1 Introduction
In this paper, we are concerned with the existence of multiple solutions of the p-Laplacian equation in ℝN,
($P_{0}$)
{
-
Δ
p
u
+
a
(
x
)
|
u
|
p
-
2
u
=
|
u
|
q
-
2
u
in
ℝ
N
,
u
∈
W
1
,
p
(
ℝ
N
)
,
where 1<p<N, p<q<p*=NpN-p, Δp is the p-Laplacian operator, Δpu=∇(|∇u|p-2∇u), and the potential function a is positive, finite and verifies suitable decay assumptions.
Problem ($P_{0}$) has a variational structure, given by the functional I: W1,p(ℝN)→ℝ defined by
I
(
u
)
=
1
p
∫
ℝ
N
(
|
∇
u
|
p
+
a
(
x
)
|
u
|
p
)
𝑑
x
-
1
q
∫
ℝ
N
|
u
|
q
𝑑
x
.
The embedding W1,p(ℝN)↪Lq(ℝN) is continuous, but not compact due to the translations, therefore the Palais–Smale condition is not satisfied by I, that is, problem ($P_{0}$) lacks the compactness property. Since the pioneering work [3] of H. Brezis and L. Nirenberg, significant progress has been made in recent decades. In particular, Devillanova and Solimini [7] dealt with the critical growth problem
($P_{a}$)
{
-
Δ
u
=
|
u
|
2
*
-
2
u
+
λ
u
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is an open regular domain of ℝN (N≥3) and 2*=2NN-2 is the critical exponent, and proved a multiplicity result. The solutions are found as limits of solutions of approximated problems with subcritical growth. The lack of compactness due to the scaling does not allow to deduce that a sequence of approximated solutions must have a converging subsequence but the fact that they solve the approximated problems gives, with the use of a local Pohožaev identity, some extra estimates which lead to a proof of the desired compactness.
This argument was adapted in [5] to problem ($P_{0}$) with p=2. Cerami, Devillanova and Solimini [5] considered the following related problem in a sequence of balls in ℝN:
($P_{b}$)
{
-
Δ
u
+
a
(
x
)
u
=
|
u
|
q
-
2
u
in
B
R
,
u
=
0
on
∂
B
R
,
where BR is a ball in ℝN, BR={x∣x∈ℝN,|x|<R}. Problem ($P_{b}$) possesses infinitely many solutions, which are used as approximation solutions for the problem in the whole space.
Another way to deal with the problem in ℝN is to introduce a coercive potential. Namely we first consider the equation
($P_{W}$)
{
-
Δ
u
+
(
ε
W
(
x
)
+
a
(
x
)
)
u
=
|
u
|
q
-
2
u
in
ℝ
N
,
u
∈
H
W
,
where ε>0, W is a coercive function, say W(x)=1+|x|2, and HW is the Hilbert space
H
W
=
{
u
|
u
∈
H
1
(
ℝ
N
)
,
∫
ℝ
N
W
(
x
)
u
2
𝑑
x
<
+
∞
}
equipped with the norm
∥
u
∥
H
W
=
(
∫
ℝ
N
(
|
∇
u
|
2
+
W
(
x
)
u
2
)
𝑑
x
)
1
2
.
The embedding from HW to Lq(ℝN) (2≤q<2*=2NN-2) is compact and hence problem ($P_{W}$) possesses infinitely many solutions. By taking the limit ε→0, we obtain solutions of the original problem with finite potential. This approach was introduced in [8]. In fact, Liu and Wang [8] considered the existence of solutions of some quasilinear elliptic equations, a model problem is the so-called modified nonlinear Schrödinger equation
($P_{m}$)
Δ
u
+
1
2
u
Δ
u
2
-
V
(
x
)
u
+
|
u
|
p
-
2
u
=
0
in
ℝ
N
with p∈(4,4NN-2).
For the p-Laplacian problems with critical nonlinearities, one may meet with different technical difficulties for different problems, and therefore there are different ways to deal with these kinds of problems. In [4], Cao, Peng and Yan considered the following p-Laplacian problem with critical Sobolev exponent:
{
-
Δ
p
u
=
|
u
|
p
*
-
2
u
+
μ
|
u
|
p
-
2
u
in
Ω
,
u
=
0
on
∂
Ω
,
where 1<p<N, p*=NpN-p, μ>0 and Ω is an open bounded domain in ℝN. They first investigated the equations of subcritical growth and then proved a convergence theorem to obtain solutions of the original problem, following the argument in [7].
As for the existence of multiple solutions of the original problem we need to check that the multiple solutions of the approximated problems do not converge all to the same solution of the limit problem. In both [5] and [7], some estimates on the Morse index are employed, which has been used as one of possible devices to distinguish the limits of the multiple approximated solutions by their original variational characterization. For general p-Laplacian equations with p≠2 we have no information on the Morse index, therefore the approaches in [7, 5, 8] can not be extended in a straightforward way to problems involving Δp with p≠2.
In [4], Cao, Peng and Yan proved the existence of multiple solutions, with the help of a claim that if the functional corresponding to the original problem has only finitely many critical values, then the critical set has genus at least two. At this last step the argument seems to be not so easy to follow (see [4, (4.20)]). Actually this is one of the motivations for us to use the truncation method.
Note that every solution of problem ($P_{0}$) decays exponentially. Inspired by this fact, in this paper we are going to use the truncation technique to obtain infinitely many solutions of problem ($P_{0}$). We first consider truncated problems, solutions of which will be used as approximating solutions. By a compactness argument, similar to that in [7, 5], in particular with the use of the Pohožaev formula, theorems of convergence of solutions of approximating problems and existence of multiple solutions of the original problem ($P_{0}$) are proved. The advantage of the truncation method is that the original problems and the approximating problems share some common solutions, more and more as the parameter tends to zero.
Moreover, the truncation method mentioned in this paper may be used to deal with the p-Laplacian problem with critical Sobolev exponent. Although the problem involves critical Sobolev exponent, solutions of the problem are still bounded. So we can use the truncation technique to reduce the critical problem to a subcritical one, see Section 4 for some details.
