Bifurcation Results for the Mean Curvature Equations Defined on all ℝ N

Abstract

We prove certain local bifurcation results for the mean curvature problem. $$ - \nabla \left( {\frac{{\nabla u}}{{\sqrt {1 + |\nabla u|^2 } }}} \right) = \lambda f(x,u),{\text{ }}x \in \mathbb{R}^N .$$ . This is achieved by applying standard local bifurcation theory. The use of certain equivalent weighted and homogeneous Sobolev spaces was proved to be crucial.