An elliptic partial differential equation and its application

This paper deals with the following elliptic equation − 2 σ 2 Δ z + ∇ z 2 + 4 α z = 4 x 2 for x ∈ R N , ( N ≥ 1 ) , where α > 0 , and σ > 0 are some real constants. The solution method is based on the sub- and super-solutions approach. The case N > 1 seemed not considered before. This equation models a stochastic production planning problem.