Let us describe the truncation method in more details. Let φ∈C0∞(ℝ) be such that φ(t)=1 for |t|≤1, φ(t)=0 for |t|≥2, φ is decreasing in [1,2], and φ is even. For λ>0, (x,s)∈ℝN×ℝ, define
b
λ
(
x
,
s
)
=
φ
(
λ
e
λ
1
+
|
x
|
2
s
)
,
m
λ
(
x
,
s
)
=
∫
0
s
b
λ
(
x
,
τ
)
𝑑
τ
,
F
λ
(
x
,
s
)
=
1
q
|
s
|
r
|
m
λ
(
x
,
s
)
|
q
-
r
,
p
<
r
<
q
,
f
λ
(
x
,
s
)
=
∂
∂
s
F
λ
(
x
,
s
)
,
where r∈(p,q) is a fixed number, for example r=12(p+q). For λ=0 we understand that m0(x,s)≡s, F0(x,s)≡1q|s|q and f0(x,s)≡|s|q-2s. We consider the equation
($P_{\lambda}$)
{
-
Δ
p
u
+
a
(
x
)
|
u
|
p
-
2
u
=
f
λ
(
x
,
u
)
in
ℝ
N
,
u
∈
W
1
,
p
(
ℝ
N
)
.
The embedding from W1,p(ℝN) to Lq(ℝN) (p<q<NpN-p) is not compact, but if un⇀u in W1,p(ℝN), then mλ(⋅,un)→mλ(⋅,u) in Lq(ℝN) for λ>0. As a problem with compactness, ($P_{\lambda}$), λ>0, possesses infinitely many solutions. The truncation method has its advantages, that is, if u is a solution of ($P_{\lambda}$), λ>0, and satisfies the estimate
(1.1)
|
u
(
x
)
|
≤
1
λ
e
-
λ
1
+
|
x
|
2
for all
x
∈
ℝ
N
,
then automatically u will be a solution of the original problem ($P_{0}$), too. We can prove that as λ→0 more and more solutions u of problem ($P_{\lambda}$) satisfy the above estimate (1.1), hence we obtain infinitely many solutions of problem ($P_{0}$).
The truncation technique is not new. People use this technique to deal with various problems, because of various reasons. For example, in equation ($P_{a}$) with 0<λ<λ1, we replace the term |u|2*-2u by u+2*-1 in order to obtain a positive solution. In the quasilinear equation ($P_{m}$) we replace the quasilinear term uΔu2 by uMΔ(uuM), where uM=u if |u|≤M, uM=±M if ±u≥M. See also [9, 10] for Hamiltonian systems and for nonlinear wave equations. But as to the authors’ knowledge, the application of the truncation technique to a problem on the unbounded domains by introducing a decay function, such as 1λe-λ1+|x|2, in order to overcome the difficulty of lack of compactness seems to be new.
Now we make the following assumptions on the potential function (see [5]):
There exist a1,a0>0 such that a0≤a(x)≤a1, x∈ℝN.
For all α>0, we have lim|x|→∞∂a∂r(x)eα|x|=+∞, where ∂a∂r(x)=x|x|⋅∇a(x).
There exists c¯>1 such that |∇a(x)|≤c¯∂a∂r(x) for all x∈ℝN, |x|≥c¯.
Our main results are the following two theorems.
Theorem 1.1.
Assume (a1)–(a4). Given M>0, there exists μ=μ(M) such that if u∈W1,p(RN) is a solution of problem ($P_{\lambda}$), λ≥0 and ∥u∥≤M. Then
|
u
(
x
)
|
≤
1
μ
e
-
μ
1
+
|
x
|
2
for all
x
∈
ℝ
N
.
Theorem 1.2.
Assume (a1)–(a4). Problem ($P_{0}$) has infinitely many solutions.
Throughout the paper, |⋅|p denotes the norm in Lp(ℝN), ∥⋅∥ denotes the norm in W1,p(ℝN), → and ⇀ denote the strong convergence and the weak convergence, respectively. Furthermore, c denotes a (possibly different) positive constant.
2 Uniform Bounds
Let un∈W1,p(ℝN) be a solution of problem ($P_{\lambda}$) with λ=λn≥0, n=1,2,…. Assume ∥un∥≤M. By [12, Theorem 2.1 and Theorem 3.3] (see also [13]), {un} has a profile decomposition
(2.1)
u
n
=
u
+
∑
k
∈
Λ
U
k
(
⋅
-
x
n
,
k
)
+
r
n
,
where u∈W1,p(ℝN), Uk∈W1,p(ℝN), k∈Λ, rn∈W1,p(ℝN) and {xn,k}⊂ℝN such that the following properties hold:
u
n
⇀
u
, un(⋅+xn,k)⇀Uk in W1,p(ℝN) as n→∞, k∈Λ.
|
x
n
,
k
|
→
∞
, |xn,k-xn,l|→∞ as n→∞, k,l∈Λ, k≠l.
|
u
|
q
q
+
∑
k
∈
Λ
|
U
k
|
q
q
≤
lim
¯
n
→
∞
|
u
n
|
q
q
, p<q<p*.
r
n
→
0
in Lq(ℝN) as n→∞, p<q<p*.
Lemma 2.1.
For (x,s)∈RN×R, the following properties hold:
|
s
|
b
λ
(
x
,
s
)
≤
|
m
λ
(
x
,
s
)
|
, smλ(x,s)≥0.
|
m
λ
(
x
,
s
)
|
≤
min
{
|
s
|
,
2
λ
e
-
λ
1
+
|
x
|
2
}
and
m
λ
(
x
,
s
)
=
s
if
|
s
|
≤
1
λ
e
-
λ
1
+
|
x
|
2
.
|
f
λ
(
x
,
s
)
|
≤
|
s
|
r
-
1
|
m
λ
(
x
,
s
)
|
q
-
r
.
