An elliptic partial differential equation and its application

Dragos-Patru Covei, Traian A. Pirvu

Published 2019-07-10Version 1

This paper deals with the following elliptic equation \begin{equation*} -2\left\vert \sigma \right\vert ^{2}\Delta z+\left\vert \nabla z\right\vert ^{2}+4\alpha z=4\left\vert x\right\vert ^{2}\text{ for }x\in \mathbb{R}^{N}% \text{, (}N\geq 1\text{),} \end{equation*}% where $\alpha >0$ is a real parameter and $\sigma $ is a vector from $% \mathbb{R}^{N}$. The solution method is based on the sub- and super-solutions method. The case $N>1$ seemed not considered before. This equation models a stochastic production planning problem.