Liouville theorems for nonlinear elliptic equations on Riemannian manifolds

2021 ◽  
Vol 496 (1) ◽  
pp. 124803
Author(s):  
Nguyen Thac Dung ◽  
Pham Duc Thoan ◽  
Nguyen Dang Tuyen
Keyword(s):  
Riemannian Manifolds ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Liouville Theorems ◽  
Nonlinear Elliptic

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS

2007 ◽  
Vol 18 (09) ◽  
pp. 1071-1111 ◽  
Author(s):  
JÉRÔME VÉTOIS
Keyword(s):  
Continuous Function ◽  
Riemannian Manifolds ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Nonlinear Elliptic Equations ◽  
Beltrami Operator ◽  
Hölder Continuous ◽  
Nonlinear Elliptic

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 4, and h be a Holdër continuous function on M. We prove multiplicity of changing sign solutions for equations like Δg u + hu = |u|2* - 2 u, where Δg is the Laplace–Beltrami operator and 2* = 2n/(n - 2) is critical from the Sobolev viewpoint.

Elliptic equations on manifolds and isoperimetric inequalities

1990 ◽  
Vol 114 (3-4) ◽  
pp. 213-227 ◽  
Author(s):  
Andrea Cianchi
Keyword(s):  
Riemannian Manifolds ◽  
Elliptic Equations ◽  
A Priori ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Isoperimetric Inequalities ◽  
Divergence Form ◽  
Sharp Estimates ◽  
Nonlinear Elliptic ◽  
Homogeneous Neumann Boundary Condition

SynopsisWe consider linear and nonlinear elliptic equations in divergence form on Riemannian manifolds with or without boundary. In the former case we impose a homogeneous Neumann boundary condition. By making use of isoperimetric inequalities for manifolds, we obtain a priori sharp estimates for the decreasing rearrangement of the solutions to such equations. These estimates enable us to derive bounds for suitable norms of the solutions and of their gradients.

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