1
r
s
f
λ
(
x
,
s
)
-
F
λ
(
x
,
s
)
=
q
-
r
q
r
|
s
|
r
+
1
|
m
λ
(
x
,
s
)
|
q
-
r
-
1
b
λ
(
x
,
s
)
.
∇
x
m
λ
(
x
,
s
)
=
-
λ
x
1
+
|
x
|
2
(
m
λ
(
x
,
s
)
-
s
b
λ
(
x
,
s
)
)
and
∇
x
F
λ
(
x
,
s
)
=
-
(
1
-
r
q
)
λ
x
1
+
|
x
|
2
|
s
|
r
|
m
λ
(
x
,
s
)
|
q
-
r
-
1
|
m
λ
(
x
,
s
)
-
s
b
λ
(
x
,
s
)
|
.
Proof.
(1) By the definition of mλ(x,s) and the intermediate value theorem, we have
m
λ
(
x
,
s
)
=
∫
0
s
b
λ
(
x
,
τ
)
𝑑
τ
=
s
b
λ
(
x
,
ξ
)
,
0
<
ξ
<
s
.
Then by the definition of bλ(x,s) and the fact that φ is decreasing in [1,2], we get
|
s
|
b
λ
(
x
,
s
)
=
|
s
|
φ
(
λ
e
λ
1
+
|
x
|
2
s
)
≤
|
s
|
φ
(
λ
e
λ
1
+
|
x
|
2
ξ
)
=
|
m
λ
(
x
,
s
)
|
.
(2) We consider the case s>0. Since bλ(x,s)≤1, we have
m
λ
(
x
,
s
)
=
∫
0
s
b
λ
(
x
,
τ
)
𝑑
τ
≤
s
.
For τ>2λe-λ1+|x|2, we have λeλ1+|x|2τ≥2 and bλ(x,τ)=φ(λeλ1+|x|2τ)=0. Then
m
λ
(
x
,
s
)
=
∫
0
s
b
λ
(
x
,
τ
)
𝑑
τ
≤
∫
0
2
λ
e
-
λ
1
+
|
x
|
2
b
λ
(
x
,
τ
)
𝑑
τ
≤
2
λ
e
-
λ
1
+
|
x
|
2
.
Hence,
m
λ
(
x
,
s
)
≤
min
{
s
,
2
λ
e
-
λ
1
+
|
x
|
2
}
.
For 0<τ≤s≤1λe-λ1+|x|2 we have
λ
e
λ
1
+
|
x
|
2
τ
≤
1
and
b
λ
(
x
,
τ
)
=
φ
(
λ
e
λ
1
+
|
x
|
2
τ
)
=
1
,
hence for s≤1λe-λ1+|x|2 we obtain
m
λ
(
x
,
s
)
=
∫
0
s
b
λ
(
x
,
τ
)
𝑑
τ
=
∫
0
s
𝑑
τ
=
s
.
(3) Since
(2.2)
f
λ
(
x
,
s
)
=
∂
F
λ
(
x
,
s
)
∂
s
=
r
q
|
s
|
r
-
2
s
|
m
λ
(
x
,
s
)
|
q
-
r
+
q
-
r
q
|
s
|
r
|
m
λ
(
x
,
s
)
|
q
-
r
-
2
m
λ
(
x
,
s
)
b
λ
(
x
,
s
)
and |s|bλ(x,s)≤|mλ(x,s)|, we have
|
f
λ
(
x
,
s
)
|
≤
r
q
|
s
|
r
-
1
|
m
λ
(
x
,
s
)
|
q
-
r
+
q
-
r
q
|
s
|
r
-
1
|
m
λ
(
x
,
s
)
|
q
-
r
=
|
s
|
r
-
1
|
m
λ
(
x
,
s
)
|
q
-
r
.
(4) By (2.2) and smλ(x,s)≥0, we get
1
r
s
f
λ
(
x
,
s
)
-
F
λ
(
x
,
s
)
=
q
-
r
q
r
|
s
|
r
s
|
m
λ
(
x
,
s
)
|
q
-
r
-
2
m
λ
(
x
,
s
)
b
λ
(
x
,
s
)
=
q
-
r
q
r
|
s
|
r
+
1
|
m
λ
(
x
,
s
)
|
q
-
r
-
1
b
λ
(
x
,
s
)
.
(5) By the definition of mλ(x,s), we obtain
∇
x
m
λ
(
x
,
s
)
=
∫
0
s
∇
x
b
λ
(
x
,
τ
)
𝑑
τ
=
∫
0
s
φ
′
(
λ
e
λ
1
+
|
x
|
2
τ
)
⋅
λ
e
λ
1
+
|
x
|
2
τ
⋅
λ
x
1
+
|
x
|
2
𝑑
τ
=
∫
0
s
λ
x
1
+
|
x
|
2
⋅
τ
𝑑
φ
(
λ
e
λ
1
+
|
x
|
2
τ
)
=
λ
x
1
+
|
x
|
2
(
s
φ
(
λ
e
λ
1
+
|
x
|
2
s
)
-
∫
0
s
φ
(
λ
e
λ
1
+
|
x
|
2
τ
)
𝑑
τ
)
=
-
λ
x
1
+
|
x
|
2
(
m
λ
(
x
,
s
)
-
s
b
λ
(
x
,
s
)
)
.
Thus,
∇
x
F
λ
(
x
,
s
)
=
q
-
r
q
|
s
|
r
|
m
λ
(
x
,
s
)
|
q
-
r
-
2
m
λ
(
x
,
s
)
∇
x
m
λ
(
x
,
s
)
=
-
(
1
-
r
q
)
λ
x
1
+
|
x
|
2
|
s
|
r
|
m
λ
(
x
,
s
)
|
q
-
r
-
1
|
m
λ
(
x
,
s
)
-
s
b
λ
(
x
,
s
)
|
.
∎
Lemma 2.2.
Let u∈W1,p(RN) be a solution of ($P_{\lambda}$), λ>0. Then v=|u| satisfies the following differential inequality:
(2.3)
∫
ℝ
N
|
∇
v
|
p
-
2
∇
v
∇
φ
d
x
+
a
0
∫
ℝ
N
v
p
-
1
φ
𝑑
x
≤
∫
ℝ
N
v
q
-
1
φ
𝑑
x
,
φ
∈
W
1
,
p
(
ℝ
N
)
,
φ
≥
0
.
