Existence and Multiplicity of Radially Symmetric k-Admissible Solutions for Dirichlet Problem of k-Hessian Equations

Abstract

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 ( D 2 u ) = f ( − u ) in   B , u = 0 on   ∂ B , and the corresponding one-parameter problem, where B is a unit ball in ℝ n 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.