1 Introduction
In this paper, we study the following biharmonic equations:
(1.1)
{
Δ
2
u
-
a
0
Δ
u
+
V
(
x
)
u
=
f
(
x
,
u
)
in
ℝ
N
,
u
∈
H
2
(
ℝ
N
)
,
where N≥3, a0∈ℝ is a constant and λ>0 is a parameter. V(x) and f(x,u) satisfy some conditions to be specified later.
The biharmonic equations in a bounded domain are generally regarded as a mathematical modeling, which can describe some phenomena appearing in physics, engineering and other sciences. For example, the problem of nonlinear oscillation in a suspension bridge [18, 22] and the problem of the static deflection of an elastic plate in a fluid [1]. Due to such applications, the existence and multiplicity of nontrivial solutions for the biharmonic equations in a bounded domain have been extensively studied in the past two decades. We refer the readers to [15, 16, 20, 24, 34] and the references therein. Most of the literature was devoted to the following Dirichlet–Navier type boundary value problem:
(\emph{P}α)
{
Δ
2
u
-
α
Δ
u
=
g
(
x
,
u
)
in
Ω
,
u
=
Δ
u
=
0
on
∂
Ω
,
where Ω⊂ℝN is a bounded domain with smooth boundary, α>-μ1 is a parameter and μ1 is the first eigenvalue of -Δ in L2(Ω). In particular, the existence of sign-changing solutions to (\emph{P}α) was obtained in [20, 24, 36] when g(x,u) is subcritical and superlinear or asymptotically linear at infinity.
In recent years, the study on problem (1.1), i.e. the biharmonic equations in the whole space ℝN, has begun to attract much attention. We refer the readers to [6, 7, 8, 12, 13, 27, 29, 28, 32] and the references therein. In these works, various existence results of the nontrivial solutions to problem (1.1) were established by the variational method in the case of a0≥0. Indeed, in the case of a0≥0, under some suitable conditions on V(x) and f(x,u), one can give a variational setting to problem (1.1) in the following Hilbert space:
𝒱
=
{
u
∈
H
2
(
ℝ
N
)
:
∫
ℝ
N
V
+
(
x
)
u
2
𝑑
x
<
+
∞
}
,
where V+(x)=max{V(x),0}, the inner product and the corresponding norm are respectively given by
〈
u
,
v
〉
𝒱
=
∫
ℝ
N
(
Δ
u
Δ
v
+
a
0
∇
u
∇
v
+
V
+
(
x
)
u
v
)
𝑑
x
and
∥
u
∥
=
〈
u
,
u
〉
𝒱
1
/
2
.
Thus, the variational method can be used to find the nontrivial solutions of problem (1.1), see for example [6, 8, 29, 28, 32] and the references therein.
If a0<0 then 𝒱 with 〈u,v〉𝒱 may not be a Hilbert space, since the bilinear operator 〈u,v〉𝒱 may not be an inner product in 𝒱 for V(x)≠0 in general. This is quite different from the situation of V(x)=0. Indeed, for example, if we consider the problem (𝒫α) in a bounded Ω⊂ℝN, then α can take negative value since the operator Δ2-αΔ is compact in L2(Ω) and the spectrum of Δ2-αΔ in L2(Ω) are the eigenvalues {μk2+αμk} with the first eigenvalue μ12+αμ1, where {μk} are the eigenvalues of -Δ in L2(Ω) with the first eigenvalue μ1>0, so that if α>-μ1 then H01(Ω)∩H2(Ω) is also a Hilbert space with the following inner product:
〈
u
,
v
〉
α
=
∫
Ω
(
Δ
u
Δ
v
+
α
∇
u
∇
v
)
𝑑
x
.
Then one can study (𝒫α) by the variational method under some suitable conditions on g(x,u) in the case of α>-μ1. However, when V(x)≠0, the operator Δ2-a0Δ+V(x) in 𝒱 is much more complex due to the potential V(x) and the spectrum of the operator Δ2-a0Δ+V(x) in 𝒱 is not clear in the case of a0<0, also the variational setting of (1.1) is not clear in the case of a0<0. Due to these reasons, the only study on problem (1.1) for the case a0<0 we are aware of is [27]. There, by introducing the Gagliardo–Nirenberg inequality in studying the biharmonic equation, the existence result of a positive solution to problem (1.1) with some special nonlinearities has been established for a0<0 and |a0| small enough. Therefore, based upon the above facts, a natural question is whether the biharmonic equation as problem (1.1) does have a nontrivial solution for all a0<0. The purpose of this paper is to explore this question. Note that even the variational setting is not established for the biharmonic equation for a0<0 and |a0| large enough, thus, the methods in the literature are invalid and some new ideas are needed.
We assume V(x)=λb(x)+b0, where b0∈ℝ is a constant, λ>0 is a parameter and b(x) satisfies the following conditions:
b(x)∈C(ℝN) and b(x)≥0 on ℝN.
There exists b∞>0 such that |ℬ∞|<+∞, where ℬ∞={x∈ℝN:b(x)<b∞} and |ℬ∞| is the Lebesgue measure of the set ℬ∞.
Ω=intb-1(0) is a bounded domain having the smooth boundary ∂Ω and Ω¯=b-1(0).
We call λb(x) the steep potential well for λ sufficiently large under conditions (B1)–(B3) and the depth of the well is controlled by the parameter λ. Such potentials were first introduced by Bartsch and Wang in [4] for the scalar Schrödinger equations. An interesting phenomenon for this kind of Schrödinger equations is that one can expect to find the solutions which are concentrated at the bottom of the wells as the depth goes to infinity. Due to this interesting property, such topic for the scalar Schrödinger equations was studied extensively in the past decade. We refer the readers to [2, 5, 3, 10, 9, 21, 25, 30] and the references therein. Recently, the steep potential well was also considered for some other elliptic equations and systems, see for example [11, 14, 17, 19, 26, 28, 31, 32, 33, 35] and the references therein. In particular, the steep potential well was introduced to the biharmonic equations in [19] and was further studied in [28, 32] in the case of a0≥0. For the nonlinearity, we assume that f(x,t)=f(t) is continuous and satisfies the following conditions:
There exists 2≤p<2* such that limt→∞f(t)|t|p-2t=l∞>0, where 2*=+∞ for N=3,4 and 2*=2NN-4 for N≥5.
f(t)t is nondecreasing on ℝ∖{0}.
There exists l*∈(0,l∞] such that f(t)t-2F(t)≥l*|t|p and F(t)≥0 for all t∈ℝ, where F(t)=∫0tf(s)𝑑s.
Now, under conditions (B1)–(B3) and (F1)–(F4), we mainly study the following problem in this paper:
(\emph{P}λ)
{
Δ
2
u
-
a
0
Δ
u
+
(
λ
b
(
x
)
+
b
0
)
u
=
f
(
u
)
in
ℝ
N
,
u
∈
H
2
(
ℝ
N
)
,
In order to establish a variational framework of (\emph{P}λ) in the case of a0<0, we need to study the spectrum and Morse index of the operator Δ2-a0Δ+(λb(x)+b0) in a suitable Hilbert space under conditions (B1)–(B3). We will borrow some ideas of [9] (see also [35]) to carry on this study. Note that in the case of a0<0, the negative part of the operator Δ2-a0Δ+(λb(x)+b0) is generated not only by (λb(x)+b0)- but also by -a0Δ, where (λb(x)+b0)±=max{±(λb(x)+b0),0}. Therefore, some new ideas and modifications are needed in establishing a variational framework of (\emph{P}λ) in the case of a0<0.
Before we state our results, we need to introduce some notations. Let Ω be given in condition (B3) and let {μk} be the eigenvalues of -Δ in L2(Ω). Then it is well known that 0<μ1<μ2≤μ3≤⋯≤μk<⋯ with μk→+∞ as k→∞ and ϕk are orthogonal in L2(Ω)∩H01(Ω) and span{ϕk}¯=H01(Ω), where ϕk are the eigenfunctions of μk. Since ∂Ω is smooth due to condition (B3), it is also well known that {ϕk}⊂C0∞(Ω¯). Let H be the Hilbert space H2(Ω)∩H01(Ω) equipped with the inner product
〈
u
,
v
〉
H
=
∫
Ω
(
Δ
u
Δ
v
+
max
{
a
0
,
0
}
∇
u
∇
v
+
max
{
b
0
,
0
}
u
v
)
𝑑
x
.
Then span{ϕk}¯=H and ϕk are orthogonal in H (cf. [34]). We re-denote {μk} by {μ¯n} such that μ¯j<μ¯j+1 for all j∈ℕ. Clearly, μ1=μ¯1. In the case of min{a0,b0}<0, we denote
(1.2)
k
0
*
=
inf
{
k
∈
ℕ
:
μ
¯
k
2
+
max
{
a
0
,
0
}
μ
¯
k
+
max
{
b
0
,
0
}
max
{
-
a
0
,
0
}
μ
¯
k
+
max
{
-
b
0
,
0
}
>
1
}
and μ¯0=0. Then the main results obtained in this paper can be stated as follows.
Theorem 1.1
Theorem 1.1 (The Superlinear Case)
Suppose that conditions (B1)–(B3), (F1)–(F2) and (F4) hold with p>2. If either min{a0,b0}≥0 or min{a0,b0}<0 with
μ
¯
k
0
*
-
1
2
+
max
{
a
0
,
0
}
μ
¯
k
0
*
-
1
+
max
{
b
0
,
0
}
max
{
-
a
0
,
0
}
μ
¯
k
0
*
-
1
+
max
{
-
b
0
,
0
}
<
1
𝑎𝑛𝑑
b
0
≤
μ
¯
1
2
,
then there exist positive constants l¯0 and Λ^ such that problem (Pλ) has a nontrivial solution for λ>Λ^ in the case of l0<l¯0.