Proof.
Set vε=u2+ε2-ε, ε>0. Then vε→v in W1,p(ℝN) as ε→0. By Lemma 2.1 (2) and (3), we have
|
f
λ
(
x
,
s
)
|
≤
|
s
|
r
-
1
|
m
λ
(
x
,
s
)
|
q
-
r
≤
|
s
|
q
-
1
.
Then for φ∈C0∞(ℝN), φ≥0, we have
∫
ℝ
N
|
∇
v
|
p
-
2
∇
v
ε
∇
φ
d
x
=
∫
ℝ
N
|
∇
u
|
p
-
2
∇
u
u
(
u
2
+
ε
2
)
1
2
∇
φ
d
x
=
∫
ℝ
N
|
∇
u
|
p
-
2
∇
u
∇
(
u
(
u
2
+
ε
2
)
1
2
φ
)
d
x
-
∫
ℝ
N
|
∇
u
|
p
ε
2
(
u
2
+
ε
2
)
3
2
φ
𝑑
x
≤
∫
ℝ
N
|
∇
u
|
p
-
2
∇
u
∇
(
u
(
u
2
+
ε
2
)
1
2
φ
)
d
x
=
-
∫
ℝ
N
a
(
x
)
|
u
|
p
-
2
u
⋅
u
(
u
2
+
ε
2
)
1
2
φ
𝑑
x
+
∫
ℝ
N
f
λ
(
x
,
u
)
u
(
u
2
+
ε
2
)
1
2
φ
𝑑
x
≤
-
a
0
∫
ℝ
N
v
p
-
1
v
(
v
2
+
ε
2
)
1
2
φ
𝑑
x
+
∫
ℝ
N
v
q
-
1
v
(
v
2
+
ε
2
)
1
2
φ
𝑑
x
.
Let ε→0. By Lebesgue’s dominated convergence theorem we obtain (2.3) for φ∈C0∞(ℝN), φ≥0. By approximation, (2.3) holds for φ∈W1,p(ℝN), φ≥0. ∎
Lemma 2.3.
Let un∈W1,p(RN) be a solution of ($P_{\lambda}$) with λ=λn≥0, n=1,2,…, and {xn}⊂RN. Suppose u~n=un(⋅+xn)⇀U in W1,p(RN). Then u~n→U in Wloc1,p(RN).
Proof.
The sequence u~n satisfies the equation
∫
ℝ
N
|
∇
u
~
n
|
p
-
2
∇
u
~
n
∇
φ
d
x
+
∫
ℝ
N
a
(
x
+
x
n
)
|
u
~
n
|
p
-
2
u
~
n
φ
𝑑
x
=
∫
ℝ
N
f
λ
n
(
x
+
x
n
,
u
~
n
)
φ
𝑑
x
for all φ∈W1,p(ℝN). Let R>0, φ∈C0∞(ℝN), such that φ(x)=1 for |x|≤R and φ(x)=0 for |x|≥2R. The sequence u~n converges in Llocq(ℝN), 1≤q<p*. We have
∫
ℝ
N
(
|
∇
u
~
n
|
p
-
2
∇
u
~
n
-
|
∇
u
~
m
|
p
-
2
∇
u
~
m
,
∇
u
~
n
-
∇
u
~
m
)
φ
𝑑
x
=
-
∫
ℝ
N
(
|
∇
u
~
n
|
p
-
2
∇
u
~
n
-
|
∇
u
~
m
|
p
-
2
∇
u
~
m
,
∇
φ
)
(
u
~
n
-
u
~
m
)
𝑑
x
-
∫
ℝ
N
(
a
(
x
+
x
n
)
|
u
~
n
|
p
-
2
u
~
n
-
a
(
x
+
x
m
)
|
u
~
m
|
p
-
2
u
~
m
)
(
u
~
n
-
u
~
m
)
φ
𝑑
x
+
∫
ℝ
N
(
f
λ
n
(
x
+
x
n
,
u
~
n
)
-
f
λ
m
(
x
+
x
m
,
u
~
m
)
)
(
u
~
n
-
u
~
m
)
φ
𝑑
x
≤
c
∫
B
2
R
(
|
∇
u
~
n
|
p
-
1
+
|
∇
u
~
m
|
p
-
1
)
|
u
~
n
-
u
~
m
|
𝑑
x
+
c
∫
B
2
R
(
|
u
~
n
|
p
-
1
+
|
u
~
m
|
p
-
1
)
|
u
~
n
-
u
~
m
|
𝑑
x
+
c
∫
B
2
R
(
|
u
~
n
|
q
-
1
+
|
u
~
m
|
q
-
1
)
|
u
~
n
-
u
~
m
|
𝑑
x
≤
c
(
∫
ℝ
N
(
|
∇
u
~
n
|
p
+
|
∇
u
~
m
|
p
+
|
u
~
n
|
p
+
|
u
~
m
|
p
)
𝑑
x
)
p
-
1
p
(
∫
B
2
R
|
u
~
n
-
u
~
m
|
p
𝑑
x
)
1
p
+
(
∫
ℝ
N
(
|
u
~
n
|
q
+
|
u
~
m
|
q
)
𝑑
x
)
q
-
1
q
(
∫
B
2
R
|
u
~
n
-
u
~
m
|
q
𝑑
x
)
1
q
(2.4)
≤
c
(
∫
B
2
R
|
u
~
n
-
u
~
m
|
p
𝑑
x
)
1
p
+
c
(
∫
B
2
R
|
u
~
n
-
u
~
m
|
q
𝑑
x
)
1
q
→
0
as
n
,
m
→
∞
.
