1 Introduction and Main Result
In this paper, we are concerned with the multiplicity of positive solutions to the following Kirchhoff type problem:
(1.1)
-
(
1
+
b
∫
ℝ
3
|
∇
u
|
2
d
x
)
△
u
=
λ
f
(
x
)
u
+
|
u
|
4
u
,
x
∈
ℝ
3
,
u
∈
D
1
,
2
(
ℝ
3
)
,
where b is a positive constant, λ is a positive parameter, and the weight function f satisfies:
f∈L3/2(ℝ3)∖{0} and f(x)≥0 for all x∈ℝ3.
When problem (1.1) is set in a bounded domain and b=0, it reduces to the classic semilinear elliptic problem, first considered in the celebrated paper by Brézis and Nirenberg [4]. After that many authors are devoted to the investigation of a variety of elliptic equations with critical growth in a bounded domain or the whole space. Remarkably, (1.1) is a nonlocal problem, which means that the energy functional has totally different properties from the case b=0. This makes the study particularly interesting, see, e.g., [14, 2, 3, 17, 27, 22].
The solvability or multiplicity of the Kirchhoff type problem, with critical exponents in a bounded domain or the whole space, has been studied by means of the variational method, the genus theory, the fountain theorem, the Nehari manifold, the Ljusternik–Schnirelmann category theory, and so on, see [1, 11, 23, 15, 21, 10, 26, 24, 13, 12, 16, 20, 18]. In particular, for Ω⊂ℝ3 being a smooth bounded domain, consider the following Kirchhoff type problem:
-
(
a
+
b
∫
Ω
|
∇
u
|
2
d
x
)
△
u
=
λ
h
(
x
,
u
)
+
|
u
|
4
u
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω
.
For h(x,u) being a continuous superlinear and subcritical nonlinearity on u, Alves, Corrêa and Figueiredo [1, 11] have studied the existence and multiplicity of positive solutions, when the parameter λ is sufficiently large, using the variational method and an appropriated truncated argument. For h(x,u)=|u|q-1u and 0<q<1, Sun and Liu [23] have obtained a positive solution for small enough λ using the Nehari manifold. For h(x,u)=|u|q-1u and -1<q<0, Lei, Liao and Tang [15] have got two positive solutions for small enough λ via variational and perturbation methods. For Kirchhoff type problems in the whole space ℝ3, there are a few results about the existence and multiplicity of positive solutions. Wang et al. [24] have studied the following nonlinear Kirchhoff type problem:
-
(
ϵ
2
a
+
b
ϵ
∫
ℝ
3
|
∇
u
|
2
d
x
)
△
u
+
M
(
x
)
u
=
λ
h
(
u
)
+
|
u
|
4
u
,
u
∈
H
1
(
ℝ
3
)
,
where h is a continuous superlinear and subcritical nonlinearity. Assuming that M(x) has at least one minimum, they have proved that the problem has a positive ground state solution for λ sufficiently large and ϵ sufficiently small. In [24, 13, 12], the concentration behavior of the positive solution is considered. Liang and Zhang [16] and Liu, Liao and Tang [20] have investigated the existence and multiplicity of solutions for the perturbed Kirchhoff type problem.
Based on the above research, we think about the case where h is linear on u. If h(u)=u, as we all know, Brézis and Nirenberg [4] have proved that for b=0, problem (1.1) has no solution for all λ>λ1 in a smooth bounded domain Ω⊂ℝ3, where λ1 is the principal eigenvalue of (-△,H01(Ω)). However, Kirchhoff type elliptic problems involve a nonlocal perturbation, which means that the energy functional has totally different properties from the case b=0. Furthermore, we delicately analyze the behavior of (∫ℝ3|∇u|2dx)△u and find that, in the competing of the term (∫ℝ3|∇u|2dx)△u and the critical term, the former may dominate the situation, which implies that it is likely to obtain the existence of solutions to problem (1.1). Therefore, motivated by the works described above, particularly, by the results in [1, 16, 20, 21, 18], we try to extend the analysis to the Kirchhoff type problem with zero mass and a critical nonlinearity in ℝ3. Under assumption (f), by [8], the eigenvalue problem -△u=λf(x)u, u∈D1,2(ℝ3), has the first eigenvalue λ1>0. In this paper, we will consider the existence and multiplicity of positive solutions to problem (1.1) for λ>λ1, where
(1.2)
λ
1
:=
inf
u
∈
D
1
,
2
(
ℝ
3
)
∖
{
0
}
∥
u
∥
2
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
.
Noting that when problem (1.1) is set in a bounded domain, Naimen [21] has only obtained one solution for λ>λ1 and b>b0(λ)>0, by the concentration compactness principle, see [18, 19]. We will illustrate the multiplicity of positive solutions and the existence of positive ground state solutions when λ lies in a small right neighborhood of λ1, and problem (1.1) is set in the whole space ℝ3. To the best of our knowledge, we have not seen any results related to problem (1.1) in ℝ3. The main results can be described as follows.
Theorem 1.1.
Suppose that condition (f) holds. Then, for each λ1<λ<λ1+δ, problem (1.1) has at least one positive ground state solution, where δ=λ1b2S38, and S is the best Sobolev constant for the embedding of D1,2(R3) in L6(R3).
Theorem 1.2.
Suppose that f satisfies (f). Then there exists 0<δ*<δ such that, for each λ1<λ<λ1+δ*, problem (1.1) has at least two positive solutions.
Remark 1.3.
(1) We point out that, as far as we are concerned, this paper seems to be the first attempt to obtain the existence of positive ground state solutions and the multiplicity of positive solutions to problem (1.1) for λ in a small right neighborhood of λ1, even in a bounded domain. It is different from the result of Brezis and Nirenberg in 1983, and it can be regarded as the extension and complementary work of Naimen [21] in the whole space ℝ3.
(2) The main difficulties in obtaining the multiplicity of positive solutions for λ>λ1 are twofold. On the one hand, when λ lies in a small right neighborhood of λ1, we select a testing function u1,0=31/4(1+|x|2)1/2 and delicately analyze the behaviors of ∥u1,0∥2 and λ∫ℝ3f(x)|u1,0|2𝑑x, and find that their size relationship can not be determined, which means that the estimation of the least energy on a block of the Nehari manifold is difficult. We shall give the more accurate estimation by considering two cases: ∥u1,0∥2≥λ∫ℝ3f(x)|u1,0|2𝑑x and ∥u1,0∥2<λ∫ℝ3f(x)|u1,0|2𝑑x. On the other hand, as we deal with the critical problem, the Sobolev embedding D1.2(ℝ3)↪L6(ℝ3) is continuous and not compact. The energy functional does not satisfy the (PS)c condition at every energy level c. To overcome this difficulty, we try to recover the compactness in the spirit of the concentration compactness principle in [18, 19].
We organize this paper as follows. In Section 2, we present some notations and prove some useful preliminary lemmas. In Section 3, we obtain a positive ground state solution to problem (1.1), where we not only prove Theorem 1.1 but also prove several lemmas which pave the way for getting another positive solution. Section 4 is devoted to proving Theorem 1.2.
