In this section, we will recall and prove some lemmas which are crucial in the proof of the main result.
Lemma 3.1. Let the assumptions of Theorem 2 be satisfied. If the sequence {(u
n
, v
n
)} ⊂ B is a (PS)
c
sequence for I
λ
, then we get that c ≥ 0 and {(u
n
, v
n
)} is bounded in the space B.
Proof. One has
By the assumptions (K0) and (H3), we have
Together with I
λ
(u
n
, v
n
) → c and as n → ∞, we easily obtain that the (PS)
c
sequence is bounded in B and the energy level c ≥ 0. □
From Lemma 3.1, there exists (u, v) ∈ B such that (u
n
, v
n
) ⇀ (u, v) in B. Furthermore, passing to a subsequence, we have u
n
→ u and v
n
→ v in for any d ∈ [p, p*) and u
n
→ u, v
n
→ v a.e. in ℝ N .
Lemma 3.2. Let d ∈ [p, p*). There exists a subsequence such that for any ε > 0, there is r
ε
> 0 with
for any r ≥ r
ε
, where B
r
:= {x ∈ ℝ N : |x| ≤ r}.
Proof. The proof of Lemma 3.2 is similar to the one of Lemma 3.2 of [10], so we omit it. □
Let η ∈ C∞(ℝ+) be a smooth function satisfying 0 ≤ η(t) ≤ 1, η(t) = 1 if t ≤ 1 and η(t) = 0 if t ≥ 2. Define , . It is obvious that
(3.1)
Lemma 3.3. One has
and
uniformly in (φ, ψ) ∈ B with ‖(φ, ψ‖
B
≤ 1.
Proof. From the assumptions (H1)-(H2) and Lemma 3.2, we have
(3.2)
By Hölder inequality and Lemma 3.2, it follows that
and
Similarly, we get
and
Thus
From the similar argument, we also get
□
Lemma 3.4. One has along a subsequence
and
Proof. From the Lemma 2.1 of [15] and the argument of [16], we have
By (3.1) and the similar idea of proving the Brézis-Lieb Lemma [17], it is easy to get
and
In connection with the fact I
λ
(u
n
, v
n
) → c and , we obtain
In the following, we will verify the fact .
For any (φ, ψ) ∈ B, it follows that
Standard argument shows that
and
uniformly in ‖φ, ψ)‖
B
≤ 1.
By Lemma 3.3, we have
and
uniformly in ‖(φ, ψ)‖
B
≤ 1. From the facts above mentioned, we obtain
□
Let , , then , . From (3.1), we get (u
n
, v
n
) → (u, v) in B if and only if in B.
Observe that
where .
Thus by Lemma 3.4, we get
(3.3)
Now, we consider the energy level of the functional I
λ
below which the (PS)
c
condition hold.
Let V
b
(x):= max{V (x), b}, where b is the positive constant in the assumption (V0). Since the set ν
b
has finite measure and , in , we get
(3.4)
From (K0), (H1)-(H3) and Young inequality, there is C
b
> 0 such that
(3.5)
Let S be the best Sobolev constant of the immersion
Lemma 3.5. Let the assumptions of Theorem 2 be satisfied. There exists α0> 0 independent of λ such that, for any (PS)
c
sequence {(u
n
, v
n
)} ⊂ B for I
λ
with (u
n
, v
n
) ⇀ (u, v), either (u
n
, v
n
) → (u, v) or .
Proof. Assume that (u
n
, v
n
) ↛ (u, v), then
and
By the Sobolev inequality, (3.4) and (3.5), we get
This, together with and (3.3), gives
Set , then
This proof is completed. □
Since is not compact, I
λ
does not satisfy the (PS)
c
condition for all c > 0. But Lemma 3.5 shows that I
λ
satisfies the following local (PS)
c
condition.
Lemma 3.6. From the assumptions of Theorem 2, there exists a constant α0> 0 independent of λ such that, if a (PS)
c
sequence {(u
n
, v
n
)} ⊂ B for I
λ
satisfies , the sequence {(u
n
, v
n
)} has a strongly convergent subsequence in B.
Proof. By the fact , we have
This, together with I
λ
(u, v) ≥ 0 and Lemma 3.5, gives the desired conclusion. □
Next, we consider λ = 1. From the following standard argument, we get that I
λ
possesses the mountain-pass structure.
Lemma 3.7. Under the assumptions of Theorem 2, there exist α
λ
, ρ
λ
> 0 such that
Proof. By (3.5), we get that for any δ > 0, there is C
δ
> 0 such that
Thus
Note that . If δ ≤ (2pλC1)-1, then
The fact p* > p implies the desired conclusion. □
Lemma 3.8. Under the assumptions of Lemma 3.7, for any finite dimensional subspace
F ⊂ B, we have
Proof. By the assumption (H3), it follows that
Since all norms in a finite-dimensional space are equivalent and α, β > p, we prove the result of this Lemma. □
By Lemma 3.6, for λ larger enough and c
λ
small sufficiently, I
λ
satisfies (PS)
cλ
condition.
Thus, we will find special finite-dimensional subspaces by which we establish sufficiently small minimax levels.
Define the functional
It is apparent that Φ
λ
∈ C1(B) and I
λ
(u, v) ≤ Φ
λ
(u, v) for all (u, v) ∈ B.
Observe that
and
For any δ> 0, there are φ
δ
, with and suppφδ, such that .
Let , then . For t ≥ 0, we get
where
We easily prove that
Together with V (0) = 0 and , this implies that there is Λδ> 0 such that for all λ ≥ Λδ, we have
(3.6)
It follows from (3.6) that
Lemma 3.9. Under the assumptions of Lemma 3.7, for any ⊂ > 0, there is Λσ> 0 such that λ ≥ Λσ, there exists with and
where ρ
λ
is defined in Lemma 3.7.
Proof. This proof is similar to the one of Lemma 4.3 in [10], it can be easily proved. □