A Laplacian System With Sign-Changing Weight Function

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.