2 Notations and Preliminaries
Let D1,2(ℝ3) be the usual Sobolev space equipped with the norm ∥u∥=(∫ℝ3|∇u|2dx)1/2, let |⋅|s be the usual Lebesgue space Ls(ℝ3) norm and let C be various positive constants. Let S be the best Sobolev constant for the embedding of D1,2(ℝ3) in L6(ℝ3). In particular,
S
=
inf
u
∈
D
1
,
2
(
ℝ
3
)
∖
{
0
}
∥
u
∥
2
|
u
|
6
2
,
|
u
|
6
≤
S
-
1
/
2
∥
u
∥
.
It is well known that S is achieved by the function
u
ε
,
y
:=
(
3
ε
)
1
/
4
(
ε
+
|
x
-
y
|
2
)
1
/
2
for any
ε
>
0
,
y
∈
ℝ
3
.
We can also easily get that ∥uε,y∥2=|uε,y|66=S3/2.
Define an energy functional Jλ on the space D1,2(ℝ3) by
J
λ
(
u
)
=
1
2
∥
u
∥
2
+
b
4
∥
u
∥
4
-
λ
2
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
1
6
∫
ℝ
3
u
6
𝑑
x
.
Then Jλ is well defined on D1,2(ℝ3) and is C1. Furthermore, the weak solutions of problem (1.1) correspond to the critical points of the functional Jλ.
As Jλ(u) is not bounded from below on D1,2(ℝ3), we introduce the Nehari manifold
𝐍
λ
=
{
u
∈
D
1
,
2
(
ℝ
3
)
∖
{
0
}
:
〈
J
λ
′
(
u
)
,
u
〉
=
0
}
,
where 〈⋅,⋅〉 denotes the usual duality. Thus, u∈𝐍λ if and only if u≠0 and
∥
u
∥
2
+
b
∥
u
∥
4
-
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
∫
ℝ
3
u
6
𝑑
x
=
0
.
Then 𝐍λ contains every nonzero solution of (1.1). Moreover, we have the following results.
Lemma 2.1.
The energy functional Jλ is coercive and bounded below on Nλ for all λ>0.
Proof.
For u∈𝐍λ, we have
∥
u
∥
2
+
b
∥
u
∥
4
=
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
+
∫
ℝ
3
u
6
𝑑
x
.
Thanks to the Sobolev and Hölder inequalities, one can obtain
J
λ
(
u
)
=
1
2
∥
u
∥
2
+
b
4
∥
u
∥
4
-
λ
2
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
1
6
∫
ℝ
3
u
6
𝑑
x
=
1
3
∥
u
∥
2
+
b
12
∥
u
∥
4
-
λ
3
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
≥
1
3
∥
u
∥
2
+
b
12
∥
u
∥
4
-
λ
3
S
|
f
|
3
/
2
∥
u
∥
2
.
Therefore, Jλ is coercive and bounded below on 𝐍λ for all λ>0. ∎
For each u∈D1,2(ℝ3)∖{0}, we introduce Ku:t→Jλ(tu) for t>0, defined by
K
u
(
t
)
=
t
2
2
∥
u
∥
2
+
b
t
4
4
∥
u
∥
4
-
λ
t
2
2
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
t
6
6
∫
ℝ
3
u
6
𝑑
x
,
which is called a fibering map in [9, 6]. Then Ku′(1)=0 if and only if u∈𝐍λ. Furthermore, for each u∈𝐍λ, we observe
K
u
′′
(
1
)
=
∥
u
∥
2
+
3
b
∥
u
∥
4
-
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
5
∫
ℝ
3
u
6
𝑑
x
=
4
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
4
∥
u
∥
2
-
2
b
∥
u
∥
4
(2.1)
=
2
b
∥
u
∥
4
-
4
∫
ℝ
3
u
6
𝑑
x
.
In order to obtain the multiplicity of solutions, it is natural to split 𝐍λ into three parts:
𝐍
λ
+
=
{
u
∈
𝐍
λ
:
K
u
′′
(
1
)
>
0
}
,
𝐍
λ
0
=
{
u
∈
𝐍
λ
:
K
u
′′
(
1
)
=
0
}
,
𝐍
λ
-
=
{
u
∈
𝐍
λ
:
K
u
′′
(
1
)
<
0
}
.
For u∈D1,2(ℝ3)∖{0}, we define
(2.2)
t
max
(
u
)
=
[
b
∥
u
∥
4
2
∫
ℝ
3
u
6
𝑑
x
]
1
/
2
,
δ
0
=
λ
1
b
2
S
3
4
.
Similar to the proof of [5, Lemma 2.6], we can obtain the following lemmas.
Lemma 2.2.
Suppose that condition (f) holds. For each u∈D1,2(R3)∖{0}, we have the following cases:
If
∥
u
∥
2
≥
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
, then there exists a unique
t
-
=
t
-
(
u
)
>
t
max
such that
t
-
u
∈
𝐍
λ
-
and
J
λ
(
t
-
u
)
=
sup
t
≥
0
J
λ
(
t
u
)
.
If
λ
1
<
λ
<
λ
1
+
δ
0
and
∥
u
∥
2
<
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
, then there exist unique
t
+
=
t
+
(
u
)
and
t
-
=
t
-
(
u
)
, with
0
<
t
+
<
t
max
(
u
)
=
t
max
=
<
t
-
, such that
t
±
u
∈
𝐍
λ
±
and
J
λ
(
t
+
u
)
=
inf
0
≤
t
≤
t
max
J
λ
(
t
u
)
,
J
λ
(
t
-
u
)
=
sup
t
≥
t
max
J
λ
(
t
u
)
=
sup
t
≥
0
J
λ
(
t
u
)
.
Proof.
Let u∈D1,2(ℝ3)∖{0} and the function αλ(t):ℝ+→ℝ defined by
α
λ
(
t
)
=
∥
u
∥
2
+
b
t
2
∥
u
∥
4
-
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
t
4
∫
ℝ
3
u
6
𝑑
x
.
Then
α
λ
′
(
t
)
=
2
t
(
b
∥
u
∥
4
-
2
t
2
∫
ℝ
3
u
6
𝑑
x
)
.
Clearly, we see that tu∈𝐍λ if and only if αλ(t)=0. Furthermore, αλ(t) has a unique critical point at t=tmax=tmax(u). This implies that αλ is strictly increasing for t∈(0,tmax) with αλ(0)=∥u∥2-λ∫ℝ3f(x)u2𝑑x, and strictly decreasing for t∈(tmax,+∞) with limt→∞αλ(t)=-∞. Clearly, if tu∈𝐍λ, then tαλ′(t)=Ku′′(t). Hence, tu∈𝐍λ+ (or 𝐍λ-) if and only if αλ′(t)>0 (or <0).
If ∥u∥2≥λ∫ℝ3f(x)u2𝑑x, then αλ(0)≥0 and αλ(t)=0 has a unique solution t->tmax such that αλ′(t-)<0, and thus t-u∈𝐍λ-. Since αλ(t)>0 for all t∈(0,t-), and αλ(t)<0 for all t∈(t-,+∞), we have
J
λ
(
t
-
u
)
=
sup
t
≥
0
J
λ
(
t
u
)
.
If λ1<λ<λ1+δ0 and ∥u∥2<λ∫ℝ3f(x)u2𝑑x, then, from (1.2) and the Sobolev inequality, we conclude that
α
λ
(
t
max
)
=
∥
u
∥
2
+
b
∥
u
∥
4
2
∫
ℝ
3
u
6
𝑑
x
b
∥
u
∥
4
-
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
[
b
∥
u
∥
4
2
∫
ℝ
3
u
6
𝑑
x
]
2
∫
ℝ
3
u
6
𝑑
x
≥
(
1
-
λ
λ
1
)
∥
u
∥
2
+
b
2
∥
u
∥
8
4
∫
ℝ
3
u
6
𝑑
x
≥
(
1
-
λ
λ
1
+
b
2
S
3
4
)
∥
u
∥
2
>
0
.
