Existence and uniqueness conditions of positive solutions to semilinear elliptic equations with double power nonlinearities

TL;DR

This paper establishes the conditions under which positive solutions to semilinear elliptic equations with double power nonlinearities exist and are unique, providing a simplified proof of the relationship between these conditions.

Contribution

It offers new equivalent conditions for existence and uniqueness of solutions and simplifies the proof linking these conditions.

Findings

Derived equivalent conditions for solution existence and uniqueness

Simplified proof of the relationship between existence and uniqueness conditions

Clarified the role of double power nonlinearities in elliptic equations

Abstract

In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that the uniqueness condition comes from the existence condition.