Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911546572013568 |
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| author | Rathore, Karan Mallick, Mohan |
| author_facet | Rathore, Karan Mallick, Mohan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25627 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator Rathore, Karan Mallick, Mohan Analysis of PDEs 35J25, 35J60, 35D40, 35A01 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. |
| title | Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator |
| topic | Analysis of PDEs 35J25, 35J60, 35D40, 35A01 |
| url | https://arxiv.org/abs/2603.25627 |