A Laplacian System With Sign-Changing Weight Function

TL;DR

This paper establishes the existence of positive solutions for a Laplacian system with a sign-changing weight function on a bounded domain, using variational methods involving the Nehari manifold and fibering maps.

Contribution

It introduces a novel approach to prove positive solutions for Laplacian systems with sign-changing weights via Nehari manifold techniques.

Findings

Existence of at least one positive solution proven.

Application of Nehari manifold and fibering maps to systems with sign-changing weights.

Method extends variational techniques to more complex Laplacian systems.

Abstract

We prove the existence of at least one positive solution for the Laplacian system\\ -\Delta v=\lambda a(x)|v|^{q-2}v+\beta\frac{\beta}{\alpha+\beta}b(x)|u|^{\alpha}|v|^{\beta-2}v&$for~$x\in\Omega$$ $$ \end{array}\right.$$ On a bounded region $\Omega$ by using the Nehari manifold and the fibering maps associated with the Euler functional for the system.