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

TL;DR

This paper proves the existence and multiple positive solutions for a system of nonlinear elliptic equations involving Pucci extremal operators, using sub/supersolutions, fixed-point, and bifurcation methods for large parameter values.

Contribution

It introduces new results on solution multiplicity for coupled Pucci operator systems, extending previous work to more general nonlinearities and boundary conditions.

Findings

Multiple positive solutions exist for large parameter μ.

The number of solutions can be controlled by bifurcation analysis.

The methods apply to a broad class of nonlinear elliptic systems.

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.