Liouville type results for semilinear elliptic equation and inequality on pseudo-Hermitian manifolds

2025/07/12 by Biqiang Zhao, Zhao, Biqiang
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2507.09130

Abstract

In this paper, we study the semilinear elliptic equation and inequality on pseudo-Hermitian manifolds. In particular, we first obtain a Liouville theorem for the equation Δ_b u+F(u)=0 based on a generalized Jerison-Lee's formula. Next, we prove the nonexistence of a positive solution to the inequality Δ_b u+F(u)≤ 0 under the volume estimate.

Citations

• Liouville Theorems on pseudohermitian manifolds with nonnegative Tanaka-Webster curvature
• Optimal Liouville theorems for the Lane-Emden equation on Riemannian manifolds
• A Liouville theorem in the Heisenberg group
• On the critical p-Laplace equation
• Semilinear elliptic equations on manifolds with nonnegative Ricci curvature
• Liouville theorems for semilinear differential inequalities on sub-Riemannian manifolds
• On the Cheng-Yau gradient estimate for Carnot groups and sub-Riemannian manifolds
• Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations
• Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds
• Bishop and Laplacian comparison theorems on Sasakian manifolds
• Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Subriemannian Manifolds with Symmetry
• Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations
• Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
• Global and local behavior of positive solutions of nonlinear elliptic equations
• Differential equations on riemannian manifolds and their geometric applications

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• A New proof of Liouville type theorems for a class of semilinear elliptic equations
• A note on Liouville type theorem of elliptic inequality Δu+u^σ≤ 0 on Riemannian manifolds
• Gradient Estimates And Liouville Theorems For A Class of Nonlinear Elliptic Equations