Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 15, pp. 1-14. Title: Multiple positive solutions for quasilinear elliptic systems Authors: Qin Li (Nanjing Normal Univ., China) Zuodong Yang (Nanjing Normal Univ., China) Abstract: In this article, we investigate how the coefficient $f(z)$ affects the number of positive solutions of the quasilinear elliptic system $$\displaylines{ -\Delta_{p}u =\lambda g(z)|u|^{q-2}u+\frac{\alpha}{\alpha+\beta} f(z)|u|^{\alpha-2}u|v|^{\beta} \quad\hbox{in }\Omega,\cr -\Delta_{p}v =\mu h(z)|v|^{q-2}v +\frac{\beta}{\alpha+\beta}f(z)|u|^{\alpha}|v|^{\beta-2}v \quad\hbox{in }\Omega,\cr u=v=0\quad\hbox{on }\partial\Omega, }$$ where $0\in\Omega\subset \mathbb{R}^{N}$ is a bounded domain, $\alpha >1$, $\beta>1$ and $1
2p$. Submitted July 6, 2012. Published January 17, 2013. Math Subject Classifications: 35J65, 35J50. Key Words: Quasilinear elliptic systems; multiple positive solutions; critical point, Nehari manifold.