Theorem 1.2
Theorem 1.2 (The Asymptotically Linear Case)
Suppose that conditions (B1)–(B3) and (F1)–(F3) hold with p=2. If either min{a0,b0}≥0 or min{a0,b0}<0 with
μ
¯
k
0
*
-
1
2
+
max
{
a
0
,
0
}
μ
¯
k
0
*
-
1
+
max
{
b
0
,
0
}
max
{
-
a
0
,
0
}
μ
¯
k
0
*
-
1
+
max
{
-
b
0
,
0
}
<
1
𝑎𝑛𝑑
b
0
≤
μ
¯
1
2
,
then there exist positive constants l¯0, l¯∞ and Λ^ such that problem (Pλ) has a nontrivial solution for λ>Λ^ in the cases of l0<l¯0 and l∞>l¯∞ with l∞∉σ(Δ2-a0Δ+b0,L2(Ω)), where σ(Δ2-a0Δ+b0,L2(Ω)) is the spectrum of Δ2-a0Δ+b0 in L2(Ω).
Since problem (𝒫λ) depends on the parameter λ, it is natural to investigate the concentration behavior of the solutions for λ→+∞. Our result in this topic can be stated as follows.
Theorem 1.3
Suppose that uλ is the nontrivial solution of problem (\emph{P}λ) obtained by Theorem 1.1 or Theorem 1.2. Then uλ→u* strongly in H2(RN) as λ→+∞ up to a subsequence for some u*∈H01(Ω)∩H2(Ω). Furthermore, u* is a nontrivial weak solution of the following equation:
(1.3)
{
Δ
2
u
-
a
0
Δ
u
+
b
0
u
=
f
(
u
)
in
Ω
,
u
=
Δ
u
=
0
on
∂
Ω
.
Remark 1.4
The concentration behaviors obtained by Theorem 1.3 have not been obtained in the literature. Moreover, Theorem 1.3 actually gives an existence result to (1.3) in the case of a0≤-μ1, where the nonlinearities are superlinear and subcritical or asymptotically linear at infinity. To our best knowledge, such a result also has not been obtained in the literature regardless of whether the nonlinearities are superlinear and subcritical or asymptotically linear at infinity.
Through this paper, C and C′ will be indiscriminately used to denote various positive constants, on(1) and oλ(1) will always denote the quantities tending towards zero as n→∞ and λ→+∞ respectively.
2 The Variational Setting
In this section, we will give the variational setting of (𝒫λ). Let
E
λ
=
{
u
∈
H
2
(
ℝ
N
)
:
∫
ℝ
N
(
λ
b
(
x
)
+
b
0
)
+
u
2
𝑑
x
<
+
∞
}
.
Then by condition (B1), the space Eλ is a Hilbert space equipped with the inner product
〈
u
,
v
〉
λ
=
∫
ℝ
N
(
Δ
u
Δ
v
+
max
{
a
0
,
0
}
∇
u
∇
v
+
(
λ
b
(
x
)
+
b
0
)
+
u
v
)
𝑑
x
for all λ>0. The corresponding norm in Eλ is given by ∥u∥λ=〈u,u〉λ1/2.
Lemma 2.1
Suppose that conditions (B1)–(B2) hold. Then Eλ is embedded continuously into H2(RN) for λ>max{0,-b0/b∞}.
Proof.
Set 2*=2NN-2. Then it is easy to see that 2<2*<2* for all N≥3. Now, by condition (B2) and the Sobolev inequality and Hölder inequality, for λ>max{0,-b0b∞}, we have
(2.1)
∥
u
∥
L
2
(
ℝ
N
)
2
≤
|
ℬ
∞
|
2
*
-
2
2
*
S
-
1
∥
∇
u
∥
L
2
(
ℝ
N
)
2
+
1
λ
b
∞
+
b
0
∥
u
∥
λ
2
,
where ∥⋅∥Lp(ℝN) are the usual norms in Lp(ℝN) for all p≥1 and S is the best Sobolev embedding constant from D1,2(ℝN) to L2*(ℝN) and given by
S
=
inf
{
∥
∇
u
∥
L
2
(
ℝ
N
)
2
:
u
∈
D
1
,
2
(
ℝ
N
)
,
∥
u
∥
L
2
*
(
ℝ
N
)
2
=
1
}
.
Let A∞=|ℬ∞|2*-22*S-1. If a0>0 then by (2.1), we can see
(2.2)
∥
u
∥
L
2
(
ℝ
N
)
2
≤
(
A
∞
a
0
-
1
+
1
λ
b
∞
+
b
0
)
∥
u
∥
λ
2
.
If a0≤0 then by (2.1), the Young inequality and the Gagliardo–Nirenberg inequality, we have
(2.3)
∥
u
∥
L
2
(
ℝ
N
)
2
≤
(
4
A
∞
2
B
0
4
+
2
λ
b
∞
+
b
0
)
∥
u
∥
λ
2
,
where B0>0 is the constant in the Gagliardo–Nirenberg inequality
∥
∇
u
∥
L
2
(
ℝ
N
)
≤
B
0
∥
Δ
u
∥
L
2
(
ℝ
N
)
1
2
∥
u
∥
L
2
(
ℝ
N
)
1
2
for all
u
∈
H
2
(
ℝ
N
)
.
Thus, by (2.2) and (2.3), we get
(2.4)
∥
u
∥
L
2
(
ℝ
N
)
2
≤
C
λ
∥
u
∥
λ
2
,
where
C
λ
=
{
A
∞
a
0
-
1
+
1
λ
b
∞
+
b
0
if
a
0
>
0
,
4
A
∞
2
B
0
4
+
2
λ
b
∞
+
b
0
if
a
0
≤
0
.
Inequality (2.4), once more together with the Gagliardo–Nirenberg inequality, implies
(2.5)
∥
∇
u
∥
L
2
(
ℝ
N
)
≤
B
0
∥
Δ
u
∥
L
2
(
ℝ
N
)
1
2
∥
u
∥
L
2
(
ℝ
N
)
1
2
≤
B
0
C
λ
1
4
∥
u
∥
λ
.
Combining (2.4) and (2.5), we deduce that
∥
u
∥
H
2
(
ℝ
N
)
≤
(
1
+
C
λ
+
B
0
2
C
λ
1
/
2
)
1
/
2
∥
u
∥
λ
for all
u
∈
H
2
(
ℝ
N
)
,
where ∥⋅∥H2(ℝN) is the usual norm in H2(ℝN). Thus, for λ>max{0,-b0b∞}, the Hilbert space Eλ is embedded continuously into H2(ℝN). ∎
Let ℰλ(u):Eλ→ℝ be given by
ℰ
λ
(
u
)
=
1
2
𝒟
λ
(
u
,
u
)
-
∫
ℝ
N
F
(
x
,
u
)
𝑑
x
,
where
𝒟
λ
(
u
,
v
)
=
〈
u
,
v
〉
λ
-
𝒢
λ
(
u
,
v
)
with
𝒢
λ
(
u
,
v
)
=
max
{
-
a
0
,
0
}
〈
∇
u
,
∇
v
〉
L
2
(
ℝ
N
)
+
∫
ℝ
N
(
λ
b
(
x
)
+
b
0
)
-
u
v
𝑑
x
.
Then by Lemma 2.1, we have the following.
Lemma 2.2
Suppose that conditions (B1)–(B2) and (F1)–(F2) hold. Then the functional Eλ(u) is well defined and it belongs to C1 for λ>max{0,-b0b∞}. Furthermore, Eλ(u) is the corresponding functional of (Pλ).
Proof.
By (F1)–(F2), we can see that |F(t)|≤max{2l0,l0+ε}|t|2+C|t|p, where ε>0 is a small enough constant. Since p<2*, by the Sobolev embedding theorem, we can see that
|
∫
ℝ
N
F
(
u
)
𝑑
x
|
≤
∫
ℝ
N
|
F
(
u
)
|
𝑑
x
≤
max
{
2
l
0
,
l
0
+
ε
}
∥
u
∥
L
2
(
ℝ
N
)
2
+
C
∥
u
∥
L
p
(
ℝ
N
)
p
(2.6)
≤
max
{
2
l
0
,
l
0
+
ε
}
∥
u
∥
H
2
(
ℝ
N
)
2
+
C
′
∥
u
∥
H
2
(
ℝ
N
)
p
.
On the other hand, by (2.4) and conditions (B1)–(B2), we also have
(2.7)
∫
ℝ
N
(
λ
b
(
x
)
+
b
0
)
-
u
2
𝑑
x
≤
max
{
0
,
-
b
0
}
∥
u
∥
L
2
(
ℝ
N
)
2
≤
max
{
0
,
-
b
0
}
C
λ
1
2
∥
u
∥
λ
2
.
Thanks to Lemma 2.1, by (2.6) and (2.7), we can see that the functional ℰλ(u) is well defined and it belongs to C1 for λ>max{0,-b0b∞}. Furthermore, by a standard argument, we also have that ℰλ(u) is the corresponding functional of (𝒫λ). ∎
3 The Functional 𝒟λ(u,u)
If min{a0,b0}≥0 then 𝒢λ(u,v)=0, which gives that 𝒟λ(u,v)=〈u,v〉λ for all (u,v)∈Eλ and then 𝒟λ(u,u) is definite on Eλ. Let us consider the case min{a0,b0}<0. Let
(3.1)
𝒩
j
=
span
{
ϕ
i
:
ϕ
i
is the corresponding eigenfunction of
μ
¯
j
}
.