The following elementary inequalities are very useful (see [6]). For p>1 there exists a constant cp such that for ξ,η∈ℝN we have
(2.5)
(
|
ξ
|
p
-
2
ξ
-
|
η
|
p
-
2
η
,
ξ
-
η
)
≥
c
p
|
ξ
-
η
|
p
if
p
≥
2
,
(2.6)
(
|
ξ
|
p
-
2
ξ
-
|
η
|
p
-
2
η
,
ξ
-
η
)
≥
c
p
|
ξ
-
η
|
2
(
|
ξ
|
2
-
p
+
|
η
|
2
-
p
)
-
1
if
1
<
p
<
2
.
For p≥2, by (2.4) and (2.5) we have
∫
B
R
|
∇
u
~
n
-
∇
u
~
m
|
p
𝑑
x
≤
c
∫
ℝ
N
(
|
∇
u
~
n
|
p
-
2
∇
u
~
n
-
|
∇
u
~
m
|
p
-
2
∇
u
~
m
,
∇
u
~
n
-
∇
u
~
m
)
φ
𝑑
x
→
0
as
n
,
m
→
∞
.
For 1<p≤2, by (2.4) and (2.6) we have
∫
B
R
|
∇
u
~
n
-
∇
u
~
m
|
p
𝑑
x
≤
c
∫
B
R
|
(
|
∇
u
~
n
|
p
-
2
∇
u
~
n
-
|
∇
u
~
m
|
p
-
2
∇
u
~
m
,
∇
u
~
n
-
∇
u
~
m
)
|
p
2
(
|
∇
u
~
n
|
2
-
p
+
|
∇
u
~
m
|
2
-
p
)
p
2
𝑑
x
≤
c
(
∫
B
R
(
|
∇
u
~
n
|
p
-
2
∇
u
~
n
-
|
∇
u
~
m
|
p
-
2
∇
u
~
m
,
∇
u
~
n
-
∇
u
~
m
)
𝑑
x
)
p
2
(
∫
B
R
(
|
∇
u
~
n
|
p
+
|
∇
u
~
m
|
p
)
𝑑
x
)
2
-
p
2
≤
c
(
∫
ℝ
N
(
|
∇
u
~
n
|
p
-
2
∇
u
~
n
-
|
∇
u
~
m
|
p
-
2
∇
u
~
m
,
∇
u
~
n
-
∇
u
~
m
)
φ
𝑑
x
)
p
2
→
0
as
n
,
m
→
∞
,
since φ=1 in BR and φ≥0. Hence {u~n} converges in Wloc1,p(ℝN). ∎
Lemma 2.4.
Let the decomposition (2.1) for un hold. Then we have the following results:
v
=
|
u
|
and
V
k
=
|
U
k
|
(
k
∈
Λ
)
satisfy the differential inequality
(2.7)
∫
ℝ
N
|
∇
v
|
p
-
2
∇
v
∇
φ
d
x
+
a
0
∫
ℝ
N
v
p
-
1
φ
𝑑
x
≤
∫
ℝ
N
v
q
-
1
φ
𝑑
x
for all
φ
∈
W
1
,
p
(
ℝ
N
)
,
φ
≥
0
.
The index set Λ is finite.
There exists
α
>
0
such that
v
(
x
)
≤
1
α
e
-
α
1
+
|
x
|
2
,
V
k
(
x
)
≤
1
α
e
-
α
1
+
|
x
|
2
for all
x
∈
ℝ
N
,
k
∈
Λ
.
Proof.
(1) Denote vn=|un|. By Lemma 2.2, the sequence vn satisfies the differential inequality
(2.8)
∫
ℝ
N
|
∇
v
n
|
p
-
2
∇
v
n
∇
φ
d
x
+
a
0
∫
ℝ
N
v
n
p
-
1
φ
𝑑
x
≤
∫
ℝ
N
v
n
q
-
1
φ
𝑑
x
for all
φ
∈
W
1
,
p
(
ℝ
N
)
,
φ
≥
0
.
We have un⇀u in W1,p(ℝN). By Lemma 2.3, we know that un→u in Wloc1,p(ℝN), consequently we obtain vn=|un|→|u|=v in Wloc1,p(ℝN). Take the limit n→∞ in (2.8). For φ∈C0∞(ℝN), φ≥0, we obtain
∫
ℝ
N
|
∇
v
|
p
-
2
∇
v
∇
φ
d
x
+
a
0
∫
ℝ
N
v
p
-
1
φ
𝑑
x
≤
∫
ℝ
N
v
q
-
1
φ
𝑑
x
.
By approximation this inequality holds for φ∈W1,p(ℝN), φ≥0. Similarly, we can prove that Vk, for k∈Λ, satisfies (2.7).
(2) By (2.7) and the Sobolev embedding theorem, we have
(
∫
ℝ
N
V
k
q
𝑑
x
)
p
q
≤
S
p
,
q
-
1
∫
ℝ
N
(
|
∇
V
k
|
p
+
V
k
p
)
𝑑
x
≤
c
0
∫
ℝ
N
V
k
q
𝑑
x
,
where Sp,q is the Sobolev constant of the embedding from W1,p(ℝN) to Lq(ℝN):
S
p
,
q
=
inf
u
∈
W
1
,
p
(
ℝ
N
)
∖
{
0
}
∫
ℝ
N
(
|
∇
u
|
p
+
u
p
)
𝑑
x
(
∫
ℝ
N
u
q
𝑑
x
)
p
q
.
Hence ∫ℝN|Uk|q𝑑x=∫ℝNVkq𝑑x≥m for some m>0. By property (3) of the decomposition (2.1), Λ is finite.
(3) This fact is well known (see [5, 8]). For the convenience of the readers, we give the proof for the function v.
