On a power-type coupled system of k-Hessian equations

Abstract

We deal with a coupled system of k-Hessian equations:

Sk( μ(D²u₁)) = (-u₁))  ^∝   in B,

Sk( μ(D²u₁)) = (-u2))        in B,

u₁ < 0,     u₂ < 0               in B,

u₁  = u₂ = 0                      on ∂B,

where k = 1,2, ⋯ , N, B is a unit ball in R^N, N ≥ 2, α and β  are positive constants. By using the fixed-point index theory in cone, we obtain the existence, uniqueness and nonexistence of radial convex solutions for some suitable constants α and β. Furthermore, by using a generalized Krein-Rutman theorem, we also obtain a necessary and sufficient existence condition of the convex solutions to a nonlinear eigenvalue problem.