Then it is well known that dim(𝒩j)<+∞ for all j∈ℕ. In particular, dim(𝒩1)=1 and ϕ1 is positive on Ω. Since min{a0,b0}<0, the set
ℳ
0
=
{
u
∈
H
:
∫
Ω
(
max
{
-
a
0
,
0
}
|
∇
u
|
2
+
max
{
-
b
0
,
0
}
|
u
|
2
)
𝑑
x
=
1
}
is well defined and it is easy to see that ℳ0 is a natural constraint in H. Let
β
1
0
=
inf
ℳ
0
∫
Ω
(
|
Δ
u
|
2
+
max
{
a
0
,
0
}
|
∇
u
|
2
+
max
{
b
0
,
0
}
|
u
|
2
)
𝑑
x
and
(3.2)
β
j
0
=
inf
(
ℳ
j
-
1
)
⊥
∫
Ω
(
|
Δ
u
|
2
+
max
{
a
0
,
0
}
|
∇
u
|
2
+
max
{
b
0
,
0
}
|
u
|
2
)
𝑑
x
,
j
=
2
,
3
,
…
,
where
(
ℳ
j
-
1
)
⊥
=
{
u
∈
ℳ
0
:
〈
u
,
v
〉
H
=
0
for all
v
∈
⊕
i
=
1
j
-
1
𝒩
i
}
.
Lemma 3.1
Suppose min{a0,b0}<0 and b0≤μ¯12. Then we have
β
j
0
=
μ
¯
j
2
+
max
{
a
0
,
0
}
μ
¯
j
+
max
{
b
0
,
0
}
max
{
-
a
0
,
0
}
μ
¯
j
+
max
{
-
b
0
,
0
}
for all
j
∈
ℕ
and βj0 can be attained by u∈H if and only if u∈Nj∩M0.
Proof.
Consider the function
g
(
x
)
=
x
2
+
max
{
a
0
,
0
}
x
+
max
{
b
0
,
0
}
max
{
-
a
0
,
0
}
x
+
max
{
-
b
0
,
0
}
.
By a direct calculation, we can see that
g
′
(
x
)
=
max
{
-
a
0
,
0
}
x
2
+
2
max
{
-
b
0
,
0
}
x
(
max
{
-
a
0
,
0
}
x
+
max
{
-
b
0
,
0
}
)
2
+
max
{
a
0
,
0
}
max
{
-
b
0
,
0
}
-
max
{
-
a
0
,
0
}
max
{
b
0
,
0
}
(
max
{
-
a
0
,
0
}
x
+
max
{
-
b
0
,
0
}
)
2
.
Note that
max
{
a
0
,
0
}
max
{
-
b
0
,
0
}
-
max
{
-
a
0
,
0
}
max
{
b
0
,
0
}
≥
min
{
0
,
a
0
b
0
}
for
min
{
a
0
,
b
0
}
<
0
,
thus, g′(x)≥0 for x≥μ¯1. Now, by the fact that span{ϕk}=H and the orthogonality of {ϕk} in H, we can easily see from the Ekeland’s principle and the method of Lagrange multiplier that
β
j
0
=
μ
¯
j
2
+
max
{
a
0
,
0
}
μ
¯
j
+
max
{
b
0
,
0
}
max
{
-
a
0
,
0
}
μ
¯
j
+
max
{
-
b
0
,
0
}
for all
j
∈
ℕ
and βj0 can be attained by u∈H if and only if u∈𝒩j∩ℳ0. ∎
Let
ℳ
λ
=
{
u
∈
E
λ
:
𝒢
λ
(
u
,
u
)
=
1
}
.
Lemma 3.2
Suppose that conditions (B1)–(B3) hold, min{a0,b0}<0 and b0≤μ¯12. Then there exists Λ1>0 such that β1(λ)=infMλ∥u∥λ2 can be attained by some e1(λ)∈Mλ for λ>Λ1. Moreover, (e1(λ),β1(λ)) satisfies the following equation:
(3.3)
{
Δ
2
u
-
max
{
a
0
,
0
}
Δ
u
+
(
λ
b
(
x
)
+
b
0
)
+
u
=
β
(
max
{
-
a
0
,
0
}
Δ
u
+
(
λ
b
(
x
)
+
b
0
)
-
u
)
in
ℝ
N
,
u
∈
H
2
(
ℝ
N
)
,
and (e1(λ),β1(λ))→(ϕ1,β10) strongly in H2(RN) as λ→+∞ up to a subsequence.
Proof.
We first prove that β1(λ) can be attained for λ sufficiently large. Indeed, by the Ekeland principle, there exists {un}⊂ℳλ such that the following hold:
∥v∥λ2≥∥un∥λ2-1n∥v-un∥λ for all v∈ℳλ.
For each n∈ℕ and w∈Eλ, by applying the implicit function theorem in a standard way and noting that conditions (B1)–(B2) hold, we can see that there exist εn>0 and tn(l)∈C1([-εn,εn]) with tn(0)=1 and tn′(0)=-𝒢λ(un,w) such that tn(l)un+lw∈ℳλ for l∈(0,εn]. It follows from (2) that
0
≥
∥
u
n
∥
λ
2
-
∥
t
n
(
l
)
u
n
+
l
w
∥
λ
2
-
1
n
∥
(
t
n
(
l
)
-
1
)
u
n
+
l
w
∥
λ
≥
(
1
-
t
n
(
l
)
2
)
∥
u
n
∥
λ
2
-
2
t
n
(
l
)
l
〈
u
n
,
w
〉
λ
-
l
2
∥
w
∥
λ
2
-
|
t
n
(
l
)
-
1
|
n
∥
u
n
∥
λ
-
l
n
∥
w
∥
λ
.
Since we have by (1) that {un} is bounded in Eλ, multiplying this inequality with l-1 on both sides and letting l→0+, we deduce
0
≥
2
∥
u
n
∥
λ
2
𝒢
λ
(
u
n
,
w
)
+
|
𝒢
λ
(
u
n
,
w
)
|
n
∥
u
n
∥
λ
-
2
〈
u
n
,
w
〉
λ
-
1
n
∥
w
∥
λ
=
2
(
β
1
(
λ
)
𝒢
λ
(
u
n
,
w
)
-
〈
u
n
,
w
〉
λ
)
+
o
n
(
1
)
.
Since w∈Eλ is arbitrary, we must have
(3.4)
o
n
(
1
)
=
β
1
(
λ
)
𝒢
λ
(
u
n
,
w
)
-
〈
u
n
,
w
〉
λ
for all w∈Eλ. Let Jλ(u)=12∥u∥λ2-β1(λ)2𝒢λ(u,u). Then by (3.4), we have Jλ′(un)w=on(1) for all w∈Eλ. In particular, by the choice of {un}, we can also see that Jλ′(un)un=on(1). On the other hand, by (1), we have that {un} is bounded in Eλ. Thus, without loss of generality, we may assume that un⇀e1(λ) weakly in Eλ as n→∞. Note that Jλ(u) is C2 in Eλ for λ>max{0,-b0/b∞} due to (2.7). We have Jλ′(e1(λ))=0. It follows from the Sobolev embedding theorem, conditions (B1)–(B2) and similar arguments used in the proofs of (2.4) and (2.5) that
o
n
(
1
)
=
(
J
λ
′
(
u
n
)
-
J
λ
′
(
e
1
(
λ
)
)
)
(
u
n
-
e
1
(
λ
)
)
=
∥
u
n
-
e
1
(
λ
)
∥
λ
2
-
β
1
(
λ
)
(
𝒢
λ
(
u
n
-
e
1
(
λ
)
,
u
n
-
e
1
(
λ
)
)
)
≥
∥
u
n
-
e
1
(
λ
)
∥
λ
2
-
β
1
(
λ
)
max
{
-
a
0
,
0
}
∥
∇
(
u
n
-
e
1
(
λ
)
)
∥
L
2
(
ℝ
N
)
2
-
β
1
(
λ
)
max
{
-
b
0
,
0
}
∥
u
n
-
e
1
(
λ
)
∥
L
2
(
ℝ
N
)
2
(3.5)
≥
(
1
-
β
1
(
λ
)
max
{
-
a
0
,
0
}
B
0
2
(
λ
b
∞
+
b
0
)
1
2
-
β
1
(
λ
)
max
{
-
b
0
,
0
}
λ
b
∞
+
b
0
)
∥
u
n
-
e
1
(
λ
)
∥
λ
2
+
o
n
(
1
)
.
Note that by condition (B3) and b0≤μ¯12, we get from Lemma 3.1 that
(3.6)
β
1
(
λ
)
≤
∥
ϕ
1
∥
λ
2
𝒢
λ
(
ϕ
1
,
ϕ
1
)
=
β
1
0
.
It follows from (3.5) that there exists Λ1>max{0,-b0/b∞} such that un→e1(λ) strongly in Eλ as n→∞ for λ>Λ1. Thus, e1(λ)∈ℳλ and β1(λ) can be attained by e1(λ) for λ>Λ1. By (3.4), we also see that (e1(λ),β1(λ)) satisfies (3.3).
To complete the proof of this lemma, we shall show that (e1(λ),β1(λ))→(ϕ1,β10) strongly in H2(ℝN) as λ→+∞ up to a subsequence. Indeed, by (3.6), we know that ∥e1(λ)∥λ≤β10 for all λ>Λ1, which, together with Lemma 2.1, implies that {e1(λ)} is bounded in H2(ℝN). Without loss of generality, we may assume that e1(λ)⇀e1* weakly in H2(ℝN) as λ→+∞. Since ∥e1(λ)∥λ≤β10 for all λ>Λ1, by condition (B1) and the Fatou lemma, we have
(3.7)
0
=
lim inf
λ
→
+
∞
β
1
0
λ
≥
lim inf
λ
→
+
∞
∫
ℝ
N
(
b
(
x
)
+
b
0
λ
)
+
[
e
1
(
λ
)
]
2
𝑑
x
≥
∫
ℝ
N
b
(
x
)
[
e
1
*
]
2
𝑑
x
.
It follows from condition (B3) that e1*∈H01(Ω). Thanks to condition (B2) and Lemma 2.1, we can see from (3.7) and the Sobolev embedding theorem that e1(λ)→e1* strongly in L2(ℝN) as λ→+∞. Then the Gagliardo–Nirenberg inequality yields that e1(λ)→e1* strongly in H1(ℝN) as λ→+∞. On the other hand, by conditions (B1)–(B3), we obtain from a variant of the Lebesgue dominated convergence theorem (cf. [23, Theorem 2.2]) and the fact that e1(λ)→e1* strongly in H1(ℝN) as λ→+∞ that e1*∈ℳ0, which, together with the definition of β10 and conditions (B1)–(B3), yields
lim inf
λ
→
+
∞
β
1
(
λ
)
≥
∫
Ω
(
|
Δ
e
1
*
|
2
+
max
{
a
0
,
0
}
|
∇
e
1
*
|
2
+
max
{
b
0
,
0
}
|
e
1
*
|
2
)
𝑑
x
≥
β
1
0
.