Since αλ(0)=∥u∥2-λ∫ℝ3f(x)u2𝑑x<0, the equation αλ(t)=0 has exactly two solutions 0<t+<tmax<t- such that αλ′(t+)>0 and αλ′(t-)<0. Hence, there are exactly two multiples of u such that t+u∈𝐍λ+ and t-u∈𝐍λ-, by tαλ′(t)=Ku′′(t). Furthermore, αλ(t)<0 for all t∈(0,t+)∪(t-,+∞), and αλ(t)>0 for all t∈(t+,t-), and thus we obtain
J
λ
(
t
+
u
)
=
inf
0
≤
t
≤
t
max
J
λ
(
t
u
)
and
J
λ
(
t
-
u
)
=
sup
t
≥
t
max
J
λ
(
t
u
)
=
sup
t
≥
0
J
λ
(
t
u
)
.
∎
Lemma 2.3.
Assume that λ1<λ<λ1+δ0 and that uλ is a local minimizer for Jλ on Nλ. Under condition (f), we have that Jλ′(uλ)=0 in D*, where D* is the dual space of D1,2(R3), that is, uλ is a nonzero weak solution of problem (1.1).
Proof.
Firstly, we declare that if λ1<λ<λ1+δ0, then 𝐍λ0=∅. By way of contradiction, suppose that there exists a number λ0, with λ1<λ0<λ1+δ0, such that u0∈𝐍λ00. Thanks to (1.2), (2.1) and the Sobolev inequality, we get
4
∥
u
0
∥
2
+
2
b
∥
u
0
∥
4
=
4
λ
0
∫
ℝ
3
f
(
x
)
u
0
2
𝑑
x
≤
4
λ
0
λ
1
∥
u
0
∥
2
,
2
b
∥
u
0
∥
4
=
4
∫
ℝ
3
u
0
6
𝑑
x
≤
4
S
-
3
∥
u
0
∥
6
,
from which it follows that
b
S
3
2
≤
∥
u
0
∥
2
≤
2
λ
0
-
2
λ
1
b
λ
1
.
Thus,
λ
0
≥
λ
1
+
λ
1
b
2
S
3
4
=
λ
1
+
δ
0
,
which contradicts our assumption.
Furthermore, let φ(u)=〈Jλ′(u),u〉 for each u∈D1,2(ℝ3)∖{0}. Then from Lemma 2.2 we deduce that
𝐍
λ
=
{
u
∈
D
1
,
2
(
ℝ
3
)
∖
{
0
}
:
φ
(
u
)
=
0
}
≠
∅
.
As in [6], if 𝐍λ0=∅, then we have φ′(u)≠0 for all u∈𝐍λ. It follows that there exists a number θ∈ℝ such that
J
λ
′
(
u
λ
)
=
θ
φ
′
(
u
λ
)
.
Since uλ∈𝐍λ and uλ∉𝐍λ0, we get 〈Jλ′(uλ),uλ〉=0 and 〈φ′(uλ),uλ〉≠0, and thus θ=0, which implies that Jλ′(uλ)=0 in D*. ∎
3 Proof of Theorem 1.1
For c∈ℝ, a sequence {un} is a (PS)c-sequence for Jλ on D1,2(ℝ3) if Jλ(un)=c+o(1) and Jλ′(un)=o(1) in D* as n→∞. Furthermore, if every (PS)c-sequence for Jλ on D1,2(ℝ3) has a strongly convergent subsequence, then Jλ satisfies the (PS)c-condition.
For λ1<λ<λ1+δ0, we know that 𝐍λ=𝐍λ+∪𝐍λ-, by the proof of Lemma 2.3, and we define
α
λ
=
inf
u
∈
𝐍
λ
J
λ
(
u
)
,
α
λ
+
=
inf
u
∈
𝐍
λ
+
J
λ
(
u
)
,
α
λ
-
=
inf
u
∈
𝐍
λ
-
J
λ
(
u
)
.
Then we get the following results.
Lemma 3.1.
If λ1<λ<λ1+δ0, then one has αλ≤αλ+<0.
If λ1<λ<λ1+12δ0, then one has αλ->C>0, and C depends on b,S,λ,λ1.
Proof.
(i) Let u∈𝐍λ+. By (2.1) we find that
4
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
>
4
∥
u
∥
2
+
2
b
∥
u
∥
4
.
Then, since u∈𝐍λ, it follows that
J
λ
(
u
)
=
1
2
∥
u
∥
2
+
b
4
∥
u
∥
4
-
λ
2
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
-
1
6
∫
ℝ
3
u
6
𝑑
x
=
1
3
∥
u
∥
2
+
b
12
∥
u
∥
4
-
λ
3
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
<
1
3
∥
u
∥
2
+
b
12
∥
u
∥
4
-
(
1
3
∥
u
∥
2
+
b
6
∥
u
∥
4
)
<
0
.
Therefore, Jλ(u)<0 for all u∈𝐍λ+, and we can deduce, by definition, that αλ≤αλ+<0.
(ii) Let u∈𝐍λ-. By (2.1) we have
2
b
∥
u
∥
4
<
4
∫
ℝ
3
u
6
𝑑
x
≤
4
S
-
3
∥
u
∥
6
,
∥
u
∥
2
>
b
S
3
2
.
Combining (1.2) with the fact that u∈𝐍λ gives
J
λ
(
u
)
=
1
3
∥
u
∥
2
+
1
12
b
∥
u
∥
4
-
1
3
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
≥
1
3
∥
u
∥
2
[
(
1
-
λ
λ
1
)
+
1
4
b
∥
u
∥
2
]
>
b
S
3
6
λ
1
(
λ
1
-
λ
+
1
2
δ
0
)
,
from which it follows easily that αλ->0 for λ1<λ<λ1+12δ0. ∎
Lemma 3.2.
If λ1<λ<λ1+δ0, then there exists a (PS)αλ-sequence {un}⊂𝐍λ for Jλ.
If λ1<λ<λ1+12δ0, then there exists a (PS)αλ--sequence {un}⊂𝐍λ- for Jλ.
Proof.
(i) Let {un}⊂𝐍λ be a minimizing sequence for αλ. Applying Ekeland’s variational principle (see [25]), there exists a sequence {vn}⊂𝐍λ such that ∥vn-un∥<1n, Jλ(vn)=αλ+o(1), and (Jλ|𝐍λ)′(vn)=o(1) as n→∞; and thanks to Lemma 2.1, {vn} is bounded in D1,2(ℝ3). Let φ(u)=〈Jλ′(u),u〉 for each u∈D1,2(ℝ3)∖{0}. Then we find
𝐍
λ
=
{
u
∈
D
1
,
2
(
ℝ
3
)
∖
{
0
}
:
φ
(
u
)
=
0
}
.
Proposition 5.12 in [25] implies that for each vn∈𝐍λ,
∥
(
J
λ
|
𝐍
λ
)
′
(
v
n
)
∥
=
min
θ
∈
ℝ
∥
J
λ
′
(
v
n
)
-
θ
φ
′
(
v
n
)
∥
.
