Nodal solutions with synchronous sign changing components and Constant sign solutions for singular Gierer-Meinhardt type system

TL;DR

This paper proves the existence of multiple solutions, including constant-sign and nodal solutions with synchronous sign components, for a singular Gierer-Meinhardt type elliptic system using topological and perturbation methods.

Contribution

It introduces a novel combination of sub-supersolutions and Leray-Schauder degree techniques to handle strong singularities in elliptic systems.

Findings

Existence of three solutions, including two with constant signs.

Presence of a nodal solution with synchronous sign-changing components.

Application of combined analytical methods to singular systems.

Abstract

We establish the existence of three solutions for singular semilinear elliptic system, two of which are of opposite constant-sign. Under a strong singularity effect, the third solution is nodal with synchronous sign components. The approach combines sub-supersolutions method and Leray-Schauder topological degree involving perturbation argument.