Thus, we must have limλ→+∞β1(λ)=β10 due to (3.6). Now, once more thanks to conditions (B1)–(B3), we can see from the weak convergence of {e1(λ)} in H2(ℝN) and the Fatou lemma that
(3.8)
β
1
0
=
lim
λ
→
+
∞
∥
e
1
(
λ
)
∥
λ
2
≥
∫
Ω
|
Δ
e
1
*
|
2
+
max
{
a
0
,
0
}
|
∇
e
1
*
|
2
+
max
{
b
0
,
0
}
|
e
1
*
|
2
d
x
≥
β
1
0
.
Thus, Δe1(λ)→Δe1* strongly in L2(ℝN) as λ→+∞ since e1(λ)→e1* strongly in L2(ℝN)∩H1(ℝN) as λ→+∞, so that e1(λ)→e1* strongly in H2(ℝN) as λ→+∞. Note that e1*∈H which attains β10, thus we finally have e1*=ϕ1 by Lemma 3.1. ∎
Let 𝒩λ,1=span{u∈ℳλ:∥u∥λ2=β1(λ)}. Then by Lemma 3.2, we can see that e1(λ)∈𝒩λ,1 for λ>Λ1.
Lemma 3.3
Suppose that conditions (B1)–(B3) hold, min{a0,b0}<0 and b0≤μ¯12. Then there exists Λ1*≥Λ1 such that Nλ,1=span{e1(λ)} for λ>Λ1*.
Proof.
Suppose on the contrary that there exists {λn} satisfying λn→+∞ as n→∞ such that
𝒩
λ
n
,
1
≠
span
{
e
1
(
λ
n
)
}
⊂
H
2
(
ℝ
N
)
.
It follows that there exists u(λn)∈𝒩λn,1 satisfying u(λn)∉span{e1(λn)} for all n∈ℕ. Without loss of generality, we may assume that 〈u(λn),e1(λn)〉λn=0 for all n∈ℕ. Similarly as in the proof of Lemma 3.2, going if necessary to a subsequence, we may get that u(λn)=ϕ1+on(1) strongly in H2(ℝN) and
∫
ℝ
N
(
λ
n
b
(
x
)
+
b
0
)
+
[
u
(
λ
n
)
]
2
𝑑
x
=
∫
Ω
max
{
b
0
,
0
}
ϕ
1
2
𝑑
x
+
o
n
(
1
)
,
which, together with Lemma 3.2, implies
∥
e
1
(
λ
n
)
-
u
(
λ
n
)
∥
λ
n
2
≤
∥
u
(
λ
n
)
-
ϕ
1
∥
λ
n
2
+
∥
e
1
(
λ
n
)
-
ϕ
1
∥
λ
n
2
=
o
n
(
1
)
.
Therefore, we have
0
<
2
β
1
0
=
lim
n
→
∞
(
∥
e
1
(
λ
n
)
∥
λ
n
2
+
∥
u
(
λ
n
)
∥
λ
n
2
)
=
lim
n
→
∞
∥
e
1
(
λ
n
)
-
u
(
λ
n
)
∥
λ
n
2
=
0
,
which is a contradiction. ∎
Let
β
2
(
λ
)
=
inf
(
ℳ
λ
1
)
⊥
∥
u
∥
λ
2
,
where (ℳλ1)⊥={u∈ℳλ:〈u,v〉λ=0 for all v∈𝒩λ,1}. Then it is easy to see that β2(λ)≥β1(λ) for λ>Λ1. Thus, β2(λ) is well defined.
Lemma 3.4
Suppose that conditions (B1)–(B3) hold, min{a0,b0}<0 and b0≤μ¯12. Then there exists Λ2≥Λ1 such that β2(λ) can be attained by some e2(λ)∈Eλ for λ>Λ2. Moreover, (e2(λ),β2(λ)) satisfies (3.3) and (e2(λ),β2(λ))→(e20,β20) strongly in H2(RN) as λ→+∞ up to a subsequence for some e20∈N2, where N2 is defined in (3.1) and β20 is given by (3.2).
Proof.
For the sake of clarity, the proof will be performed in the following three steps.
Step 1. We prove that lim supλ→+∞β2(λ)≤β20.
Let φ∈𝒩2. Then φ=φλ-+φλ+, where φλ- and φλ+ are the projections of φ in 𝒩λ,1 and (ℳλ1)⊥ respectively. It follows from 𝒩2⊂(ℳ1)⊥, condition (B3) and Lemmas 3.2 and 3.3 that
lim
λ
→
+
∞
∥
φ
λ
-
∥
λ
2
=
lim
λ
→
+
∞
〈
φ
λ
-
,
φ
〉
λ
=
0
up to a subsequence. By Lemmas 2.1 and 2.2, we have limλ→+∞𝒢λ(φλ-,φλ-)=0 up to a subsequence. Now, using the definitions of β2(λ) and β20, conditions (B1)–(B3), b0≤μ¯12 and Lemmas 3.1–3.3, we have
lim sup
λ
→
+
∞
β
2
(
λ
)
≤
lim sup
λ
→
+
∞
∥
φ
λ
+
∥
λ
2
𝒢
λ
(
φ
λ
+
,
φ
λ
+
)
=
lim sup
λ
→
+
∞
∥
φ
-
φ
λ
-
∥
λ
2
𝒢
λ
(
φ
-
φ
λ
-
,
φ
-
φ
λ
-
)
=
lim sup
λ
→
+
∞
∥
φ
∥
λ
2
-
2
〈
φ
,
φ
λ
-
〉
λ
+
∥
φ
λ
-
∥
λ
2
𝒢
λ
(
φ
,
φ
)
-
2
𝒢
λ
(
φ
,
φ
λ
-
)
+
𝒢
λ
(
φ
λ
-
,
φ
λ
-
)
=
∫
Ω
|
Δ
φ
|
2
+
max
{
a
0
,
0
}
|
∇
φ
|
2
+
max
{
b
0
,
0
}
|
φ
|
2
d
x
∫
Ω
max
{
-
a
0
,
0
}
|
∇
φ
|
2
+
max
{
-
b
0
,
0
}
|
φ
|
2
d
x
=
β
2
0
.
Step 2. We prove that there exists Λ2≥Λ1 such that β2(λ) can be attained by some e2(λ)∈H2(ℝN) for λ>Λ2 and (e2(λ),β2(λ)) satisfies (3.3).
Indeed, by a similar argument used in the proof of Lemma 3.2, we can show that there exists {un}⊂(ℳλ1)⊥ such that the following hold:
on(1)=β2(λ)𝒢λ(un,w)-〈un,w〉λ for all w∈(ℳλ1)⊥.
Clearly, (1) gives the boundedness of {un} in Eλ, hence we may assume that un⇀e2(λ) weakly in Eλ as n→∞. Since {un}⊂(ℳλ1)⊥, we must have e2(λ)∈(ℳλ1)⊥. By (2), we obtain
0
=
β
2
(
λ
)
𝒢
λ
(
e
2
(
λ
)
,
e
2
(
λ
)
)
-
∥
e
2
(
λ
)
∥
λ
2
,
which, together with (1) and similar arguments in the proof of (3.5), implies
o
n
(
1
)
≥
(
1
-
β
2
(
λ
)
max
{
-
a
0
,
0
}
B
0
2
(
λ
b
∞
+
b
0
)
1
2
-
β
2
(
λ
)
max
{
-
b
0
,
0
}
λ
b
∞
+
b
0
)
∥
u
n
-
e
2
(
λ
)
∥
λ
2
+
o
n
(
1
)
.
It follows from step 1 and Lemma 3.1 that there exists Λ2≥Λ1 such that un→e2(λ) strongly in Eλ as n→∞ for λ>Λ2. Hence, by (1)–(2) and Lemma 3.3, we have that β2(λ) is attained by e2(λ) for λ>Λ2 and (e2(λ),β2(λ)) satisfies (3.3).
Step 3. We prove that (e2(λ),β2(λ))→(e20,β20) strongly in H2(ℝN) as λ→+∞ up to a subsequence for some e20∈𝒩2.
Indeed, by step 1, as in the proof of Lemma 3.2, we can show that (e2(λ),β2(λ))→(e20,β*) strongly in H1(ℝN)∩L2(ℝN) as λ→+∞ up to a subsequence for some e20∈H and β*≤β20. It follows from conditions (B1)–(B3) and a variant of the Lebesgue dominated convergence theorem (cf. [23, Theorem 2.2]) that e20∈ℳ0. Once more by step 1, we see that either e20∈𝒩1∩ℳ0 or e20∈𝒩2∩ℳ0. If e20∈𝒩1∩ℳ0 then e20=ϕ1. Moreover, by step 2 and condition (B3), we can see that
β
1
0
=
〈
ϕ
1
,
ϕ
1
〉
H
=
〈
e
2
(
λ
)
,
ϕ
1
〉
λ
+
o
λ
(
1
)
=
β
2
(
λ
)
𝒢
λ
(
e
2
(
λ
)
,
ϕ
1
)
+
o
λ
(
1
)
=
β
*
+
o
λ
(
1
)
.
Thus, by an argument similar to that used in (3.8), we can obtain that e2(λ)→ϕ1 strongly in H2(ℝN) and
∫
ℝ
N
(
λ
b
(
x
)
+
b
0
)
+
[
e
2
(
λ
)
]
2
𝑑
x
=
∫
Ω
max
{
b
0
,
0
}
ϕ
1
2
𝑑
x
+
o
λ
(
1
)
.