Step 1. We use Moser’s iteration. For a function v and any T>0 define vT as vT=v if v(x)≤T and vT=T if v(x)≥T. For 1≤ρ<R≤2 let ψ∈C0∞(ℝN) such that ψ(x)=1 for |x|≤ρ, ψ(x)=0 for |x|≥R and |∇ψ|≤2R-ρ for all x∈ℝN. Choose T>0, r≥1 and take φ=vvTp(r-1)ψp as the test function in (2.7). We have
∫
v
≤
T
(
(
p
(
r
-
1
)
+
1
)
v
p
(
r
-
1
)
ψ
p
|
∇
v
|
p
+
p
v
p
(
r
-
1
)
+
1
ψ
p
-
1
|
∇
v
|
p
-
2
∇
v
∇
ψ
)
𝑑
x
+
∫
v
≥
T
(
T
p
(
r
-
1
)
ψ
p
|
∇
v
|
p
+
p
T
p
(
r
-
1
)
v
ψ
p
-
1
|
∇
v
|
p
-
2
∇
v
∇
ψ
)
𝑑
x
+
a
0
∫
ℝ
N
v
p
v
T
p
(
r
-
1
)
ψ
p
𝑑
x
(2.9)
≤
∫
ℝ
N
v
q
v
T
p
(
r
-
1
)
ψ
p
𝑑
x
.
By (2.9), we get
(2.10)
∫
ℝ
N
v
T
p
(
r
-
1
)
ψ
p
|
∇
v
|
p
𝑑
x
≤
p
∫
ℝ
N
v
T
p
(
r
-
1
)
v
ψ
p
-
1
|
∇
v
|
p
-
1
|
∇
ψ
|
𝑑
x
+
∫
ℝ
N
v
q
v
T
p
(
r
-
1
)
ψ
p
𝑑
x
.
By (2.10), the Hölder inequality and the Young inequality, we have
(2.11)
∫
ℝ
N
v
T
p
(
r
-
1
)
ψ
p
|
∇
v
|
p
𝑑
x
≤
c
∫
ℝ
N
(
v
v
T
r
-
1
)
p
|
∇
ψ
|
p
𝑑
x
+
c
∫
ℝ
N
v
q
v
T
p
(
r
-
1
)
ψ
p
𝑑
x
.
Then by (2.11), we obtain
∫
ℝ
N
|
∇
(
v
v
T
r
-
1
ψ
)
|
p
𝑑
x
≤
c
∫
ℝ
N
|
∇
(
v
v
T
r
-
1
)
|
p
ψ
p
𝑑
x
+
c
∫
ℝ
N
(
v
v
T
r
-
1
)
p
|
∇
ψ
|
p
𝑑
x
≤
c
r
p
∫
ℝ
N
|
∇
v
|
p
(
v
T
r
-
1
)
p
ψ
p
𝑑
x
+
c
∫
ℝ
N
(
v
v
T
r
-
1
)
p
|
∇
ψ
|
p
𝑑
x
(2.12)
≤
c
r
p
(
∫
ℝ
N
(
v
v
T
r
-
1
)
p
|
∇
ψ
|
p
𝑑
x
+
∫
ℝ
N
v
q
v
T
p
(
r
-
1
)
ψ
p
𝑑
x
)
.
Thus,
1
r
p
∫
ℝ
N
|
∇
(
v
v
T
r
-
1
ψ
)
|
p
𝑑
x
≤
c
∫
ℝ
N
(
v
v
T
r
-
1
)
p
|
∇
ψ
|
p
𝑑
x
+
c
∫
ℝ
N
v
q
-
p
(
v
v
T
r
-
1
ψ
)
p
𝑑
x
≤
c
(
R
-
ρ
)
p
(
∫
B
R
(
v
v
T
r
-
1
)
p
*
⋅
p
p
+
p
*
-
q
𝑑
x
)
p
+
p
*
-
q
p
*
+
c
(
∫
ℝ
N
v
p
*
𝑑
x
)
q
-
p
p
*
⋅
(
∫
ℝ
N
(
v
v
T
r
-
1
ψ
)
p
*
⋅
p
p
+
p
*
-
q
𝑑
x
)
p
+
p
*
-
q
p
*
(2.13)
≤
c
(
R
-
ρ
)
p
(
∫
B
R
(
v
v
T
r
-
1
)
p
*
⋅
d
𝑑
x
)
p
p
*
d
,
where d=pp+p*-q<1. Then by the Sobolev embedding theorem and (2.13), we have
(
∫
B
ρ
(
v
v
T
r
-
1
)
p
*
𝑑
x
)
p
p
*
≤
c
∫
ℝ
N
|
∇
(
v
v
T
r
-
1
ψ
)
|
p
𝑑
x
≤
c
r
p
(
R
-
ρ
)
p
(
∫
B
R
(
v
v
T
r
-
1
)
p
*
⋅
d
𝑑
x
)
p
p
*
d
.
So
(
∫
B
ρ
(
v
v
T
r
-
1
)
p
*
𝑑
x
)
1
p
*
≤
c
r
R
-
ρ
(
∫
B
R
(
v
v
T
r
-
1
)
p
*
⋅
d
𝑑
x
)
1
p
*
d
.
If ∫BRvrp*d𝑑x<+∞, then
(2.14)
(
∫
B
ρ
v
r
p
*
𝑑
x
)
1
r
p
*
≤
(
c
r
R
-
ρ
)
1
r
(
∫
B
R
v
r
p
*
d
𝑑
x
)
1
r
p
*
d
.
Let χ=1d, rj=χj,j=0,1,2,…, and ρj=ρ+12j(R-ρ). Then by (2.14) we get
(
∫
B
ρ
j
v
r
j
p
*
𝑑
x
)
1
r
j
p
*
≤
(
c
r
j
ρ
j
-
1
-
ρ
j
)
1
r
j
(
∫
B
ρ
j
-
1
v
p
*
r
j
-
1
𝑑
x
)
1
r
j
-
1
p
*
=
(
c
2
j
r
j
R
-
ρ
)
1
r
j
(
∫
B
ρ
j
-
1
v
p
*
r
j
-
1
𝑑
x
)
1
r
j
-
1
p
*
.
We use Moser’s iteration to obtain
(
∫
B
ρ
j
v
r
j
p
*
𝑑
x
)
1
r
j
p
*
≤
∏
j
=
1
(
c
2
j
χ
j
R
-
ρ
)
1
χ
j
(
∫
B
R
v
p
*
𝑑
x
)
1
p
*
.