Then there exists a sequence {θn}⊂ℝ such that
0
≤
∥
J
λ
′
(
v
n
)
-
θ
n
φ
′
(
v
n
)
∥
≤
∥
(
J
λ
|
𝐍
λ
)
′
(
v
n
)
∥
+
1
n
for all n∈ℕ. Therefore, since (Jλ|𝐍λ)′(vn)→0 as n→∞, we have
lim
n
→
∞
∥
J
λ
′
(
v
n
)
-
θ
n
φ
′
(
v
n
)
∥
=
0
.
Since {vn} is bounded in D1,2(ℝ3) and {vn}⊂𝐍λ, it follows that
(3.1)
|
θ
n
〈
φ
′
(
v
n
)
,
v
n
〉
|
=
|
〈
J
λ
′
(
v
n
)
-
θ
n
φ
′
(
v
n
)
,
v
n
〉
|
≤
∥
J
λ
′
(
v
n
)
-
θ
n
φ
′
(
v
n
)
∥
⋅
∥
v
n
∥
→
0
as
n
→
∞
.
Now, we claim that lim infn→∞|〈φ′(vn),vn〉|≠0. Otherwise, there exists a subsequence (still denoted by {vn}) such that limn→∞〈φ′(vn),vn〉=0. We define ∥vn∥→d≥0. From (2.1) we have
(3.2)
4
λ
λ
1
d
2
≥
4
λ
lim
n
→
∞
∫
ℝ
3
f
(
x
)
v
n
2
𝑑
x
=
4
d
2
+
2
b
d
4
,
(3.3)
4
S
-
3
d
6
≥
4
lim
n
→
∞
∫
ℝ
3
v
n
6
𝑑
x
=
2
b
d
4
,
which means Jλ(vn)→0 if d=0, but this is impossible, since Jλ(vn)→αλ<0. It follows that d>0. Consequently, from (3.2) and (3.3), we obtain
b
S
3
2
≤
d
2
≤
2
λ
-
2
λ
1
b
λ
1
.
Thus,
λ
≥
λ
1
+
λ
1
b
2
S
3
4
=
λ
1
+
δ
0
,
which contradicts our assumption.
Finally, by (3.1), we obtain θn→0 as n→∞. Furthermore, from the Sobolev and Hölder inequalities, we have
∥
φ
′
(
v
n
)
∥
=
sup
∥
w
∥
=
1
w
∈
D
1
,
2
(
ℝ
3
)
|
〈
φ
′
(
v
n
)
,
w
〉
|
=
sup
∥
w
∥
=
1
w
∈
D
1
,
2
(
ℝ
3
)
|
2
(
1
+
2
b
∥
v
n
∥
2
)
∫
ℝ
3
∇
v
n
⋅
∇
w
d
x
-
2
λ
∫
ℝ
3
f
(
x
)
v
n
w
d
x
-
6
∫
ℝ
3
|
v
n
|
4
v
n
w
d
x
|
≤
2
(
1
+
2
b
∥
v
n
∥
2
)
∥
v
n
∥
+
2
λ
|
f
|
3
/
2
S
-
1
∥
v
n
∥
+
6
S
-
3
∥
v
n
∥
5
.
Since {vn} is bounded in D1,2(ℝ3), we have that ∥φ′(vn)∥ is bounded. This implies that Jλ′(vn)→0 in D* as n→∞, that is, {vn}⊂𝐍λ is a (PS)αλ-sequence for Jλ.
The proof of assertion (ii) is closely similar to that of (i), and thus is omitted. ∎
Proof of Theorem 1.1.
Let δ=12δ0 and λ1<λ<λ1+δ. It suffices to prove that the functional Jλ has a nonnegative minimizer u* in 𝐍λ by Lemma 2.3 and the strong maximum principle. Now, we will establish the existence of a minimizer for Jλ on 𝐍λ.
By Lemmas 3.1 and 3.2, assume that {un}⊂𝐍λ is a minimizing sequence such that
J
λ
(
u
n
)
=
α
λ
+
o
(
1
)
and
J
λ
′
(
u
n
)
=
o
(
1
)
in
D
*
.
We have already observed that Jλ is coercive on 𝐍λ, by Lemma 2.1. This implies that the sequence {un} is bounded in D1,2(ℝ3), and going if necessary to a subsequence, still denoted by {un}, we can assume that for n sufficiently large, ∥un∥→d and
u
n
⇀
u
*
weakly in
D
1
,
2
(
ℝ
3
)
,
u
n
(
x
)
→
u
*
(
x
)
a.e. in
ℝ
3
,
u
n
→
u
*
strongly in
L
loc
s
(
ℝ
3
)
for
1
≤
s
<
6
.
Since {un} is bounded in D1,2(ℝ3), we have that {un} is bounded in L6(ℝ3), then |un|4un is bounded in L6/5(ℝ3), which implies that |un|4un⇀|u*|4u* weakly in L6/5(ℝ3). Therefore, for any ψ∈D1,2(ℝ3), one obtains that
∫
ℝ
3
|
u
n
|
4
u
n
ψ
d
x
→
∫
ℝ
3
|
u
*
|
4
u
*
ψ
d
x
as
n
→
∞
.
As {un} is bounded in L6(ℝ3), by condition (f), for each ψ∈D1,2(ℝ3), we deduce
∫
ℝ
3
f
(
x
)
u
n
ψ
𝑑
x
→
∫
ℝ
3
f
(
x
)
u
*
ψ
𝑑
x
as
n
→
∞
.
Next, since Jλ′(un)→0, for n large enough, we have
o
(
1
)
∥
u
*
∥
=
(
1
+
b
∥
u
n
∥
2
)
∫
ℝ
3
∇
u
n
⋅
∇
u
*
d
x
-
λ
∫
ℝ
3
f
(
x
)
u
n
u
*
d
x
-
∫
ℝ
3
|
u
n
|
4
u
n
u
*
d
x
=
(
1
+
b
∥
u
n
∥
2
)
∥
u
*
∥
2
-
λ
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
-
∫
ℝ
3
u
*
6
𝑑
x
+
o
(
1
)
.
In particular,
(3.4)
∥
u
*
∥
2
+
b
d
2
∥
u
*
∥
2
-
λ
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
-
∫
ℝ
3
u
*
6
𝑑
x
=
0
.
Thanks to ∥u*∥≤d, we get
(3.5)
∥
u
*
∥
2
+
b
∥
u
*
∥
4
≤
λ
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
+
∫
ℝ
3
u
*
6
𝑑
x
.
Next, we will prove that u* is the required minimizer. From the fact that Φ(u)=∫ℝ3f(x)u2𝑑x is weakly continuous on D1,2(ℝ3), together with {un}⊂𝐍λ, we conclude
α
λ
=
lim
n
→
∞
J
λ
(
u
n
)
=
lim
n
→
∞
(
1
3
∥
u
n
∥
2
+
b
12
∥
u
n
∥
4
-
λ
3
∫
ℝ
3
f
(
x
)
u
n
2
𝑑
x
)
(3.6)
=
1
3
d
2
+
b
12
d
4
-
λ
3
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
.
