In this section, we give the proof of main theorems in this paper. We first compute the critical groups of Φ at both infinity and zero.
Lemma 5.1 (see [5])
Let F satisfy ()-(), then for any fixed ,
(5.1)
Proof The idea of the proof comes from the famous paper [5]. We include a sketched proof in an abstract version. Given , denote , . Modifying the arguments in [5], we get the following facts:
The following arguments are from [10]. As , it follows from (5.2) and (5.3) that for each , there is a unique such that
(5.4)
By (5.4) and the implicit function theorem, we have that . Define
Then . Define a map by
(5.5)
Clearly, ϱ is continuous, and for all with , by (5.4),
Therefore,
and so is a strong deformation retract of . Hence,
since is contractible, which follows from the fact that . □
Lemma 5.2 Let F satisfy ()-().
-
(1)
For , .
-
(2)
For , .
-
(3)
For , if for small, then .
-
(4)
For , if for small, then .
Proof By (), we have
-
(1)
When , is a nondegenerate critical point of Φ with the Morse index , thus .
-
(2)
When , is a nondegenerate critical point of Φ with the Morse index , thus .
-
(3)
When , is a degenerate critical point of Φ with the Morse index and the nullity , .
Assume that for with small. We will show that Φ has a local linking structure at with respect to . If this has been done, then by Proposition 2.2, we have .
Now, Φ can be written as
By () and (), for , there is such that
Hence, for , we have that
Since is finite dimensional, all norms on are equivalent, hence for small,
By (), we have that for some ,
(5.6)
For , we write where and . Then
(5.7)
For with , we have . Hence, by (5.6) and the Poincaré inequality, we have for various constants ,
For with , . Therefore,
(5.9)
Since , for small,
(5.10)
For , it must hold that
(5.11)
Here we use a potential convention that (GS)
λ
has finitely many solutions and then 0 is isolated. Otherwise, one would have that as small, implies for all , for all . Thus, 0 would not be an isolated critical point of Φ and (GS)
λ
would have infinitely many nontrivial solutions. By (5.10) and (5.11), we verify that
Applying Proposition 2.2, we obtain
(4) When for small, a similar argument shows that Φ has a local linking structure at with respect to . By Proposition 2.2, it follows that . □
Finally, we prove the theorems.
Proof of Theorem 1.1 It follows from () that for small. By Theorem 3.4 for the part , (GS)
λ
has a nontrivial solution satisfying
(5.12)
By Theorem 4.2(1), (GS)
λ
has two nontrivial solutions () with their Morse indices satisfying
From Proposition 2.1(2), we have that
(5.13)
From (5.12) and (5.13), we see that (). The proof is complete. □
Proof of Theorem 1.2 With the same argument as above, it follows from Theorem 4.2(2) and Theorem 3.4 for the part . We omit the details. □
Proof of Theorem 1.3 By Theorem 3.4 for the part , (GS)
λ
has a solution with its energy and
(5.14)
By Lemma 5.1 and Lemma 5.2(3), we have that
Assume that (GS)
λ
has only two solutions 0 and . Choose such that . Then by the deformation and excision properties of singular homology (see [12]), we have
(5.17)
By (5.17), the long exact sequences for the topological triple read as
(5.18)
We deduce by (5.15) and (5.18) that
(5.19)
Take in (5.19), then
which contradicts (5.14). The proof is complete. □
We finally remark that Theorem 1.1 is valid for , from which one sees that is a local minimizer of Φ.