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?