Existence of radial solutions to biharmonic k-Hessian equations

This work presents the construction of the existence theory of radial solutions to the elliptic equationΔ2u=(−1)kSk[u]+λf(x),x∈B1(0)⊂RN, provided either with Dirichlet boundary conditionsu=∂nu=0,x∈∂B1(0), or Navier boundary conditionsu=Δu=0,x∈∂B1(0), where the k  -Hessian Sk[u]Sk[u] is the k  -th elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum f∈L1(B1(0))f∈L1(B1(0)) while λ∈Rλ∈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||λ| is small enough. Moreover, we prove that the solvability set of λ is finite, giving an explicity bound of the extreme value.