Yesterday, I read a recent paper by S. Yijing and Z. Duanzhi published in the journal *Calculus of Variations* in 2013. In one of their results, they proved the following

**Theorem 2**. Let , be bounded open set with smooth boundary. If , then

Since , it is clear that . By definition, which gives us some comparison between and its inverse power.

In order to prove the above theorem, the authors first proved the following

**Theorem 1**. Let , be bounded open set with smooth boundary, the function is positive a.e. in , and . Then the equation

admits a unique -solution if and only if there exists such that

The proof of Theorem 1 is variational which makes use of the Nehari manifold and the fibering map. Basically, the authors tried to minimize the functional

over the closed set (in -topology)

To examine the best minimizing sequence for , the authors used the Ekeland variational principle. Making use of the fibering map, the authors was successfully locate the critical point which gives nothing but a solution to the problem. In view of this setting, if the integral , the set is non-empty. Conversely, the solution of the problem gives the existence of such an . It turns out that the difficulty is when the limiting function, say , lies in the boundary of , say

It is worth noticing that by -solution we mean that

Since in general it is not true that

where with in , the validity of

is preferred to solutions.

In order to prove Theorem 2, the authors made use of a result of Lazer and McKenna which says that the problem

has a -solution if and only if .

We are now in a position to prove Theorem 2. Indeed, we argue by contradiction that there exists some satisfying

with . Then in view of Theorem 1, one can obtain some solving the following

while by the Lazer-McKenna result this problem has if and only if . The proof follows.

In my opinion, the result in Theorem 2 is interesting which provides a necessary and sufficient condition for the question: why becomes such an obstruction to solvability in the study of elliptic equations with negative exponents?

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