Now, by Lemma 3.2, we can obtain that
2
β
1
0
≤
∥
e
2
(
λ
)
∥
λ
2
+
∥
e
1
(
λ
)
∥
λ
2
+
o
λ
(
1
)
=
∥
e
2
(
λ
)
-
e
1
(
λ
)
∥
λ
2
+
o
λ
(
1
)
≤
∥
e
2
(
λ
)
-
ϕ
1
∥
λ
2
+
∥
e
1
(
λ
)
-
ϕ
1
∥
λ
2
+
o
λ
(
1
)
(3.9)
=
o
λ
(
1
)
.
This is impossible. Thus, we must have e20∈𝒩2∩ℳ0, which implies that
lim inf
λ
→
+
∞
β
2
(
λ
)
≥
β
2
0
.
This, together with step 1, yields that limλ→+∞β2(λ)=β20 and e20 attains β20. Now, by an argument similar to that we used in the proof of Lemma 3.2, we get that e2(λ)→e20 strongly in H2(ℝN) as λ→+∞ up to a subsequence. ∎
Let 𝒩λ,2=span{u∈ℳλ:∥u∥λ2=β2(λ)}. Then by Lemma 3.4, we can see that e2(λ)∈𝒩λ,2 for λ>Λ2.
Lemma 3.5
Suppose that conditions (B1)–(B3) hold, min{a0,b0}<0 and b0≤μ¯12. Then there exists Λ2*≥Λ2 such that Nλ,1⊥Nλ,2 and dim(Nλ,2)≤dim(N2) for λ>Λ2*. Here, by Nλ,1⊥Nλ,2 we mean that 〈u,v〉λ=0 for all u∈Nλ,1 and v∈Nλ,2. Moreover, we also have β1(λ)<β2(λ).
Proof.
Let u(λ) and v(λ)∈𝒩λ,2 with u(λ)∉span{v(λ)}. Without loss of generality, we may assume that 〈u(λ),v(λ)〉λ=0. By Lemma 3.4, we can see that u(λ)→e′ and v(λ)→e′′ strongly in H2(ℝN) as λ→+∞ up to a subsequence for some e′,e′′∈𝒩2. If e′=e′′, then by an argument similar to that used in (3.9), we can show that β20=0, which is a contradiction. Hence, we must have 〈e′,e′′〉H=0 since 〈u(λ),v(λ)〉λ=0. It follows from Lemmas 3.2 and 3.4 that there exists Λ2*≥Λ2 such that β1(λ)<β2(λ), 𝒩λ,1⊥𝒩λ,2 and dim(𝒩λ,2)≤dim(𝒩2) for λ>Λ2*. ∎
Now, define βk(λ) as
β
k
(
λ
)
=
inf
(
ℳ
λ
k
-
1
)
⊥
∥
u
∥
λ
2
(
k
=
3
,
4
,
…
)
,
where
(
ℳ
λ
k
-
1
)
⊥
=
{
u
∈
ℳ
λ
:
〈
u
,
v
〉
λ
=
0
for all
v
∈
⊕
i
=
1
k
-
1
𝒩
λ
,
i
}
and 𝒩λ,i=span{u∈ℳλ:∥u∥λ2=βi(λ)}. Then by iteration, we can obtain the following lemma.
Lemma 3.6
Suppose that conditions (B1)–(B3) hold, min{a0,b0}<0 and b0≤μ¯12 and k∈N with k≥3. Then the following hold:
There exists
λ
k
≥
Λ
k
-
1
such that
β
k
(
λ
)
is well defined and can be attained by some
e
k
(
λ
)
∈
E
λ
for
λ
>
Λ
k
. Moreover,
(
e
k
(
λ
)
,
β
k
(
λ
)
)
satisfies (
3.3
) and
(
e
k
(
λ
)
,
β
k
(
λ
)
)
→
(
e
k
0
,
β
i
0
)
strongly in
H
2
(
ℝ
N
)
as
λ
→
+
∞
up to a subsequence for some
e
k
0
∈
⊕
i
=
1
k
𝒩
i
and some
i
=
2
,
…
,
k
, where
β
k
0
is given by (
3.2
).
There exists
Λ
k
*
≥
Λ
k
such that
β
k
-
1
(
λ
)
<
β
k
(
λ
)
,
⊕
i
=
1
k
-
1
𝒩
λ
,
i
⊥
𝒩
λ
,
k
𝑎𝑛𝑑
dim
(
⊕
i
=
1
k
𝒩
λ
,
i
)
≤
dim
(
⊕
i
=
1
k
𝒩
i
)
for λ>Λk*.
Recall k0* from (1.2). Since dim(⊕i=1k0*-1𝒩i)<+∞, by Lemmas 3.2, 3.4 and 3.6, there exists k0**≥k0* such that
lim inf
λ
→
+
∞
β
k
0
*
*
(
λ
)
=
β
k
0
*
0
,
lim sup
λ
→
+
∞
β
k
0
*
*
-
1
(
λ
)
=
β
k
0
*
0
and
lim inf
λ
→
+
∞
β
k
0
*
*
-
1
(
λ
)
=
β
k
0
*
-
1
0
.
Moreover,
𝒩
λ
,
k
0
*
=
𝒩
λ
,
k
0
*
(
1
)
⊕
𝒩
λ
,
k
0
*
(
2
)
,
where u(λ)→ek0*-1 and v(λ)→ek0* as λ→+∞ for some ek0*-1∈𝒩k0*-1 and ek0*∈𝒩k0* and for all u(λ)∈𝒩λ,k0*(1) and v(λ)∈𝒩λ,k0*(2). Without loss of generality and for simplicity, we assume
k
0
*
*
=
k
0
*
,
lim sup
λ
→
+
∞
β
k
0
*
-
1
(
λ
)
=
β
k
0
*
-
1
0
and
v
(
λ
)
→
e
k
0
*
as
λ
→
+
∞
for some ek0*∈𝒩k0* and for all v(λ)∈𝒩λ,k0*.
Lemma 3.7
Suppose that conditions (B1)–(B3) hold, min{a0,b0}<0 and b0≤μ¯12. If βk0*-10<1, then we have
𝒟
λ
(
u
,
u
)
≤
(
1
-
1
β
k
0
*
-
1
(
λ
)
)
∥
u
∥
λ
2
=
(
1
-
1
β
k
0
*
-
1
0
+
o
λ
(
1
)
)
∥
u
∥
λ
2
in
⊕
i
=
1
k
0
*
-
1
𝒩
λ
,
i
and
𝒟
λ
(
u
,
u
)
≥
(
1
-
1
β
k
0
*
(
λ
)
)
∥
u
∥
λ
2
=
(
1
-
1
β
k
0
*
0
+
o
λ
(
1
)
)
∥
u
∥
λ
2
in
(
ℳ
λ
k
0
*
-
1
)
⊥
.
Proof.
The proof follows immediately from Lemmas 3.2, 3.4 and 3.6. ∎
Remark 3.8
By Lemmas 3.3, 3.5 and 3.6, we also have ⊕i=1k0*-1𝒩λ,i=∅ in the case of β10>1 whereas we have
⊕
i
=
1
k
0
*
-
1
𝒩
λ
,
i
≠
∅
and
dim
(
⊕
i
=
1
k
0
*
-
1
𝒩
λ
,
i
)
≤
dim
(
⊕
i
=
1
k
0
*
-
1
𝒩
i
)
in the case of β10<1.
4 The Existence of Nontrivial Solutions
We first consider the case of min{a0,b0}<0. Due to the decomposition of Eλ in this case, we will obtain the nonzero critical points of ℰλ(u) by using the linking method.
Lemma 4.1
Suppose that conditions (B1)–(B3) and (F1)–(F2) and (F4) hold with p>2 and min{a0,b0}<0 and b0≤μ¯12. If βk0*-10<1 and l0d0<(1-1/βk0*0) then there exists Λ¯0>0 such that
inf
(
ℳ
λ
k
0
*
-
1
)
⊥
∩
𝔹
λ
,
ρ
0
ℰ
λ
(
u
)
≥
κ
0
𝑎𝑛𝑑
sup
∂
𝒬
λ
,
R
0
ℰ
λ
(
u
)
≤
0
for all λ>Λ¯0 with some κ0>0 and R0>ρ0>0 independent of λ, where
𝔹
λ
,
ρ
0
=
{
u
∈
E
λ
:
∥
u
∥
λ
=
ρ
0
}
,
𝒬
λ
,
R
0
=
{
u
=
v
+
t
e
k
0
*
(
λ
)
:
v
∈
⊕
i
=
1
k
0
*
-
1
𝒩
λ
,
i
,
t
≥
0
,
∥
u
∥
λ
≤
R
0
}
and
d
0
=
{
A
∞
a
0
-
1
if
a
0
>
0
,
4
A
∞
2
B
0
4
if
a
0
≤
0
.
Proof.
Since l0d0<(1-1/βk0*0), there exist δ>0 and ε>0 such that
max
{
(
1
+
δ
)
l
0
,
l
0
+
ε
}
d
0
<
(
1
-
1
β
k
0
*
0
)
.