Since αλ<0, it follows that
∥
u
*
∥
2
≤
d
2
<
λ
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
,
which implies u*(x)≢0 in ℝ3. Hence, by Lemma 2.2 (ii), there exist unique t+ and t-, with 0<t+<tmax<t-, such that t±u*∈𝐍λ±,
J
λ
(
t
+
u
*
)
=
inf
0
≤
t
≤
t
max
J
λ
(
t
u
*
)
and
J
λ
(
t
-
u
*
)
=
sup
t
≥
t
max
J
λ
(
t
u
*
)
.
By (3.5) we have Ku*′(1)≤0, and thus either 1≤t+<tmax<t- or 0<t+<tmax<t-≤1. In fact, if 0<t+<tmax<t-≤1, then by (2.2) we obtain
b
∥
u
*
∥
4
<
2
∫
ℝ
3
u
*
6
𝑑
x
≤
2
S
-
3
∥
u
*
∥
6
.
Furthermore, according to (3.6), together with (1.2), we get
1
3
∥
u
*
∥
2
+
b
12
∥
u
*
∥
4
≤
λ
3
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
≤
λ
3
λ
1
∥
u
*
∥
2
.
Repeating the same argument as in the proof of Lemma 2.3, we obtain a contradiction with our assumption, λ1<λ<λ1+δ, and thus 1≤t+<tmax<t-. From (3.4), (3.6) and ∥u*∥≤d, it follows that
J
λ
(
t
+
u
*
)
≤
J
λ
(
u
*
)
≤
1
2
∥
u
*
∥
2
+
b
4
d
2
∥
u
*
∥
2
-
λ
2
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
-
1
6
∫
ℝ
3
u
*
6
𝑑
x
=
(
1
3
∥
u
*
∥
2
+
b
12
d
2
∥
u
*
∥
2
-
λ
3
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
)
+
1
6
(
∥
u
*
∥
2
+
b
d
2
∥
u
*
∥
2
-
λ
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
-
∫
ℝ
3
u
*
6
𝑑
x
)
≤
1
3
d
2
+
b
12
d
4
-
λ
3
∫
ℝ
3
f
(
x
)
u
*
2
𝑑
x
=
α
λ
,
which implies that t+=1, u*∈𝐍λ+ and Jλ(u*)=αλ=αλ+. Note that since Jλ(u*)=Jλ(|u*|) and |u*|∈𝐍λ+⊂𝐍λ, we can assume that u* is nonnegative, and the proof is complete. ∎
4 Proof of Theorem 1.2
In this section, we will establish the existence of the second positive solution of (1.1) by finding a local minimizer for Jλ on 𝐍λ-.
Lemma 4.1.
We have Jλ(u)≥-16(λ-λ1)3/2|f|3/23/2 for all u∈Nλ and all λ>λ1.
Proof.
Let u∈𝐍λ. Then from (1.2) we obtain
J
λ
(
u
)
=
1
4
(
∥
u
∥
2
-
λ
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
)
+
1
12
∫
ℝ
3
u
6
𝑑
x
≥
1
12
∫
ℝ
3
u
6
𝑑
x
-
1
4
(
λ
-
λ
1
)
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
.
By the Hölder and Young inequalities, and λ>λ1, one has
(4.1)
λ
-
λ
1
4
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
≤
λ
-
λ
1
4
|
f
|
3
/
2
|
u
|
6
2
≤
1
12
∫
ℝ
3
u
6
𝑑
x
+
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
Thus, we get Jλ(u)≥-16(λ-λ1)3/2|f|3/23/2. ∎
Lemma 4.2.
Let δ be as in Theorem 1.1, let λ1<λ<λ1+δ and let (f) hold. If {un} is a minimizing sequence for Jλ on Nλ- and
α
λ
-
<
b
S
3
4
+
b
3
S
6
24
+
S
b
2
S
4
+
4
S
6
+
b
2
S
4
b
2
S
4
+
4
S
24
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
,
then there exists u∈D1,2(R3) such that, up to a subsequence, un→u strongly in L6(R3).
Proof.
Lemma 3.1 implies that αλ->0. By Lemma 3.2, we can assume that {un}⊂𝐍λ- satisfies
J
λ
(
u
n
)
=
α
λ
-
+
o
(
1
)
and
J
λ
′
(
u
n
)
=
o
(
1
)
in
D
*
.
It is easy to see that {un} is bounded in D1,2(ℝ3) by Lemma 2.1. Going if necessary to a subsequence, still denoted by {un}, we can assume that for n sufficiently large,
u
n
⇀
u
weakly in
D
1
,
2
(
ℝ
3
)
,
u
n
(
x
)
→
u
(
x
)
a.e. in
ℝ
3
,
u
n
→
u
strongly in
L
loc
s
(
ℝ
3
)
for
1
≤
s
<
2
*
.
By the concentration compactness principle in [18, 19], there exists a subsequence, still denoted by {un}, such that
(4.2)
|
∇
u
n
|
2
⇀
d
μ
≥
|
∇
u
|
2
+
∑
i
∈
Γ
μ
i
σ
a
i
,
|
u
n
|
6
⇀
|
u
|
6
+
∑
i
∈
Γ
ν
i
σ
a
i
,
where Γ is an at most countable index set, {ai}i∈Γ⊂ℝ3 is a sequence, {μi}i∈Γ, {νi}i∈Γ⊂ℝ+ are values, and σai is the Dirac mass at ai. Moreover, we have
(4.3)
S
ν
i
1
/
3
≤
μ
i
.
We claim that Γ=∅. Suppose, on the contrary, that Γ≠∅. Fix i∈Γ. For ϵ>0, assume that φϵi∈C0∞(ℝ3) is such that φϵi(x)∈[0,1],
φ
ϵ
i
(
x
)
=
1
for
|
x
-
a
i
|
≤
ϵ
2
,
φ
ϵ
i
(
x
)
=
0
for
|
x
-
a
i
|
≥
ϵ
,
|
∇
φ
ϵ
i
|
≤
3
ϵ
.
Since {φϵiun} is bounded, we have 〈Jλ′(un),φϵiun〉→0, which gives
(
1
+
b
∥
u
n
∥
2
)
(
∫
ℝ
3
u
n
∇
u
n
⋅
∇
φ
ϵ
i
d
x
+
∫
ℝ
3
|
∇
u
n
|
2
φ
ϵ
i
d
x
)
=
λ
∫
ℝ
3
f
(
x
)
u
n
2
φ
ϵ
i
d
x
+
∫
ℝ
3
u
n
6
φ
ϵ
i
d
x
+
o
(
1
)
.
Recalling that {un} is bounded in D1,2(ℝ3), we can get
lim
ϵ
→
0
lim sup
n
→
∞
(
1
+
b
∥
u
n
∥
2
)
∫
ℝ
3
u
n
∇
u
n
⋅
∇
φ
ϵ
i
d
x
=
0
.
