Consider the following quasilinear Schrödinger equation in
():
We have the same change of variables and the same notation as in the previous sections. Define the corresponding Orlicz space by
with the norm
The space is defined by
with the norm
The following Lemma 4.1 is a counterpart of Lemma 3.1.
Lemma 4.1 There exist two constants , such that
for any .
We denote by the closure of in . We define the functional on by
(4.1)
and we define the Nehari manifold by
Let
We recall that is a least energy solution of (D) if such that is achieved.
Lemma 4.2 Suppose . Then .
Proof It is easy to see that for . We claim that is monotone increasing with respect to λ. In fact, for , we assume that , are achieved for , . Obviously,
We first prove that there exists such that . This is sufficient to prove that
That is,
Let
Then by (), we can obtain and
Hence, there exists such that
, i.e., . Thus
In the following, we will prove that
In fact, we consider the function defined by
By for , we have . It follows that
Obviously,
and hence it is easy to check that
On the other hand,
by , it is easy to check that for any ,
which implies
for any , thus we have proved that is monotone increasing for .
Now we consider the function defined by
Then
for . Therefore, is monotone increasing with respect to . Thus, we deduce that
Assume that . If , then for any sequence (), we have .
We assume that is such that is achieved, by Lemma 3.4, is bounded in . Since , is bounded in , as a result, we have in
, in
for , in
for , a.e. in
.
We claim that , where
. Indeed, it is sufficient to prove . If not, then there exists a compact subset with such that and
Moreover, there exists such that for any .
By the choice of , we have
hence,
This contradiction shows that and so does v.
Now we show that
Suppose that (4.3) is not true, then by the concentration compactness principle of Lions (see [12]), there exist , and
with such that
On the other hand, by the choice of , we have
which shows that in
for . In the above proof, we have used the fact that as and the bounded property of .
Now, since is bounded, by the Fatou lemma, we obtain
But, by the choice of , we have
hence,
In the following, we will prove that
Indeed,
Since , one can easily see that as , and
by using in
for . It follows from (4.4) that
thus, there exists such that and
hence . A contradiction. Thus we have proved that as . □
Proof of Theorem 2.1 Suppose that is a sequence such that , , by the proof of Lemma 3.2, we have in
, in
for and . Moreover, , and if , then . Hence, in the following, we need only to prove that . To do this, it is sufficient to prove that
and
In fact, if one of the above three limits does not hold, by the Fatou lemma, we have
Similar to above, there exists such that and . A contradiction, and thus we complete the proof of Theorem 2.1. □