Letting j→∞, we have
(2.15)
∥
v
∥
L
∞
(
B
ρ
)
≤
c
(
R
-
ρ
)
d
1
-
d
∥
v
∥
L
p
*
(
B
R
)
.
For R<2 we prove ∥v∥Lp*(BR)≤c∥v∥Lq(B2). Indeed, take r=1 in (2.12), ψ(x)=1 for |x|≤R, ψ(x)=0 for |x|≥2, and |∇ψ|≤22-R for all x∈ℝN. Then
(
∫
B
R
v
p
*
𝑑
x
)
p
p
*
≤
c
∫
ℝ
N
|
∇
(
v
ψ
)
|
p
𝑑
x
≤
c
∫
ℝ
N
v
p
|
∇
ψ
|
p
𝑑
x
+
c
∫
ℝ
N
v
q
ψ
p
𝑑
x
≤
c
(
2
-
R
)
p
(
∫
B
2
v
q
𝑑
x
)
p
q
+
c
(
∫
B
2
v
q
𝑑
x
)
p
q
≤
c
(
2
-
R
)
p
(
∫
B
2
v
q
𝑑
x
)
p
q
,
that is,
(2.16)
∥
v
∥
L
p
*
(
B
R
)
≤
c
2
-
R
∥
v
∥
L
q
(
B
2
)
.
Combining (2.15) and (2.16), we obtain
|
v
|
L
∞
(
B
1
(
x
)
)
≤
c
|
v
|
L
q
(
B
2
(
x
)
)
for all
x
∈
ℝ
N
,
where the constant c is independent of x, hence v(x)→0 as |x|→∞.
Step 2. Choose ε and R0 such that vq-p(x)≤εq-p<12a0 for |x|≥R0. Then for R>R0 we have
(2.17)
∫
ℝ
N
∖
B
R
|
∇
v
|
p
-
2
∇
v
∇
φ
d
x
+
1
2
a
0
∫
ℝ
N
∖
B
R
v
p
-
1
φ
𝑑
x
≤
0
for all
φ
∈
W
0
1
,
p
(
ℝ
N
∖
B
R
)
,
φ
≥
0
.
For any R>0 define ηR as ηR(x)=0 for |x|≤R, ηR(x)=1 for |x|≥R+1 and |∇ηR|≤2 for all x∈ℝN. Taking φ=ηR2v in (2.17), we have
∫
ℝ
N
∖
B
R
|
∇
v
|
p
-
2
∇
v
(
η
R
2
∇
v
+
2
v
η
R
∇
η
R
)
𝑑
x
+
1
2
a
0
∫
ℝ
N
∖
B
R
v
p
η
R
2
𝑑
x
≤
0
.
Thus,
∫
ℝ
N
∖
B
R
+
1
(
|
∇
v
|
p
+
v
p
)
η
R
2
𝑑
x
≤
c
|
∫
ℝ
N
∖
B
R
+
1
η
R
∇
η
R
⋅
v
|
∇
v
|
p
-
2
∇
v
d
x
|
≤
c
∫
B
R
+
1
∖
B
R
(
|
∇
v
|
p
+
v
p
)
𝑑
x
.
For R0>0 let Rn=R0+n. Then for any R>0 we have Rn<R<Rn+1. Define a sequence
a
n
=
∫
ℝ
N
\
B
R
0
+
n
(
|
∇
v
|
p
+
v
p
)
𝑑
x
.
Then an+1≤c(an-an+1), which implies that an+1≤θan, where θ=cc+1<1. Thus there exists c (independent of n) such that an≤cθn. Hence,
(2.18)
∫
ℝ
N
∖
B
R
(
|
∇
v
|
p
+
v
p
)
𝑑
x
≤
c
e
-
α
R
,
and we have
∫
ℝ
N
∖
B
R
v
q
𝑑
x
≤
c
e
-
α
R
.
Step 3. Repeat the argument in Step 1. By (2.18) we have
v
(
x
)
≤
c
e
-
α
|
x
|
for all
x
∈
ℝ
N
.
∎
Remark 2.5.
Suppose that vn≥0 and vn satisfies the differential inequality (2.3), n=1,2,…. Suppose vn→v in Lq(ℝN). By checking the proof of Lemma 2.4, the sequence vn is uniformly bounded, that is, for some α>0 we have
v
n
(
x
)
≤
1
α
e
-
α
|
x
|
for all
x
∈
ℝ
N
.
Lemma 2.6.
Denote
Ω
R
(
n
)
=
ℝ
N
∖
{
B
R
∪
⋃
k
∈
Λ
B
R
(
x
n
,
k
)
}
.
Then there exists α>0 such that
∫
Ω
R
(
n
)
(
|
∇
u
n
|
p
+
|
u
n
|
p
)
𝑑
x
≤
1
α
e
-
α
R
,
|
u
n
(
x
)
|
≤
1
α
e
-
α
R
for all
x
∈
Ω
R
(
n
)
.
The proof of Lemma 2.6 is similar to that of Lemma 2.4, so we only sketch it.
Proof.
One can prove the lemma in four steps. Step 1. By Lemma 2.4 and property (4) of the decomposition (2.1), we have
∫
Ω
R
(
n
)
v
n
q
𝑑
x
≤
c
∫
ℝ
N
∖
B
R
v
q
𝑑
x
+
c
∑
k
∈
Λ
∫
ℝ
N
∖
B
R
V
k
q
𝑑
x
+
c
∫
ℝ
N
|
r
n
|
q
𝑑
x
≤
c
e
-
α
R
+
o
n
(
1
)
.
Step 2. We use Moser’s iteration to prove the L∞-estimate
|
v
n
(
x
)
|
≤
c
e
-
α
R
+
o
n
(
1
)
for all
x
∈
Ω
R
(
n
)
.
In particular, for any ε>0 there exist n0, R0 such that for n≥n0 there holds
|
v
n
(
x
)
|
<
ε
for all
x
∈
Ω
R
0
(
n
)
.
In the following, we assume n≥n0, R≥R0.
