Let us consider the following equation
in . We assume that both and . We are interested in finding positive solutions for the above equation.
By sup- and super-solutions to the above equation we mean functions and such that
The key point of sup- and super-solutions method is to tell us that having the existence of and we can prove the existence of one solution satisfying
We also note that this method says nothing about the uniqueness. We will try to construct such that .
Construction of sup-solution. We first consider equation (when )
This equation admits a constant sub-solution but no finite constant super-solution. However, it admits a non-constant super-solution, namely, the function
with a solution of the linear equation
Indeed, the maximum principle shows that , hence and
Thus, we can prove the existence of a super-solution .
Construction of super-solution. We next consider equation (when )
This equation admits the sub-solution and the super-solution . And thus, there exists a solution to the equation. In fact we can prove (see the proof of Brill-Canton theorem).
The proof completed. We now consider the given equation, clear and are sup- and super-solutions which proves the existence of solution to the equation.