Stable Solutions of Elliptic Equations on Riemannian Manifolds
نویسندگان
چکیده
Abstract. This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincaré inequality which allows to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.
منابع مشابه
Some Elliptic Pdes on Riemannian Manifolds with Boundary
The goal of this paper is to investigate some rigidity properties of stable solutions of elliptic equations set on manifolds with boundary. We provide several types of results, according to the dimension of the manifold and the sign of its Ricci curvature.
متن کاملNodal solutions to quasilinear elliptic equations on compact Riemannian manifolds
We show the existence of nodal solutions to perturbed quasilinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. A nonexistence result is also given.
متن کاملSecond Order Estimates and Regularity for Fully Nonlinear Elliptic Equations on Riemannian Manifolds
We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under structure conditions which are close to optimal. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary. These estimates yield regularity and existence results some of ...
متن کاملSome Nonlinear Elliptic Equations Have Only Constant Solutions *
We study some nonlinear elliptic equations on compact Riemannian manifolds. Our main concern is to find conditions which imply that such equations admit only constant solutions.
متن کاملExistence and mutiplicity of solutions to elliptic equations of fourth order on compact manifolds
Existence and mutiplicity of solutions to elliptic equations of fourth order on compact manifolds. Abstract. This paper deals with a fourth order elliptic equation on compact Riemannian manifolds.We establish the existence of solutions to the equation with critical Sobolev growth which is the subject of the first theorem. In the second one, we prove the multiplicity of solutions in the subcriti...
متن کامل