ON THE MULTIPLICITY OF NON-CONSTANT POSITIVE SOLUTIONS TO CERTAIN SEMI-LINEAR ELLIPTIC EQUATIONS

Many phenomena occurring in the natural environ- ment have been modeled and studied using mathematical meth- ods. In particular, investigating the existence and multiplicity of positive solutions, which represent the coexistence of equations, is always an intriguing research topic. To study the multiplicity of these positive solutions, it is necessary to analyze the behavior of positive solutions concerning a given parameter in the equation. In this research, we present a semi-linear partial differential equation to explain a series of natural phenomena through the study of pos- itive solution behavior. We aim to investigate the existence and multiplicity of positive solutions that are not constant under homo- geneous Neumann boundary conditions. Specifically, we apply the Mountain Pass theorem to demonstrate the existence of positive solutions for this equation, and further, we use the Leray-Schauder degree theory to explore sufficient conditions for the existence of two or more positive solutions