This is because, from the Hölder inequality and |∇φϵi|≤3ϵ, we have
lim
ϵ
→
0
lim sup
n
→
∞
(
1
+
b
∥
u
n
∥
2
)
|
∫
ℝ
3
u
n
∇
u
n
⋅
∇
φ
ϵ
i
d
x
|
≤
lim
ϵ
→
0
lim sup
n
→
∞
C
(
∫
B
ϵ
(
a
i
)
|
∇
u
n
|
2
d
x
)
1
/
2
(
∫
B
ϵ
(
a
i
)
|
∇
φ
ϵ
i
|
2
|
|
u
n
|
2
d
x
)
1
/
2
≤
lim
ϵ
→
0
C
(
∫
B
ϵ
(
a
i
)
|
∇
φ
ϵ
i
|
2
|
|
u
|
2
d
x
)
1
/
2
≤
lim
ϵ
→
0
C
(
∫
B
ϵ
(
a
i
)
|
∇
φ
ϵ
i
|
3
d
x
)
1
/
3
(
∫
B
ϵ
(
a
i
)
|
u
|
6
d
x
)
1
/
6
≤
lim
ϵ
→
0
C
(
∫
B
ϵ
(
a
i
)
|
u
|
6
d
x
)
1
/
6
=
0
,
where Bϵ(ai)={x∈ℝ3:|x-ai|<ϵ}. Moreover, from (4.2), we can also get that
lim
ϵ
→
0
lim sup
n
→
∞
(
1
+
b
∥
u
n
∥
2
)
∫
ℝ
3
|
∇
u
n
|
2
φ
ϵ
i
d
x
≥
lim
ϵ
→
0
lim sup
n
→
∞
[
∫
ℝ
3
|
∇
u
n
|
2
φ
ϵ
i
d
x
+
b
(
∫
ℝ
3
|
∇
u
n
|
2
φ
ϵ
i
d
x
)
2
]
≥
μ
i
+
b
μ
i
2
,
lim
ϵ
→
0
lim sup
n
→
∞
∫
ℝ
3
u
n
6
φ
ϵ
i
𝑑
x
=
lim
ϵ
→
0
∫
ℝ
3
u
6
φ
ϵ
i
𝑑
x
+
ν
i
=
ν
i
,
lim
ϵ
→
0
lim sup
n
→
∞
∫
ℝ
3
f
(
x
)
u
n
2
φ
ϵ
i
𝑑
x
=
lim
ϵ
→
0
∫
ℝ
3
f
(
x
)
u
2
φ
ϵ
i
𝑑
x
=
0
.
Consequently, νi≥μi+bμi2. Combining this with (4.3), we have
(4.4)
ν
i
≥
(
b
S
2
+
b
2
S
4
+
4
S
2
)
3
.
Now, we shall prove that this is impossible, and therefore Γ=∅. If (4.4) holds, for R>0, assume that ψR∈C0∞(ℝ3) is such that ψR(x)∈[0,1],
ψ
R
(
x
)
=
1
for
|
x
|
<
R
,
ψ
R
(
x
)
=
0
for
|
x
|
>
2
R
,
|
∇
ψ
R
|
≤
2
R
.
Consequently, from (1.2), (4.1), (4.2) and since {un}⊂𝐍λ-, we deduce
lim
n
→
∞
J
λ
(
u
n
)
=
lim
n
→
∞
(
1
4
∫
ℝ
3
|
∇
u
n
|
2
d
x
+
1
12
∫
ℝ
3
u
n
6
d
x
-
λ
4
∫
ℝ
3
f
(
x
)
u
n
2
d
x
)
≥
lim
R
→
∞
lim
n
→
∞
(
1
4
∫
ℝ
3
|
∇
u
n
|
2
ψ
R
d
x
+
1
12
∫
ℝ
3
u
n
6
ψ
R
d
x
-
λ
4
∫
ℝ
3
f
(
x
)
u
n
2
d
x
)
≥
lim
R
→
∞
(
1
4
∫
ℝ
3
|
∇
u
|
2
ψ
R
d
x
+
1
12
∫
ℝ
3
u
6
ψ
R
d
x
-
1
4
λ
∫
ℝ
3
f
(
x
)
u
2
d
x
)
+
1
4
μ
i
+
1
12
ν
i
≥
1
4
∫
ℝ
3
|
∇
u
|
2
d
x
-
λ
4
∫
ℝ
3
f
(
x
)
u
2
d
x
+
1
12
∫
ℝ
3
u
6
d
x
+
1
4
μ
i
+
1
12
ν
i
≥
1
12
∫
ℝ
3
u
6
𝑑
x
-
1
4
(
λ
-
λ
1
)
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
+
1
4
μ
i
+
1
12
ν
i
≥
1
4
μ
i
+
1
12
ν
i
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
,
which, in view of (4.3) and (4.4), yields
α
λ
-
≥
b
S
3
4
+
b
3
S
6
24
+
S
b
2
S
4
+
4
S
6
+
b
2
S
4
b
2
S
4
+
4
S
24
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
Therefore, it leads to a contradiction, and thus Γ=∅.
Now, for R>0, we can define
(4.5)
μ
∞
=
lim
R
→
∞
lim sup
n
→
∞
∫
|
x
|
>
R
|
∇
u
n
|
2
d
x
,
ν
∞
=
lim
R
→
∞
lim sup
n
→
∞
∫
|
x
|
>
R
u
n
6
d
x
.
By Proposition 2 in [7], we know that μ∞ and ν∞ satisfy
(4.6)
lim sup
n
→
∞
∫
ℝ
3
|
∇
u
n
|
2
d
x
=
∫
ℝ
3
d
μ
+
μ
∞
,
lim sup
n
→
∞
∫
ℝ
3
u
n
6
d
x
=
∫
ℝ
3
d
ν
+
ν
∞
and Sν∞1/3≤μ∞. Assume that ϕR∈C∞(ℝ3) is such that ϕR(x)∈[0,1],
ϕ
R
(
x
)
=
0
for
|
x
|
<
R
2
,
ϕ
R
(
x
)
=
1
for
|
x
|
>
R
,
|
∇
ϕ
R
|
<
3
R
.
Since {ϕRun} is bounded, we have 〈Jλ′(un),ϕRun〉→0, and so
(
1
+
b
∥
u
n
∥
2
)
(
∫
ℝ
3
u
n
∇
u
n
⋅
∇
ϕ
R
d
x
+
∫
ℝ
3
|
∇
u
n
|
2
ϕ
R
d
x
)
=
λ
∫
ℝ
3
f
(
x
)
u
n
2
ϕ
R
d
x
+
∫
ℝ
3
u
n
6
ϕ
R
d
x
+
o
(
1
)
.
Due to the boundness of {un} and the Hölder inequality, one has
lim
R
→
∞
lim sup
n
→
∞
(
1
+
b
∥
u
n
∥
2
)
|
∫
ℝ
3
u
n
∇
u
n
⋅
∇
ϕ
R
d
x
|
≤
lim
R
→
∞
lim sup
n
→
∞
C
(
∫
R
2
≤
|
x
|
≤
R
|
∇
u
n
|
2
d
x
)
1
/
2
(
∫
R
2
≤
|
x
|
≤
R
|
∇
ϕ
R
|
2
|
|
u
n
|
2
d
x
)
1
/
2
≤
lim
R
→
∞
C
(
∫
R
2
≤
|
x
|
≤
R
|
∇
ϕ
R
|
2
|
|
u
|
2
d
x
)
1
/
2
≤
lim
R
→
∞
C
(
∫
R
2
≤
|
x
|
≤
R
|
∇
ϕ
R
|
3
d
x
)
1
/
3
(
∫
R
2
≤
|
x
|
≤
R
|
u
|
6
d
x
)
1
/
6
≤
lim
R
→
∞
C
(
∫
|
x
|
≥
R
2
|
u
|
6
d
x
)
1
/
6
=
0
.
