Spectrum, bifurcation and hypersurfaces of prescribed k-th mean curvature in Minkowski space

Abstract

By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k-th mean curvature problem in Minkowski spacetime r N − k v ′ 1 − v ′ 2 k ′ = λ N C N k r N − 1 H k ( r , v ) in  ( 0 , R ) , | v ′ | < 1 in  ( 0 , R ) , v ′ ( 0 ) = v ( R ) = 0 . As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ.