Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator

In this work, we investigate the existence of multiple positive solutions for a weakly coupled system of nonlinear elliptic equations governed by Pucci extremal operators. Specifically, we consider the system: $$\{\begin{array}{ll}-M_{{\lambda }_1,{\mathrm{\Lambda }}_1}^{+}(D^2{u}_1)=\mu f_1(u_1,u_2,\dots ,u_n),& \text{in }\mathrm{\Omega },\\ -M_{{\lambda }_2,{\mathrm{\Lambda }}_2}^{+}(D^2{u}_2)=\mu f_2(u_1,u_2,\dots ,u_n),& \text{in }\mathrm{\Omega },\mathrm{⋮}\\ -M_{{\lambda }_n,{\mathrm{\Lambda }}_n}^{+}(D^2{u}_n)=\mu f_n(u_1,u_2,\dots ,u_n),& \text{in }\mathrm{\Omega },\\ u_1=u_2=\cdots =u_n=0,& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $M_{\lambda ,\mathrm{\Lambda }}^{+}$ represents the Pucci extremal operator, $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with smooth boundary, and the nonlinear functions $f_i:[0,\mathrm{\infty }{)}^n\to [0,\mathrm{\infty })$ belong to the $C^{1,\alpha }$ class. Our main results establish the existence and multiplicity of solutions for sufficiently large values of the parameter $\mu >0$. The analysis relies on the method of sub and supersolutions, in conjunction with fixed-point arguments and bifurcation techniques.