Moreover, from (4.5) we can get that
lim
R
→
∞
lim sup
n
→
∞
(
1
+
b
∥
u
n
∥
2
)
∫
ℝ
3
|
∇
u
n
|
2
ϕ
R
d
x
≥
lim
R
→
∞
lim sup
n
→
∞
[
∫
ℝ
3
|
∇
u
n
|
2
ϕ
R
d
x
+
b
(
∫
ℝ
3
|
∇
u
n
|
2
ϕ
R
d
x
)
2
]
≥
lim
R
→
∞
lim sup
n
→
∞
[
∫
|
x
|
≥
R
|
∇
u
n
|
2
d
x
+
b
(
∫
|
x
|
≥
R
|
∇
u
n
|
2
d
x
)
2
]
=
μ
∞
+
b
μ
∞
2
and
lim
R
→
∞
lim sup
n
→
∞
∫
ℝ
3
u
n
6
ϕ
R
𝑑
x
=
lim
R
→
∞
lim sup
n
→
∞
∫
|
x
|
≥
R
2
u
n
6
ϕ
R
𝑑
x
≤
lim
R
→
∞
lim sup
n
→
∞
∫
|
x
|
≥
R
2
u
n
6
𝑑
x
=
ν
∞
,
and from the fact Φ(u)=∫ℝ3f(x)u2𝑑x is weakly continuous on D1,2(ℝ3), we can also obtain that
lim
R
→
∞
lim sup
n
→
∞
∫
ℝ
3
f
(
x
)
u
n
2
ϕ
R
𝑑
x
=
lim
R
→
∞
∫
ℝ
3
f
(
x
)
u
2
ϕ
R
𝑑
x
=
0
.
It follows that ν∞≥μ∞+bμ∞2. We may therefore assume that
ν
∞
=
0
or
ν
∞
≥
(
b
S
2
+
b
2
S
4
+
4
S
2
)
3
.
If the later expression occurs, then from (1.2), (4.1), (4.6) and since {un}⊂𝐍λ-, we deduce
lim
n
→
∞
J
λ
(
u
n
)
=
lim
n
→
∞
(
1
4
∫
ℝ
3
|
∇
u
n
|
2
d
x
+
1
12
∫
ℝ
3
u
n
6
d
x
-
λ
4
∫
ℝ
3
f
(
x
)
u
n
2
d
x
)
≥
1
4
∫
ℝ
3
𝑑
μ
-
λ
4
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
+
1
12
∫
ℝ
3
𝑑
ν
+
1
4
μ
∞
+
1
12
ν
∞
≥
1
12
∫
ℝ
3
u
6
𝑑
x
-
1
4
(
λ
-
λ
1
)
∫
ℝ
3
f
(
x
)
u
2
𝑑
x
+
1
4
μ
∞
+
1
12
ν
∞
≥
1
4
μ
∞
+
1
12
ν
∞
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
Consequently,
α
λ
-
≥
b
S
3
4
+
b
3
S
6
24
+
S
b
2
S
4
+
4
S
6
+
b
2
S
4
b
2
S
4
+
4
S
24
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
,
which leads to a contradiction, and thus ν∞=0.
Lastly, combining (4.2), (4.6) with Γ=∅ gives
lim sup
n
→
∞
∫
ℝ
3
u
n
6
𝑑
x
=
∫
ℝ
3
u
6
𝑑
x
.
Furthermore, by the Fatou lemma, we get
∫
ℝ
3
u
6
𝑑
x
≤
lim inf
n
→
∞
∫
ℝ
3
u
n
6
𝑑
x
≤
lim sup
n
→
∞
∫
ℝ
3
u
n
6
𝑑
x
=
∫
ℝ
3
u
6
𝑑
x
.
Consequently,
lim
n
→
∞
∫
ℝ
3
u
n
6
𝑑
x
=
∫
ℝ
3
u
6
𝑑
x
,
and we have proved that un→u strongly in L6(ℝ3). ∎
Choosing
u
1
,
0
=
3
1
/
4
(
1
+
|
x
|
2
)
1
/
2
,
one has that S is achieved by u1,0, and ∥u1,0∥2=|u1,0|66=S3/2. Let δ be as in Theorem 1.1. By Lemma 2.2, there exists a unique t->0 satisfying t-u1,0∈𝐍λ- for λ1<λ<λ1+δ. In order to give a more accurate energy estimation for the functional Jλ on 𝐍λ-, we need to consider the following two cases:
∥u1,0∥2<λ0∫ℝ3f(x)|u1,0|2𝑑x for some λ1<λ0<λ1+δ,
∥u1,0∥2≥λ∫ℝ3f(x)|u1,0|2𝑑x for all λ1<λ<λ1+δ.
We can also divide case (1) into two kinds of situations. One is that there exists δ~∈(0,δ) such that
∥
u
1
,
0
∥
2
=
(
λ
1
+
δ
~
)
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
<
λ
0
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
.
In this case, we can choose a new δ=δ~, and have ∥u1,0∥2≥λ∫ℝ3f(x)|u1,0|2𝑑x for all λ1<λ<λ1+δ, which means that it can be classified as case (2). The other is that ∥u1,0∥2<λ∫ℝ3f(x)|u1,0|2𝑑x for all λ>λ1+, which means ∥u1,0∥2=λ1∫ℝ3f(x)|u1,0|2𝑑x. Next, we will give the energy estimation of αλ- for the preceding two cases.
Lemma 4.3.
Suppose that (f) holds.
If
∥
u
1
,
0
∥
2
=
λ
1
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
, then we have
(4.7)
α
λ
-
<
b
3
S
6
12
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
for all
λ
1
<
λ
<
λ
1
+
δ
1
*
,
where
δ
1
*
=
min
{
λ
1
b
2
S
3
8
,
9
b
2
S
6
4
λ
1
2
|
f
|
3
/
2
3
}
.
If
∥
u
1
,
0
∥
2
≥
λ
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
for each
λ
1
<
λ
<
λ
1
+
δ
, then there exists a positive number
δ
2
*
<
δ
such that
α
λ
-
<
b
S
3
4
+
b
3
S
6
24
+
S
b
2
S
4
+
4
S
6
+
b
2
S
4
b
2
S
4
+
4
S
24
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
for all
λ
1
<
λ
<
λ
1
+
δ
2
*
.
Proof.
(i) Clearly, ∥u1,0∥2<λ∫ℝ3f(x)|u1,0|2𝑑x for all λ>λ1. By Lemma 2.2 (ii) and (2.2), we know that there exists a unique t->tmax=(12bS3/2)1/2 such that t-u1,0∈𝐍λ-, where
(
t
-
)
2
=
b
S
3
/
2
+
b
2
S
3
-
4
(
λ
λ
1
-
1
)
2
<
b
S
3
/
2
.
To prove (4.7), it is enough to prove that
J
λ
(
t
-
u
1
,
0
)
<
b
3
S
6
12
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
Consider the function
g
(
t
)
=
b
S
3
4
t
4
-
S
3
/
2
6
t
6
.