By conditions (F1)–(F2), we have
|
F
(
u
)
|
≤
max
{
(
1
+
δ
)
l
0
,
l
0
+
ε
}
2
|
u
|
2
+
C
|
u
|
2
*
for all u∈ℝ. It follows from the Sobolev inequality and Lemma 2.1 that
ℰ
λ
(
u
)
≥
1
2
(
𝒟
λ
(
u
,
u
)
-
max
{
(
1
+
δ
)
l
0
,
l
0
+
ε
}
∥
u
∥
L
2
(
ℝ
N
)
2
)
-
C
∥
u
∥
L
2
*
(
ℝ
N
)
2
*
≥
1
2
(
𝒟
λ
(
u
,
u
)
-
max
{
(
1
+
δ
)
l
0
,
l
0
+
ε
}
C
λ
∥
u
∥
λ
2
)
-
(
C
′
+
o
λ
(
1
)
)
∥
u
∥
λ
2
*
=
1
2
(
𝒟
λ
(
u
,
u
)
-
max
{
(
1
+
δ
)
l
0
,
l
0
+
ε
}
(
d
0
+
o
λ
(
1
)
)
∥
u
∥
λ
2
)
-
(
C
′
+
o
λ
(
1
)
)
∥
u
∥
λ
2
*
for all u∈Eλ. Using Lemma 3.7, we get
ℰ
λ
(
u
)
≥
1
2
(
1
-
1
β
k
0
*
0
+
o
λ
(
1
)
-
max
{
(
1
+
δ
)
l
0
,
l
0
+
ε
}
d
0
)
∥
u
∥
λ
2
-
(
C
′
+
o
λ
(
1
)
)
∥
u
∥
λ
2
*
for all u∈(ℳλk0*-1)⊥. Now, by a standard argument, it is easy to check that there exists Λ10>Λk0** such that
inf
(
ℳ
λ
k
0
*
-
1
)
⊥
∩
𝔹
λ
,
ρ
0
ℰ
λ
(
u
)
≥
κ
0
for all λ>Λ10 with some κ0>0 and ρ0>0 independent of λ. It remains to show that there exists a positive constant R0(>ρ0) so large that
sup
∂
𝒬
λ
,
R
0
ℰ
λ
(
u
)
≤
0
for λ sufficient large. Indeed, let uλ∈∂𝒬λ,R be such that uλ=Ru~λ with u~λ∈𝒬λ,1, then one of the following two cases must happen:
u~λ∈⊕i=1k0*-1𝒩λ,i and ∥u~λ∥λ≤1;
u~λ∈𝒬λ,1∖⊕i=1k0*-1𝒩λ,i and ∥u~λ∥λ=1.
In case (1), it follows from Lemma 3.7 and condition (F4) that ℰλ(Ru~λ)≤0 for all R≥0 and λ>Λ10. In case (2), also using Lemma 3.7, we deduce
(4.1)
ℰ
λ
(
R
u
~
λ
)
=
R
2
(
1
2
𝒟
λ
(
u
~
λ
,
u
~
λ
)
-
∫
ℝ
N
F
(
R
u
~
λ
)
(
R
u
~
λ
)
2
u
~
λ
2
𝑑
x
)
≤
R
2
(
1
2
(
1
-
1
β
k
0
*
)
+
o
λ
(
1
)
-
∫
ℝ
N
F
(
R
u
~
λ
)
(
R
u
~
λ
)
2
u
~
λ
2
𝑑
x
)
.
On the other hand, since u~λ∈⊕i=1k0*𝒩λ,i, by Lemmas 3.2, 3.4 and 3.6, we have u~λ=u~+oλ(1) strongly in H2(ℝN) for some u~∈⊕i=1k0*𝒩i with ∥u~∥H2=1, where ∥⋅∥H is the norm in H given by ∥⋅∥H=〈⋅,⋅〉H1/2. This together with conditions (F1)–(F2) gives that
(4.2)
∫
ℝ
N
F
(
R
u
~
λ
)
(
R
u
~
λ
)
2
u
~
λ
2
𝑑
x
=
∫
ℝ
N
F
(
R
u
~
)
(
R
u
~
)
2
u
~
2
𝑑
x
+
o
λ
(
1
)
.
Since dim(⊕i=1k0*𝒩i)<+∞, we see that there exists d*>0 such that
(4.3)
∥
u
∥
H
≤
d
*
∥
u
∥
L
p
(
Ω
)
for all
u
∈
⊕
i
=
1
k
0
*
𝒩
i
.
By conditions (F1) and (F4), we can see that
F
(
t
)
t
2
=
l
0
2
+
∫
0
t
(
F
(
s
)
s
2
)
′
𝑑
s
≥
l
0
2
+
l
*
p
-
2
|
t
|
p
-
2
.
It follows from p>2 that
∫
ℝ
N
F
(
R
u
~
)
R
2
u
~
2
u
~
2
𝑑
x
≥
l
*
R
p
-
2
p
-
2
∥
u
~
∥
L
p
(
Ω
)
p
≥
R
p
-
2
l
*
d
*
p
(
p
-
2
)
.
Now, by (4.1) and (4.2), there exist Λ¯0>Λ10 and R0>ρ0 independent of λ such that
sup
∂
𝒬
λ
,
R
0
ℰ
λ
(
u
)
≤
0
for all λ>Λ¯0. ∎
By arguments similar to those used for (4.3), we can obtain that
(4.4)
∥
u
∥
Ω
,
0
≤
d
*
*
∥
u
∥
L
2
(
Ω
)
for all
u
∈
⊕
i
=
1
k
0
*
𝒩
i
.
Then, we have the following.
Lemma 4.2
Suppose that conditions (B1)–(B3) and (F1)–(F3) hold with p=2, min{a0,b0}<0 and b0≤μ¯12. If
l
0
d
0
<
(
1
-
1
β
k
0
*
0
)
<
l
∞
d
*
*
,
then there exists Λ~0>0 such that the conclusions of Lemma 4.1 hold for λ>Λ~0, where d** is given by (4.4).
Proof.
As in the proof of Lemma 4.1, we can show that there exists Λ20>Λk0** such that
inf
(
ℳ
λ
k
0
*
-
1
)
⊥
∩
𝔹
λ
,
ρ
~
0
ℰ
λ
(
u
)
≥
κ
~
0
for all λ>Λ20 with some κ~0>0 and ρ~0>0 independent of λ. In what follows, we will prove that there exists a positive constant R~0 (>ρ~0) so large that
sup
∂
𝒬
λ
,
R
~
0
ℰ
λ
(
u
)
≤
0
for λ sufficient large. In fact, if uλ∈∂𝒬λ,R is such that uλ=Ru~λ with u~λ∈𝒬λ,1 then one of the following two cases must happen:
u~λ∈⊕i=1k0*-1𝒩λ,i and ∥u~λ∥λ≤1;
u~λ∈𝒬λ,1∖⊕i=1k0*-1𝒩λ,i and ∥u~λ∥λ=1.
In case (1), it follows from Lemma 3.7 and condition (F3) that ℰλ(Ru~λ)≤0 for all R≥0 and λ>Λ20. In case (2), by arguments similar to those used in the proofs of (4.1) and Lemma 3.7, we have
(4.5)
ℰ
λ
(
R
u
~
λ
)
≤
R
2
(
1
2
(
1
-
1
β
k
0
*
)
+
o
λ
(
1
)
-
∫
ℝ
N
F
(
R
u
~
λ
)
(
R
u
~
λ
)
2
u
~
λ
2
𝑑
x
)
and
(4.6)
∫
ℝ
N
F
(
R
u
~
λ
)
(
R
u
~
λ
)
2
u
~
λ
2
𝑑
x
=
∫
ℝ
N
F
(
R
u
~
)
(
R
u
~
)
2
u
~
2
𝑑
x
+
o
λ
(
1
)
for some u~∈⊕i=1k0*𝒩i with ∥u~∥H2=1. Note that ∫ℝNF(Ru~)(Ru~)2u~2𝑑x is continuous on ⊕i=1k0*𝒩i. By conditions (F1)–(F3), we have
inf
⊕
i
=
1
k
0
*
𝒩
i
∩
𝔹
1
∫
ℝ
N
F
(
R
u
~
)
(
R
u
~
)
2
u
~
2
𝑑
x
=
∫
ℝ
N
F
(
R
u
~
R
)
(
R
u
~
R
)
2
u
~
R
2
𝑑
x
>
0
for some
u
~
R
∈
⊕
i
=
1
k
0
*
𝒩
i
∩
𝔹
1
,
where
𝔹
1
=
{
u
∈
H
0
1
(
Ω
)
:
∥
u
∥
H
2
=
1
}
.
Since dim(⊕i=1k0*𝒩i)<+∞, we also have that u~R→u~*∈⊕i=1k0*𝒩i with ∥u~*∥H2=1 strongly as R→+∞. Now, thanks to the Fatou lemma, we deduce from condition (F2) and (4.4) that
lim
R
→
+
∞
∫
ℝ
N
F
(
R
u
~
R
)
(
R
u
~
R
)
2
u
~
R
2
𝑑
x
≥
∫
ℝ
N
lim
R
→
+
∞
F
(
R
u
~
R
)
R
2
u
~
R
2
u
~
R
2
d
x
≥
l
∞
2
d
*
*
.
Since l∞/d**>1-1/βk0*, by (4.5) and (4.6), there exist Λ~0≥Λ2* and R~0>ρ~0 independent of λ such that sup∂𝒬λ,R0ℰλ(u)≤0 for all λ>Λ~0. ∎
By Lemmas 4.1 and 4.2, we know that ℰλ(u) has a linking structure in Eλ for all λ>max{Λ¯0,Λ~0} in the case of min{a0,b0}<0 and b0≤μ¯12. By the well-known linking theorem, ℰλ(u) has a Cerami sequence at level cλ ((C)cλ sequence for short) for all λ>max{Λ¯0,Λ~0}, that is, there exists {uλ,n}⊂Eλ with λ>max{Λ¯0,Λ~0} such that
ℰ
λ
(
u
λ
,
n
)
=
c
λ
+
o
n
(
1
)
and
(
1
+
∥
u
λ
,
n
∥
λ
)
ℰ
λ
′
(
u
λ
,
n
)
=
o
n
(
1
)
strongly in
E
λ
*
.
In the special case k0*=1, the linking structure is actually the mountain pass geometry due to Remark 3.8 and the linking theorem can be replaced by the well known mountain pass theorem. Moreover, due to conditions (F3) for p=2 and (F4) for p>2, we can see that cλ∈[min{κ0,κ~0},12max{R02,R~02}].
We next consider the case of min{a0,b0}≥0.
