In this section, we will consider the asymptotically periodic case and prove Theorems 1.2 and 1.3. As in the proof of Theorem 1.1, we first take advantage of the minimizing sequence of Ψ to find a sequence of Φ. In what follows, the important thing is to recover the compactness for the sequence. For this purpose, we need to estimate the functional levels of the problem (NLS) and those of a related periodic problem of (NLS) (roughly speaking, the limit system of (NLS) by (V3))
Hence, first we introduce some definitions and look for solutions for the problem (NLS)
p
. The functional of (NLS)
p
is defined by
The Nehari manifold of (NLS)
p
is
and is the least energy of (NLS)
p
on . Note that
(5.1)
As for c, we have .
Lemma 5.1 Suppose that , satisfy (V1) and (V2). Let (F1)-(F6) hold. Then the problem (NLS)
p
has a positive ground state such that .
Proof As a corollary of Theorem 1.1, we infer that the problem (NLS)
p
has a positive ground state. Moreover, from the argument of Theorem 1.1, we find that the ground state of the problem (NLS)
p
we obtained is a minimizer of on . □
The existence of a positive ground state for the problem (NLS)
p
implies that (NLS) has a positive ground state when and . So, it remains to consider
Next, we prove that under some conditions.
Lemma 5.2 Suppose that , satisfy (V2). Let (V1), (V4), (5.2) and (F1)-(F6) hold. Then .
Proof Let be a positive ground state of (NLS)
p
such that . Assume satisfies . By (V4), we get
Then .
Replacing Φ and M by and respectively, (3.3) also holds. Noting that , we infer that
(5.3)
Therefore,
(5.4)
If , we are done. Otherwise, . Then by (5.3) and (5.4), we get and . Then is a ground state for (NLS). Note that is a solution of (NLS)
p
. From the first equations of (NLS) and (NLS)
p
, we infer that . Similarly, contrary to (5.2). The proof is now complete. □
Lemma 5.3 Suppose that , satisfy (V1) and (V2). Let (V1), (V5), (5.2) and (F1)-(F7) hold. Then .
Proof Let be a positive ground state of (NLS)
p
such that . By (V5) and (F7), we find that is also a minimizer of on . Let be such that . Using (V5), we have . Then
Without loss of generality, we assume that
Then . Below we argue analogously with the proof of Lemma 5.2 to infer that . This ends the proof. □
Now, we are ready to prove Theorems 1.2 and 1.3. The proof is partially inspired by [17], where the authors dealt with Schrödinger-Poisson equations.
Proof of Theorem 1.2 As the argument of Theorem 1.1, we infer that there exists a sequence such that and .
We claim that is bounded in H. Otherwise, suppose up to a subsequence. Set . As in the proof of Theorem 1.1, taking a subsequence, we suppose and exclude the case that . So, , then we can assume that
in . From the Lions compactness lemma, it follows that there exist and such that
Set and . We assume that in H, in and a.e. on up to a subsequence. Then by
we obtain . So, Lemma 3.2 implies that
Then by (2.1), we get
This is a contradiction.
Hence, is bounded in H. Up to a subsequence, we assume that in H, in and a.e. on . By Lemma 2.2, we have . Namely, is a solution of (NLS).
Below we prove that . We argue by contradiction. Suppose that . By (V3), for any , there exists such that
(5.5)
Note that , after passing to a subsequence, we assume in . So, for the above ϵ, there exists such that for , we have
Combining with (5.5), for , we get
Then . Similarly, . Therefore,
Hence,
(5.6)
Let be such that . We claim that for large n and .
First, we prove that
(5.7)
Otherwise, there exist and a subsequence of , still denoted by , such that for all . From (5.6) we have
Moreover, by , we get
Hence,
By and (F3), we obtain
(5.8)
Similar to the proof of Theorem 1.1, if in , then in H. Contrary to (3.2), since , therefore,
in . Suppose
in . Then from the Lions compactness lemma, it follows that there exist and such that
(5.9)
We denote and by and . Similarly, we assume that in H, in and a.e. on up to a subsequence. By (5.9), we have
So, . From (5.8), (F3) and the Fatou lemma, we obtain
which is impossible. Consequently, (5.7) holds.
Now, we show that for large n. Indeed, on the contrary, passing to a subsequence, we assume that . Using (3.1) and (5.1), we have
(5.10)
where (5.10) follows from the fact that α is increasing in by Lemma 2.1. Then , contrary to Lemma 5.2. Therefore, combining with (5.7), we may assume that
(5.11)
For and , using (2.2) we get
(5.12)
Combining (5.11) with (5.12), one easily has that
Since and is bounded, we get
Hence, . Then using (5.6), we have . Then . However, Lemma 5.2 implies that . This is a contradiction. Note that this contradiction follows from the hypothesis that . So, . Then .
It suffices to show that . By (3.1) we have
(5.13)
where the inequality (5.13) holds by (2.3) and the Fatou lemma. Then . According to , we have . Then is a ground state for (NLS). Below we argue analogously with the proof of Theorem 1.1 to get a positive ground state for (NLS). The proof is complete. □
Proof of Theorem 1.3 By Lemma 5.3, repeating the argument of Theorem 1.2, we show the existence of a ground state for (NLS) and then look for a positive ground state as the argument of Theorem 1.1. □