Existence of radial solutions to biharmonic k-Hessian equations.

This work presents the construction of the existence theory of radial solutions to the elliptic equation Δ 2 u = ( − 1 ) k S k [ u ] + λ f ( x ) , x ∈ B 1 ( 0 ) ⊂ R N , provided either with Dirichlet boundary conditions u = ∂ n u = 0 , x ∈ ∂ B 1 ( 0 ) , or Navier boundary conditions u = Δ u = 0 , x ∈ ∂ B 1 ( 0 ) , where the k -Hessian S k [ u ] is the k -th elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum f ∈ L 1 ( B 1 ( 0 ) ) while λ ∈ R . We prove the existence of a Carathéodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided | λ | is small enough. Moreover, we prove that the solvability set of λ is finite, giving an explicity bound of the extreme value. [ABSTRACT FROM AUTHOR]