Liouville theorem for the inequality Δ_m u+f(u)≤ 0 on Riemannian manifolds

2025/09/20 by Zhao, Biqiang
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.16659

Abstract

In this paper, we study the quasilinear inequality Δ_m u+f(u)≤ 0 on a complete Riemannian manifold, where mgt;1,αgt;m-1 and f(t)gt; 0,αf(t)-tf^'(t)≥ 0, ∀ tgt;0. If for some point x₀ and large enough r, vol B_r(x₀)≤ C r^p ln^q r, where p=(mα)/(α-(m-1)),q=(m-1)/(α-(m-1)) and B_r(x₀) is a geodesic ball of radius r centered at x₀, then the inequality possesses no positive weak solution. This generalizes the result in \citeAS,Sun.

Citations

• Gradient estimate for solutions of the equation Δ_p v+av^q=0 on a complete Riemannian manifold
• Gradient estimates for singular p-Laplace type equations with measure data
• Superlinear elliptic inequalities on manifolds
• Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds
• Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds
• Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities
• Global and local behavior of positive solutions of nonlinear elliptic equations
• Differential equations on riemannian manifolds and their geometric applications

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• Liouville theorem for one kind of elliptic equations on complete Riemannian manifold