k-convex Solution to a System of k-Hessian Equations with Power-type Nonlinearities

Abstract

We consider a system of k-Hessian equations

$$\left\{{\matrix{{{S_k}({\lambda ({{D^2}u})}) = {{({- u})}^{{\alpha _1}}} + {{({- v})}^{{\beta _1}}},} \hfill & {{\rm{in}}\,B,} \hfill \cr {{S_k}({\lambda ({{D^2}v})}) = {{({- u})}^{{\alpha _2}}},} \hfill & {{\rm{in}}\,B,} \hfill \cr {u = v = 0,} \hfill & {{\rm{on}}\,\partial B,} \hfill}} \right.$$

where 1 ≤ k ≤ n (n ≥ 2), α₁, α₂ and β₁ are positive constants, B = {x ∈ ℝⁿ: ∣x∣ < 1}. By giving the complete classification for the constants α₁, α₂ and β₁ according to the value of k, some sharp conditions are obtained for the existence, uniqueness and nonexistence results of k-convex solutions to the above problem.