Lemma 4.3
Suppose that conditions (F1)–(F2) hold and min{a0,b0}≥0. Then there exists l∞*>0 such that
inf
𝔹
ρ
0
*
ℰ
λ
(
u
)
≥
κ
0
*
𝑎𝑛𝑑
ℰ
λ
(
R
0
*
ϕ
1
)
≤
0
hold for λ>max{0,-b0/b∞} with some R0*>ρ0*>0 and κ0*>0 independent of λ in the following two cases:
Proof.
Since 𝒟λ(u,v)=〈u,v〉λ for all (u,v)∈Eλ and 𝒟λ(u,u) is definite on Eλ for λ>max{0,-b0/b∞} in the case of min{a0,b0}≥0, we can get the conclusions by arguments similar to but simpler than those used in the proofs of Lemmas 4.1 and 4.2. ∎
Due to Lemma 4.3, we can see that ℰλ(u) has a mountain pass geometry for λ>max{0,-b0/b∞} in the case of min{a0,b0}≥0. It follows from the well-known mountain pass theorem that ℰλ(u) has a (C)cλ sequence for all λ>max{0,-b0/b∞}. Furthermore,
c
λ
∈
[
κ
0
*
,
1
2
(
R
0
*
)
2
(
μ
¯
1
2
+
a
0
μ
¯
1
+
b
0
)
]
.
In other words, there exists Λ¯*,0>0 such that ℰλ(u) always has a (C)cλ sequence for λ>Λ¯*,0 and cλ∈[C,C′] in both cases of min{a0,b0}≥0 and min{a0,b0}<0 with b0≤μ¯12.
Lemma 4.4
Suppose that conditions (B1)–(B3) and (F1)–(F2) and (F4) hold with p>2. Then there exists Λ¯1>Λ¯*,0 such that ∥uλ,n∥λ≤C+on(1) uniformly for all λ>Λ¯1.
Proof.
By condition (F4), we have
o
n
(
1
)
+
c
λ
≥
ℰ
λ
(
u
λ
,
n
)
-
1
2
ℰ
λ
(
u
λ
,
n
)
u
λ
,
n
=
1
2
∫
ℝ
N
(
f
(
u
λ
,
n
)
u
λ
,
n
-
2
F
(
u
λ
,
n
)
)
𝑑
x
≥
l
*
2
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
p
.
On the other hand, due to conditions (B1)–(B2), for all λ>Λ¯*,0, we get from the Hölder inequality and the Gagliardo–Nirenberg inequality that
𝒢
λ
(
u
λ
,
n
,
u
λ
,
n
)
≤
max
{
-
a
0
,
0
}
B
0
2
∥
Δ
u
λ
,
n
∥
L
2
(
ℝ
N
)
∥
u
λ
,
n
∥
L
2
(
ℝ
N
)
+
max
{
-
b
0
,
0
}
|
ℬ
∞
|
p
-
2
p
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
2
≤
max
{
-
a
0
,
0
}
B
0
2
|
ℬ
∞
|
p
-
2
2
p
∥
Δ
u
λ
,
n
∥
L
2
(
ℝ
N
)
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
+
max
{
-
a
0
,
0
}
B
0
2
(
1
λ
b
∞
+
b
0
)
1
2
∥
u
λ
,
n
∥
λ
2
+
max
{
-
b
0
,
0
}
|
ℬ
∞
|
p
-
2
p
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
2
≤
(
1
2
+
max
{
-
a
0
,
0
}
B
0
2
(
1
λ
b
∞
+
b
0
)
1
2
)
∥
u
λ
,
n
∥
λ
2
+
(
2
max
{
-
a
0
,
0
}
2
B
0
4
+
max
{
-
b
0
,
0
}
)
|
ℬ
∞
|
p
-
2
p
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
2
.
It follows from conditions (B2), (F1)–(F2) and the Hölder inequality that for all λ>Λ¯*,0, we have
|
∫
ℝ
N
F
(
u
λ
,
n
)
𝑑
x
|
≤
max
{
2
l
0
,
l
0
+
ε
}
∥
u
λ
,
n
∥
L
2
(
ℝ
N
)
2
+
C
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
p
≤
max
{
2
l
0
,
l
0
+
ε
}
λ
b
∞
+
b
0
∥
u
λ
,
n
∥
λ
2
+
max
{
2
l
0
,
l
0
+
ε
}
|
ℬ
∞
|
p
-
2
p
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
2
+
C
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
p
.
Now, we can see from ℰλ(uλ,n)=cλ+on(1) that
(
1
2
+
o
λ
(
1
)
)
∥
u
λ
,
n
∥
λ
2
≤
C
′
c
λ
+
o
n
(
1
)
.
Note that by cλ∈[C,C′], there exist Λ¯1>Λ¯*,0 such that ∥uλ,n∥λ≤C+on(1) uniformly for all λ>Λ¯1. ∎
Now, we can give the proof of Theorem 1.1.
Proof of Theorem 1.1.
By Lemma 4.4, we have uλ,n⇀uλ,0 weakly in Eλ with λ>Λ¯1 as n→∞ up to a subsequence. Without loss of generality, we may assume that uλ,n⇀uλ,0 weakly in Eλ with λ>Λ¯1 as n→∞. Since ℰλ(u) is C1, it is easy to see from the fact that {uλ,n} is a (C)cλ sequence that ℰλ′(uλ,0)=0 with λ>Λ¯1. It remains to show that uλ,0≠0 in Eλ for λ sufficiently large. Indeed, if uλ,0=0, then by conditions (B1)–(B2) and Lemma 2.1, we can see from the Sobolev embedding theorem and the Gagliardo–Nirenberg inequality that
𝒢
λ
(
u
λ
,
n
,
u
λ
,
n
)
≤
max
{
-
a
0
,
0
}
B
0
2
∥
Δ
u
λ
,
n
∥
L
2
(
ℝ
N
)
∥
u
λ
,
n
∥
L
2
(
ℝ
N
)
+
o
n
(
1
)
≤
max
{
-
a
0
,
0
}
B
0
2
(
(
1
λ
b
∞
+
b
0
)
1
2
+
o
n
(
1
)
)
∥
u
λ
,
n
∥
λ
2
+
o
n
(
1
)
.
On the other hand, by conditions (F1)–(F2), we have
|
∫
ℝ
N
F
(
u
λ
,
n
)
𝑑
x
|
≤
max
{
2
l
0
,
l
0
+
ε
}
∥
u
λ
,
n
∥
L
2
(
ℝ
N
)
2
+
C
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
p
.
Thanks to the Sobolev embedding theorem and the Hölder inequality, we can see that
∥
u
λ
,
n
∥
L
p
(
ℝ
N
)
p
≤
{
C
∥
u
λ
,
n
∥
H
2
(
ℝ
N
)
p
-
2
∥
u
λ
,
n
∥
L
2
(
ℝ
N
)
2
for
N
=
3
,
C
∥
u
λ
,
n
∥
H
2
(
ℝ
N
)
p
p
-
1
∥
u
λ
,
n
∥
L
2
(
ℝ
N
)
p
(
p
-
2
)
p
-
1
for
N
=
4
,
C
∥
u
λ
,
n
∥
H
2
(
ℝ
N
)
2
*
(
p
-
2
)
2
*
-
2
∥
u
λ
,
n
∥
L
2
(
ℝ
N
)
2
*
(
2
*
-
p
)
2
*
-
2
for
N
≥
5
.
It follows from Lemma 2.1 and once more from the Sobolev embedding theorem that
|
∫
ℝ
N
F
(
u
λ
,
n
)
𝑑
x
|
≤
o
λ
(
1
)
∥
u
λ
,
n
∥
λ
2
+
o
n
(
1
)
,
which, together with ℰλ(uλ,n)uλ,n=on(1) and Lemma 4.4, yields that there exists Λ^>Λ¯1 such that uλ,n→0 strongly in Eλ with λ>Λ^ as n→∞. It follows that cλ=0 for λ>Λ^. This is impossible since cλ≥C>0 for all λ>Λ¯1. ∎
Let us consider the case p=2 in what follows.
Lemma 4.5
Suppose that conditions (B1)–(B3) and (F1)–(F3) hold with p=2. If l∞∉σ(Δ2-a0Δ+b0,L2(Ω)), then there exists Λ~1>Λ¯*,0 such that {uλ,n} is bounded in Eλ for all λ>Λ~1.
Proof.
Suppose on the contrary that there exists a subsequence of {uλ,n}, which is still denoted by {uλ,n}, such that ∥uλ,n∥λ→+∞ as n→+∞. Let wλ,n=uλ,n/∥uλ,n∥λ. Then without loss of generality, we may assume that wλ,n⇀wλ,0 weakly in Eλ for some wλ,0∈Eλ as n→∞.
Claim 1: There exists Λ30>Λ¯*,0 such that wλ,0≠0.
Indeed, if wλ,0=0, then by Remark 3.8, we can see that wλ,n(1)→0 strongly in Eλ with λ>Λ¯*,0 as n→∞, where wλ,n(1) is the projection of wλ,n in ⊕i=1k0*-1𝒩λ,i. It follows from Lemma 3.7 that
(4.7)
𝒟
λ
(
w
λ
,
n
,
w
λ
,
n
)
≥
(
1
-
1
β
k
0
*
0
+
o
λ
(
1
)
)
∥
w
λ
,
n
(
2
)
∥
λ
2
+
o
n
(
1
)
,
where wλ,n(2)=wλ,n-wλ,n(1). On the other hand, thanks to conditions (F2)–(F3), we see from the fact that (1+∥uλ,n∥λ)ℰλ′(uλ,n)=on(1) strongly in Eλ* that
𝒟
λ
(
w
λ
,
n
,
w
λ
,
n
)
=
o
n
(
1
)
+
∫
ℝ
N
f
(
u
λ
,
n
)
u
λ
,
n
w
λ
,
n
2
𝑑
x
≤
o
n
(
1
)
+
l
∞
∥
w
λ
,
n
∥
L
2
(
ℝ
N
)
2
,
which, together with the Sobolev embedding theorem, Lemma 2.1 and condition (B2), implies that
(4.8)
𝒟
λ
(
w
λ
,
n
,
w
λ
,
n
)
≤
l
∞
λ
b
∞
+
b
0
∥
w
λ
,
n
(
2
)
∥
λ
2
+
o
n
(
1
)
.
