Multiple positive solutions for semilinear elliptic systems

This article investigates the effect of the coefficient $f(z)$ of the critical nonlinearity. For sufficiently small $\lambda ,\mu >0$, there are at least k positive solutions of the semilinear elliptic systems$\{\begin{array}{c}?\mathrm{\Delta }u=\lambda g(z){|u|}^{p?2}u+\frac{\alpha }{\alpha +\beta }f(z){|u|}^{\alpha ?2}u{|v|}^{\beta }\text{in}\Omega ;\hfill \\ ?\mathrm{\Delta }v=\mu h(z){|v|}^{p?2}v+\frac{\beta }{\alpha +\beta }f(z){|u|}^{\alpha }{|v|}^{\beta ?2}v\text{in}\Omega ;\hfill \\ u=v=0\text{on}?\Omega ,\hfill \end{array}$ where $0\in \Omega ?{\mathbb{R}}^N$ is a bounded domain, $\alpha >1$, $\beta >1$ and $2<p<\alpha +\beta =2^{?}$ for $N>4$.