=0$, using a blend of sub-supersolution constructions and Leray–Schauder topological degree, including a regularized problem $(\mathrm{P}^{\varepsilon})$ to manage singular terms. The approach first secures two constant-sign solutions within carefully constructed rectangles, then employs degree theory to locate a third, nodal solution via a limiting process $\varepsilon\to0$. The results illuminate the sign structure and multiplicity of GM-type systems under Neumann conditions and demonstrate how cooperative/competitive dynamics (via $\beta_1=0$) influence solution topology.

Abstract

We establish the existence of three solutions for sign-coupled Gierer-Meinhardt type system with Neumann boundary conditions. Two solutions are of opposite constant-sign while the third solution is nodal with synchronous sign components. The approach combines sub-supersolutions method and Leray-Schauder topological degree involving perturbation argument.

Three solutions with precise sign properties for Gierer-Meinhardt type system

TL;DR

This work analyzes a sign-coupled Gierer–Meinhardt elliptic system with Neumann boundary conditions on a bounded smooth domain. It proves the existence of three distinct solutions: two with opposite constant signs and a third nodal pair when , using a blend of sub-supersolution constructions and Leray–Schauder topological degree, including a regularized problem to manage singular terms. The approach first secures two constant-sign solutions within carefully constructed rectangles, then employs degree theory to locate a third, nodal solution via a limiting process . The results illuminate the sign structure and multiplicity of GM-type systems under Neumann conditions and demonstrate how cooperative/competitive dynamics (via ) influence solution topology.

Abstract

We establish the existence of three solutions for sign-coupled Gierer-Meinhardt type system with Neumann boundary conditions. Two solutions are of opposite constant-sign while the third solution is nodal with synchronous sign components. The approach combines sub-supersolutions method and Leray-Schauder topological degree involving perturbation argument.

Paper Structure

This paper contains 8 sections, 5 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Two opposite constant-sign solutions
  3. A nodal solution
  4. The regularized system
  5. The degree on $\mathcal{B}_{R_{\varepsilon }}(0)$
  6. The degree on $\mathcal{B}_{R_{\varepsilon }}(0)\backslash \overline{\mathcal{M}_{R_{\varepsilon }}}$.
  7. Proof of Theorem \ref{['T3']}.
  8. Proof of Theorem \ref{['T2']}.

Key Result

Theorem 1.1

Under assumption (alpha), problem $(\mathrm{P})$ admits at least two opposite constant-sign solutions $(u_{+},v_{+})\in int\mathcal{C}_{+}^{1}(\overline{\Omega }))\times int\mathcal{C}_{+}^{1}(\overline{\Omega }),$$(u_{-},v_{-})\in -int\mathcal{C}_{+}^{1}(\overline{\Omega })\times -int\mathcal{C}_{+

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Proposition 3.1
  • proof