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

TL;DR

This paper proves the existence of three solutions to a Gierer-Meinhardt system with specific sign properties, using a combination of sub-supersolution and topological degree methods.

Contribution

It introduces a novel approach combining sub-supersolutions and Leray-Schauder degree to find multiple solutions with prescribed sign characteristics.

Findings

Existence of two solutions with opposite signs.

Existence of a nodal solution with synchronous sign components.

Application of combined sub-supersolution and topological degree methods.

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.