The Existence of Entire Positive Solutions to Monge-Amp`ere Type Equations

Abstract

Aims/ Objectives: In this paper, we study the Monge-Amp`ere type equation det D2u + α∆u = p(|x|)f(u)(x ∈ Rn). 
 In the previous articles, the equation with Monge-Amp`ere operator or Laplace operator has been studied extensively. However, the research about the combination of two kinds of operator is scarce. We would like do some research on this topic. We obtain a sufficient condition of the existence of entire positive solutions for the equation. 
 Study Design: Study on the existence of solutions. 
 Place and Duration of Study: Department of Mathematics and Physics, North China Electric Power University, September 2019 to present. 
 Methodology: We prove the existence of the solution by constructing Euler’s break line, combining the idea of transformation and the method of mathematical analysis. Results: We obtain a sufficient condition of the existence of entire positive solutions for the equation.
 Conclusion: We prove the existence of entire positive solutions to Monge-Amp`ere type equation det D2u + α∆u = p(|x|)f(u)(x ∈ Rn) and obtain the sufficient condition for the existence of solutions.