We call a sequence a sequence if and strongly in ( is the dual space of E). is said to satisfy the condition if any sequence contains a convergent subsequence.
The main result of Section 3 is the following compactness result.
Proposition 3.1 Let the assumptions of Theorem 2 be satisfied. There exists a constant independent of λ such that, for any sequence for with , either or .
As a consequence, we obtain the following result.
Proposition 3.2 Assume that the assumptions of Proposition 3.1 hold, satisfies the condition for all .
In order to prove Proposition 3.1, we need the following lemmas.
Lemma 3.1 Let the assumptions of Theorem 2 be satisfied. is a sequence of . Then and is bounded in the space E.
, we have
Then is bounded and . □
From Lemma 3.1, we may assume in E and in for any and , a.e. in .
Proof The proof of Lemma 3.2 is similar to that of Lemma 3.2 of , so we omit it. □
be a smooth function satisfying
. Obviously, we have
uniformly in with .
The local compactness of Sobolev embedding implies that for any
, we have
. For any
, there exists
. Together with the assumption (
) and the Hölder inequality, it follows from Lemma 3.2 that
) are positive constants. Similarly, we can prove
Lemma 3.4 Let and be as defined above
. Then the following conclusions hold
By using the similar arguments of [34
], we have
By (3.1) and the similar idea of proving the Brézis-Lieb lemma [36
], it is easy to get
Furthermore, using the fact
, we obtain
In order to prove
, for any
, it follows that
It is standard to check that
. Together with Lemma 3.3, we have
Let , , then , . From (3.1), we get in E if and only if in E.
. Furthermore, we get
Now, we consider the energy level of the functional below which the condition holds.
, where b
is a positive constant in the assumption (
). Since the set
has finite measure, we get
In connection with the assumptions (
) and the Young inequality, there exists
be the best Sobolev constant of the immersion
Proof of Proposition 3.1
By the Sobolev embedding inequality and the diamagnetic inequality, we get
This, together with (3.2), gives
This completes the proof of Proposition 3.1. □
Proof of Proposition 3.2
, we have
In connection with and Proposition 3.1, we complete this proof. □