Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator
Karan Rathore, Mohan Mallick
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: \[ \begin{cases} -{M}_{λ_1,Λ_1}^+(D^2u_1) = μf_1(u_1, u_2, \dots, u_n), & \text{in } Ω, \\ -{M}_{λ_2,Λ_2}^+(D^2u_2) = μf_2(u_1, u_2, \dots, u_n), & \text{in } Ω, \vdots \\ -{M}_{λ_n,Λ_n}^+(D^2u_n) = μf_n(u_1, u_2, \dots, u_n), & \text{in } Ω, \\ u_1 = u_2 = \dots = u_n = 0, & \text{on } \partialΩ, \end{cases} \] where $ {M}_{λ,Λ}^+ $ represents the Pucci extremal operator, $ Ω$ 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,α} $ class. Our main results establish the existence and multiplicity of solutions for sufficiently large values of the parameter $ μ> 0 $. The analysis relies on the method of sub and supersolutions, in conjunction with fixed-point arguments and bifurcation techniques.