Existence and multiplicity of radially symmetric k-admissible solutions for Dirichlet problem of k-Hessian equations

Abstract

In this paper, we study the existence and multiplicity of radially symmetric k-admissible solutions for the k-Hessian equation with 0-Dirichlet boundary condition

S_k(D² u)= f(-u) in B
u=0                        on ∂B

and the corresponding one-parameter problem, where B is a unit ball in Rⁿ with n ≥ 1, k∈ {1, ... , n}, f: [0, +∞) → [0, +∞) is continuous. We show that the k-admissible solutions are not convex, so we construct a new cone and obtain the existence of triple and arbitrarily many k-admissible solutions via the Leggett-Williams' fixed point theorem.