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

Abstract

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: \[ \begin{cases} -{M}_{\lambda_1,\Lambda_1}^+(D^2u_1) = \mu f_1(u_1, u_2, \dots, u_n), & \text{in } \Omega, \\ -{M}_{\lambda_2,\Lambda_2}^+(D^2u_2) = \mu f_2(u_1, u_2, \dots, u_n), & \text{in } \Omega, \vdots \\ -{M}_{\lambda_n,\Lambda_n}^+(D^2u_n) = \mu f_n(u_1, u_2, \dots, u_n), & \text{in } \Omega, \\ u_1 = u_2 = \dots = u_n = 0, & \text{on } \partial\Omega, \end{cases} \] where $ {M}_{\lambda,\Lambda}^+ $ represents the Pucci extremal operator, $ \Omega $ is a bounded domain in $ \mathbb{R}^N $ with smooth boundary, and the nonlinear functions $ f_i: [0, \infty)^n \to [0, \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.