There exists t1=(bS3/2)1/2>t->tmax such that g(t1)=maxt≥0g(t)=b3S612. Therefore,
J
λ
(
t
-
u
1
,
0
)
=
b
(
t
-
)
4
4
∥
u
1
,
0
∥
4
-
(
t
-
)
6
6
∫
ℝ
3
|
u
1
,
0
|
6
d
x
+
(
t
-
)
2
2
(
1
-
λ
λ
1
)
∥
u
1
,
0
∥
2
=
b
S
3
4
(
t
-
)
4
-
S
3
/
2
6
(
t
-
)
6
-
S
3
/
2
2
(
λ
λ
1
-
1
)
(
t
-
)
2
<
max
t
≥
0
g
(
t
)
-
S
3
/
2
2
(
λ
λ
1
-
1
)
(
t
max
)
2
=
b
3
S
6
12
-
b
S
3
(
λ
-
λ
1
)
4
λ
1
.
Furthermore, according to the fact that
0
<
λ
-
λ
1
<
δ
1
*
≤
9
b
2
S
6
4
λ
1
2
|
f
|
3
/
2
3
,
it is easy to see that
b
S
3
(
λ
-
λ
1
)
4
λ
1
>
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
For any λ1<λ<λ1+δ1*, it follows that
α
λ
-
≤
J
λ
(
t
-
u
1
,
0
)
<
b
3
S
6
12
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
(ii) If ∥u1,0∥2≥λ∫ℝ3f(x)|u1,0|2𝑑x for each λ1<λ<λ1+δ, then, by Lemma 2.2 (i), there exists a unique t->0 such that t-u1,0∈𝐍λ-, where
(
t
-
)
2
=
b
S
3
+
b
2
S
6
+
4
S
3
/
2
(
∥
u
1
,
0
∥
2
-
λ
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
)
2
S
3
/
2
>
b
S
3
/
2
.
Similarly, it suffices to prove that
J
λ
(
t
-
u
1
,
0
)
<
b
S
3
4
+
b
3
S
6
24
+
S
b
2
S
4
+
4
S
6
+
b
2
S
4
b
2
S
4
+
4
S
24
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
We consider the function
h
(
t
)
=
S
3
/
2
2
t
2
+
b
S
3
4
t
4
-
S
3
/
2
6
t
6
.
There exists t2>0 such that
h
(
t
2
)
=
max
t
≥
0
h
(
t
)
=
b
S
3
4
+
b
3
S
6
24
+
S
b
2
S
4
+
4
S
6
+
b
2
S
4
b
2
S
4
+
4
S
24
,
where
t
2
2
=
b
S
3
/
2
+
b
2
S
3
+
4
2
>
(
t
-
)
2
>
b
S
3
/
2
.
Now, we can select δ2*∈(0,δ) such that for all λ1<λ<λ1+δ2*,
1
2
(
t
-
)
2
λ
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
>
1
2
b
S
3
/
2
λ
1
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
>
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
Hence, for any λ1<λ<λ1+δ2*, we obtain
J
λ
(
t
-
u
1
,
0
)
=
(
t
-
)
2
2
∥
u
1
,
0
∥
2
+
b
(
t
-
)
4
4
∥
u
1
,
0
∥
4
-
(
t
-
)
6
6
∫
ℝ
3
|
u
1
,
0
|
6
d
x
-
(
t
-
)
2
2
λ
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
d
x
=
S
3
/
2
2
(
t
-
)
2
+
b
S
3
4
(
t
-
)
4
-
S
3
/
2
6
(
t
-
)
6
-
1
2
(
t
-
)
2
λ
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
<
max
t
≥
0
h
(
t
)
-
1
2
b
S
3
/
2
λ
1
∫
ℝ
3
f
(
x
)
|
u
1
,
0
|
2
𝑑
x
<
b
S
3
4
+
b
3
S
6
24
+
S
b
2
S
4
+
4
S
6
+
b
2
S
4
b
2
S
4
+
4
S
24
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
.
∎
Proof of Theorem 1.2.
Let δ*=min{δ1*,δ2*}. Then we have 0<δ*<δ. Since 𝐍λ- is an open subset of 𝐍λ, Lemma 2.3 is also conform for Jλ on 𝐍λ-. Then we only need to show that the functional Jλ has a local nonnegative minimizer u** in 𝐍λ-, by the strong maximum principle, for each λ1<λ<λ1+δ*. Obviously, u**∈𝐍λ- is different from u*∈𝐍λ+. Now, we will prove the existence of a local minimizer for Jλ on 𝐍λ-.
Let {un} be a minimizing sequence for Jλ on 𝐍λ- and λ1<λ<λ1+δ*. It is easy to see that
0
<
α
λ
-
<
b
S
3
4
+
b
3
S
6
24
+
S
b
2
S
4
+
4
S
6
+
b
2
S
4
b
2
S
4
+
4
S
24
-
1
6
(
λ
-
λ
1
)
3
/
2
|
f
|
3
/
2
3
/
2
,
by Lemmas 3.1 (ii) and 4.3. Hence, by Lemma 4.2, going if necessary to a subsequence, still denoted by {un}, we can assume that un⇀u** weakly in D1,2(ℝ3) and un→u** strongly in L6(ℝ3). Then we can declare that un→u** strongly in D1,2(ℝ3). Assuming the contrary, we have ∥u**∥<lim infn→∞∥un∥. Since {un}⊂𝐍λ, we have
∥
u
n
∥
2
+
b
∥
u
n
∥
4
=
λ
∫
ℝ
3
f
(
x
)
u
n
2
𝑑
x
+
∫
ℝ
3
u
n
6
𝑑
x
.
Passing to the limit, one has
∥
u
*
*
∥
2
+
b
∥
u
*
*
∥
4
<
λ
∫
ℝ
3
f
(
x
)
|
u
*
*
|
2
d
x
+
∫
ℝ
3
|
u
*
*
|
6
d
x
,
which means that u**(x)≢0 in ℝ3. Then, according to Lemma 2.2, there exists a unique t-=t-(u**)>0 such that t-u**∈𝐍λ- and Jλ(t-u**)=supt≥0Jλ(tu**). Since {un}⊂𝐍λ-, we deduce that Jλ(t-un)≤Jλ(un) for all n. Combining this with the fact that Φ(u)=∫ℝ3f(x)u2𝑑x is weakly continuous on D1,2(ℝ3), un→u** strongly in L6(ℝ3), and ∥u**∥<lim infn→∞∥un∥, we can conclude that
α
λ
-
≤
J
λ
(
t
-
u
*
*
)
=
1
2
∥
t
-
u
*
*
∥
2
+
b
4
∥
t
-
u
*
*
∥
4
-
λ
2
∫
ℝ
3
f
(
x
)
|
t
-
u
*
*
|
2
d
x
-
1
6
∫
ℝ
3
|
t
-
u
*
*
|
6
d
x
<
lim inf
n
→
∞
[
1
2
∥
t
-
u
n
∥
2
+
b
4
∥
t
-
u
n
∥
4
-
λ
2
∫
ℝ
3
f
(
x
)
|
t
-
u
n
|
2
d
x
-
1
6
∫
ℝ
3
|
t
-
u
n
|
6
d
x
]
=
lim inf
n
→
∞
J
λ
(
t
-
u
n
)
≤
lim inf
n
→
∞
J
λ
(
u
n
)
=
α
λ
-
.
This contradiction shows that ∥u**∥<lim infn→∞∥un∥ cannot hold, hence un→u** strongly in D1,2(ℝ3) and Jλ(u**)=αλ->0, which means, from Lemma 3.1 (i), that u**∈𝐍λ-. Noting that Jλ(u**)=Jλ(|u**|) and |u**|∈𝐍λ-, we can assume that u** is a nonnegative minimizer for αλ-, and the proof is completed. ∎