Thanks to Lemma 3.6, we can deduce from (4.7) and (4.8) that there exists Λ30>Λ¯*,0 such that wλ,n(2)→0 strongly in Eλ as n→∞ for λ>Λ30, which is inconsistent with ∥wλ,n∥λ=1 for all n∈ℕ.
Claim 2: There exists Λ40>Λ30 such that wλ,n→wλ,0 strongly in Eλ as n→∞ for λ>Λ40 up to a subsequence.
In fact, let 𝒬λ,0={x∈ℝN:wλ,0≠0}; then |uλ,n|→+∞ as n→∞ on 𝒬λ,0. It follows from conditions (F2)–(F3) and a variant of the Lebesgue dominated convergence theorem (cf. [23, Theorem 2.2]) that
lim
n
→
∞
∫
ℝ
N
f
(
u
λ
,
n
)
v
∥
u
λ
,
n
∥
λ
𝑑
x
=
lim
n
→
∞
∫
ℝ
N
f
(
u
λ
,
n
)
u
λ
,
n
w
λ
,
n
v
χ
𝒬
λ
,
0
𝑑
x
=
∫
ℝ
N
l
∞
w
λ
,
0
v
𝑑
x
for every v∈H2(ℝN). Since wλ,n⇀wλ,0 weakly in Eλ as n→∞, due to Lemma 2.1, we have from condition (B2) that limn→∞𝒟λ(wλ,n,v)=𝒟λ(wλ,0,v) for all v∈H2(ℝN). Thus, wλ,0∈H2(ℝN) satisfies the following equation in the weak sense:
(4.9)
Δ
2
w
λ
,
0
-
a
0
Δ
w
λ
,
0
+
(
λ
b
(
x
)
+
b
0
)
w
λ
,
0
=
l
∞
w
λ
,
0
in
ℝ
N
.
Let
I
λ
(
u
)
=
1
2
(
𝒟
λ
(
u
,
u
)
-
l
∞
∥
u
∥
L
2
(
ℝ
N
)
2
)
.
Then by (4.9), we have Iλ′(wλ,0)=0. Now, by Remark 3.8 and an argument similar to that used in the proof of (3.5), we have
o
n
(
1
)
=
(
ℰ
λ
′
(
u
λ
,
n
)
∥
u
λ
,
n
∥
λ
-
I
λ
′
(
w
λ
,
0
)
)
(
w
λ
,
n
-
w
λ
,
0
)
≥
(
1
-
1
β
k
0
*
0
+
o
λ
(
1
)
)
∥
w
λ
,
n
-
w
λ
,
0
∥
λ
2
+
o
n
(
1
)
-
∫
ℝ
N
f
(
u
λ
,
n
)
u
λ
,
n
|
w
λ
,
n
-
w
λ
,
0
|
2
𝑑
x
(4.10)
-
∫
ℝ
N
(
f
(
u
λ
,
n
)
u
λ
,
n
-
l
∞
)
(
w
λ
,
n
-
w
λ
,
0
)
w
λ
,
0
𝑑
x
.
Note that (wλ,n-wλ,0)wλ,0=on(1) strongly in L1(ℝN). By conditions (F2)–(F3), the Sobolev embedding theorem and a variant of the Lebesgue dominated convergence theorem (cf. [23, Theorem 2.2]), we can see that
(4.11)
∫
ℝ
N
f
(
u
λ
,
n
)
u
λ
,
n
|
w
λ
,
n
-
w
λ
,
0
|
2
𝑑
x
+
∫
ℝ
N
(
f
(
u
λ
,
n
)
u
λ
,
n
-
l
∞
)
(
w
λ
,
n
-
w
λ
,
0
)
w
λ
,
0
𝑑
x
≤
l
∞
λ
b
∞
+
b
0
∥
w
λ
,
n
-
w
λ
,
0
∥
λ
2
+
o
n
(
1
)
.
Combining (4.10) and (4.11), we obtain that there exists Λ40>Λ30 such that wλ,n→wλ,0 strongly in Eλ as n→∞ for λ>Λ40.
Claim 3:wλ,0→w∞,0 strongly in H1(ℝN) as λ→+∞ up to a subsequence for some w∞,0∈H which satisfies the following equation in the weak sense:
(4.12)
Δ
2
w
∞
,
0
-
a
0
Δ
w
∞
,
0
+
b
0
w
∞
,
0
=
l
∞
w
∞
,
0
in
Ω
.
Indeed, since ∥wλ,n∥λ=1, by Lemma 2.1 and an argument similar to that used in the proof of Lemma 3.2, we can show that wλ,0⇀w∞,0 weakly in H2(ℝN) and wλ,0→w∞,0 strongly in H1(ℝN)∩L2(ℝN) for some w∞,0∈H with w∞,0=0 outside Ω as λ→+∞, up to a subsequence. Multiplying (4.9) with φ∈H, we can see once more from (4.9) that w∞,0∈H satisfies (4.12) in the weak sense.
Now, multiplying (4.12) and (4.9) with w∞,0 and wλ,0, respectively, and integrating, we can see from condition (B1) that
∥
Δ
w
∞
,
0
∥
L
2
(
ℝ
N
)
2
+
a
0
∥
∇
w
∞
,
0
∥
L
2
(
ℝ
N
)
2
+
b
0
∥
w
∞
,
0
∥
L
2
(
ℝ
N
)
2
≤
lim
λ
→
+
∞
(
∥
w
λ
,
0
∥
λ
2
+
𝒢
λ
(
w
λ
,
0
,
w
λ
,
0
)
)
=
l
∞
lim
λ
→
+
∞
∥
w
λ
,
0
∥
L
2
(
ℝ
N
)
2
=
l
∞
∥
w
∞
,
0
∥
L
2
(
ℝ
N
)
2
=
∥
Δ
w
∞
,
0
∥
L
2
(
ℝ
N
)
2
+
a
0
∥
∇
w
∞
,
0
∥
L
2
(
ℝ
N
)
2
+
b
0
∥
w
∞
,
0
∥
L
2
(
ℝ
N
)
2
.
Hence, ∫ℝNλb(x)wλ,02𝑑x=oλ(1) and wλ,0→w∞,0 strongly in H2(ℝN) as λ→+∞ up to a subsequence. Therefore, by claim 1 and claim 2, we have ∥w∞,0∥H=1, which gives w∞,0≠0. It follows from (4.12) that l∞∈σ(Δ2-a0Δ+b0,L2(Ω)), which contradicts the assumption that l∞∉σ(Δ2-a0Δ+b0,L2(Ω)). Thus, there exists Λ~1>Λ¯*,0 such that {uλ,n} is bounded in Eλ for all λ>Λ~1. ∎
Now, we give the proofs of Theorems 1.2 and 1.3.
Proof of Theorem 1.2.
By Lemma 4.5, we have that {uλ,n} is bounded in Eλ for λ>Λ~1. Without loss of generality, we may assume uλ,n⇀uλ,0 weakly in Eλ as n→∞ for λ>Λ~1. Clearly, ℰλ′(uλ,0)=0. We claim that {∥uλ,0∥λ} is bounded for λ large enough. Indeed, suppose on the contrary that there exists λm→+∞ as m→∞ such that ∥uλm,0∥λm→∞ as m→∞. Let wm=uλm,0/∥uλm,0∥λm. Then without loss of generality, we may assume that wm⇀w0 weakly in H2(ℝN) as m→∞ for some w0∈H2(ℝN). By ℰλ′(uλ,0)=0 and an argument similar to that used in the proof of claim 3 of Lemma 4.5, we can show that w0 is a nontrivial weak solution of (4.12), which is inconsistent with the assumption that l∞∉σ(Δ2-a0Δ+b0,L2(Ω)). Thus, {∥uλ,0∥λ} is bounded for λ large enough. Now, by an argument similar to that used in the proof of Theorem 1.1, we can obtain the conclusion. ∎
Proof of Theorem 1.3.
Suppose that uλ is the nontrivial solution of (𝒫λ) obtained by Theorem 1.1 or Theorem 1.2 with λ large enough. We can see from Lemma 4.4 and the proof of Theorem 1.2 that ∥uλ∥λ≤C0 for all λ with some C0>0 independent of λ. By Lemma 2.1, we may assume that uλ⇀u* weakly in H2(ℝN) as λ→+∞ for some u*∈H2(ℝN). Since condition (B2) holds, as in the proof of Lemma 4.5, we can show that uλ→u* strongly in L2(ℝN) as λ→+∞ and u*∈H with u*≡0 outside Ω. Thanks to the Gagliardo–Nirenberg inequality, we also have that uλ→u* strongly in H1(ℝN) as λ→+∞. Multiplying (𝒫λ) with φ∈H, we can see from condition (B3) that u* satisfies (1.3) in the weak sense. Now, by conditions (B1), (F1)–(F2) and Lemma 2.2, we can see that
∥
Δ
u
*
∥
L
2
(
ℝ
N
)
2
+
a
0
∥
∇
u
*
∥
L
2
(
ℝ
N
)
2
+
b
0
∥
u
*
∥
L
2
(
ℝ
N
)
2
≤
𝒟
λ
(
u
λ
,
u
λ
)
+
o
λ
(
1
)
=
∫
ℝ
N
f
(
u
λ
)
u
λ
𝑑
x
+
o
λ
(
1
)
=
∫
ℝ
N
f
(
u
*
)
u
*
𝑑
x
+
o
λ
(
1
)
.
Thus, λ∫ℝNb(x)uλ2𝑑x=oλ(1) and uλ→u* strongly in H2(ℝN) as λ→+∞. Note that ℰλ(uλ)=cλ≥C>0 for all λ>max{Λ¯1,Λ~1}, hence we must have u*≠0 in H. Thus, u* is a nontrivial weak solution of (1.3). ∎
