Partial symmetry and existence of least energy solutions to some nonlinear elliptic equations on Riemannian models

Abstract

We consider least energy solutions to the nonlinear equation $-{\Delta }_{g}u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space ${\mathbb{H}}^n$ as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is proved for quite general nonlinearities $f(r,u)$, where $r$ denotes the geodesic distance from the pole of $M$.

References in corpus (1)

• The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball