A trick on finding sup- and super-solutions for some elliptic equation involing critical exponents

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

and

.

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.