Classification of positive solutions to a class of Laplace equations with a gradient term

2025/11/25 by Dou, Jingbo, Shi, Benfeng, Wu, Tian +1
Mathematics · Materials Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Nonlocal and gradient elasticity in micro/nano structures

paper · doi:10.48550/arxiv.2511.20205

Abstract

In this paper, we investigate positive solutions to a class of Laplace equations with a gradient term on a complete, connected, and noncompact Riemannian manifold \((Mⁿ,g)\) with nonnegative Ricci curvature, namely -Δu = f(u)|∇ u|^q\quadin ~Mⁿ, where \(n\geqslant 3\), \(q>0,\) and \(f\) is a positive continuous function. We prove some Liouville theorems employing a key differential identity derived via the invariant tensor technique. In particular, for \(f(u)=u^{(2-q)/(n-2)(n+(q)/(1-q))-1}\) is the second critical case in dimension \(n=3,4,5\), without any additional conditions, such as integrable conditions on \(u\), we show the rigidity for the ambient manifold and classification result of positive solutions. To our knowledge, this is the first rigidity result for equations with gradient terms in the second critical case. Moreover, this result confirms that all solutions must be of the form found in \citeBV-GH-V2019.

Citations

• Liouville theorems for anisotropic p-Laplace equations with a semilinear term
• A New proof of Liouville type theorems for a class of semilinear elliptic equations
• Critical quasilinear equations on Riemannian manifolds
• Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: a P-function approach
• Jerison-Lee identity and Semi-linear subelliptic equation on CR manifold
• Liouville theorem for elliptic equations with a source reaction term involving the product of the function and its gradient in \mathbb Rⁿ
• Nash-Moser iteration approach to the logarithmic gradient estimates and Liouville Properties of quasilinear elliptic equations on manifolds
• On the classification of entire solutions to the critical p-Laplace equation
• On the critical p-Laplace equation
• A note on the critical Laplace Equation and Ricci curvature
• Semilinear elliptic equations on manifolds with nonnegative Ricci curvature
• Uniqueness Results on a geometric PDE in Riemannian and CR Geoemetry Revisited
• Liouville results and asymptotics of solutions of a quasilinear elliptic equation with supercritical source gradient term
• Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms
• Local and global properties of solutions of quasilinear Hamilton-Jacobi equations
• 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

Related

• Gradient estimates and Liouville type theorems for a nonlinear elliptic equation
• Gradient Estimates And Liouville Theorems For A Class of Nonlinear Elliptic Equations
• Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: a P-function approach
• Critical quasilinear equations on Riemannian manifolds
• Gradient estimates for u_t=ΔF(u) on manifolds and some Liouville-type theorems