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<p>This paper focuses on the Lane-Emden system with Dirichlet boundary conditions on Riemannian manifolds with boundary. We establish some gradient estimates, universal bounds and Harnack inequalities for positive solutions of the Lane-Emden system under Dirichlet boundary conditions, covering all non-positive exponents and certain positive exponents. As an application, we derive a Liouville theorem for the Lane-Emden system with Dirichlet boundary conditions on Riemannian manifolds with convex boundaries and non-negative Ricci curvature. The results presented herein generalize the recent work by Lu [[22], J. Geom. Anal. (2025) 35:129] to the case of Dirichlet boundary conditions.</p>

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Liouville Theorems and Universal Bounds for Lane-Emden System on Riemannian Manifolds with Boundary under Dirichlet Boundary Conditions

  • 期刊论文

  • 发表日期：

    2026年5月29日

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  • Wen Wang

摘要

This paper focuses on the Lane-Emden system with Dirichlet boundary conditions on Riemannian manifolds with boundary. We establish some gradient estimates, universal bounds and Harnack inequalities for positive solutions of the Lane-Emden system under Dirichlet boundary conditions, covering all non-positive exponents and certain positive exponents. As an application, we derive a Liouville theorem for the Lane-Emden system with Dirichlet boundary conditions on Riemannian manifolds with convex boundaries and non-negative Ricci curvature. The results presented herein generalize the recent work by Lu [[22], J. Geom. Anal. (2025) 35:129] to the case of Dirichlet boundary conditions.