Step 3. This is similar to Lemma 2.4. Choose ε, R0 such that vnq-p(x)≤εq-p<12a0 for x∈ΩR0(n). Then for R>R0 we have
(2.19)
∫
Ω
R
(
n
)
|
∇
v
n
|
p
-
2
∇
v
n
∇
φ
d
x
+
1
2
a
0
∫
Ω
R
(
n
)
v
n
p
-
1
φ
𝑑
x
≤
0
for all
φ
∈
Ω
R
(
n
)
,
φ
≥
0
.
For any R>0 define φR as φR(x)=0 for x∉ΩR(n), φR(x)=1 for x∈ΩR+1(n) and |∇φR|≤2. Taking φ=φR2vn in (2.19), we have
∫
Ω
R
(
n
)
|
∇
v
n
|
p
-
2
∇
v
n
(
φ
R
2
∇
v
n
+
2
v
n
φ
R
∇
φ
R
)
𝑑
x
+
1
2
a
0
∫
Ω
R
(
n
)
v
n
p
φ
R
2
𝑑
x
≤
0
.
Thus,
∫
Ω
R
+
1
(
n
)
(
|
∇
v
n
|
p
+
v
n
p
)
φ
R
2
𝑑
x
≤
c
|
∫
Ω
R
+
1
(
n
)
φ
R
⋅
∇
φ
R
⋅
v
n
|
∇
v
n
|
p
-
2
∇
v
n
d
x
|
≤
c
∫
Ω
R
(
n
)
∖
Ω
R
+
1
(
n
)
(
|
∇
v
n
|
p
+
v
n
p
)
𝑑
x
,
where c>0 is independent of n, R. By the definition of φR, we have
∫
Ω
R
+
1
(
n
)
(
|
∇
v
n
|
p
+
v
n
p
)
𝑑
x
≤
c
c
+
1
∫
Ω
R
(
n
)
(
|
∇
v
n
|
p
+
v
n
p
)
𝑑
x
,
which implies that there exist positive constants c (independent of n, R) and α such that
(2.20)
∫
Ω
R
(
n
)
(
|
∇
v
n
|
p
+
v
n
p
)
𝑑
x
≤
c
e
-
α
R
.
And we have
∫
Ω
R
(
n
)
v
n
q
𝑑
x
≤
c
e
-
α
R
.
Step 4. Repeat the argument in Step 2. By (2.20) we have
v
n
(
x
)
≤
c
e
-
α
R
for all
x
∈
Ω
R
(
n
)
.
∎
We follow the idea of [7] to derive a local Pohožaev-type identity with a form as in [5], which is much closer to our case. Choose ψ∈C0∞(ℝN), t∈ℝN. Taking t⋅∇uψ as test function in equation ($P_{\lambda}$) and integrating by parts, we obtain a Pohožaev-type identity
1
p
∫
ℝ
N
t
⋅
∇
a
|
u
|
p
ψ
𝑑
x
-
∫
ℝ
N
t
⋅
∇
x
F
λ
(
x
,
u
)
ψ
𝑑
x
=
-
1
p
∫
ℝ
N
|
∇
u
|
p
t
⋅
∇
ψ
d
x
+
∫
ℝ
N
|
∇
u
|
p
-
2
t
⋅
∇
u
∇
u
⋅
∇
ψ
d
x
(2.21)
-
1
p
∫
ℝ
N
a
(
x
)
|
u
|
p
t
⋅
∇
ψ
d
x
+
∫
ℝ
N
F
λ
(
x
,
u
)
t
⋅
∇
ψ
d
x
.
Assume that un∈W1,p(ℝN), n=1,2,…, is a solution of problem ($P_{\lambda}$) with λ=λn≥0, ∥un∥≤M. Then we have the decomposition (2.1).
Without loss of generality we may assume |xn,1|=min{|xn,k|,k∈Λ}. Denote xn=xn,1. According to [5] we can construct a sequence of cones Cn, having vertex 12xn and generated by a ball BRn(xn) as follows:
C
n
=
{
w
∈
ℝ
N
|
w
=
1
2
x
n
+
λ
(
x
-
1
2
x
n
)
,
x
∈
B
R
n
(
x
n
)
,
λ
∈
[
0
,
∞
)
}
,
where Rn satisfies
r
^
k
0
⋅
|
x
n
|
2
=
r
n
≤
R
n
≤
k
0
r
n
=
r
^
|
x
n
|
2
,
r
^
=
1
5
(
c
¯
+
1
)
,
and c¯ is the constant in condition (a4), Λ={1,2,…,k0}.
The cone Cn has the following property:
(2.22)
∂
C
n
∩
{
B
1
2
r
n
(
0
)
∪
⋃
k
∈
Λ
B
1
2
r
n
(
x
n
,
k
)
}
=
∅
.
We now apply the identity (2.21). Take u=un, t=tn=xn|xn| and ψ=χφR, where χ,φR∈C0∞(ℝN) such that χ(x)=0 for x∉Cn, χ(x)=1 for x∈Cn and dist(x,∂Cn)≥1, φR(x)=1 for |x|≤R, and φR(x)=0 for |x|≥2R. Letting R→∞, we obtain
1
p
∫
ℝ
N
t
n
⋅
∇
a
|
u
n
|
p
χ
𝑑
x
-
∫
ℝ
N
t
n
⋅
∇
x
F
λ
(
x
,
u
n
)
χ
𝑑
x
=
-
1
p
∫
ℝ
N
|
∇
u
n
|
p
t
n
⋅
∇
χ
d
x
+
∫
ℝ
N
|
∇
u
n
|
p
-
2
t
n
⋅
∇
u
n
∇
u
n
⋅
∇
χ
d
x
(2.23)
-
1
p
∫
ℝ
N
a
(
x
)
|
u
n
|
p
t
n
⋅
∇
χ
d
x
+
∫
ℝ
N
F
λ
(
x
,
u
n
)
t
n
⋅
∇
χ
d
x
.
By (2.22) and the definition of χ, the support of ∇χ is contained in the domain Ω=ΩR(n) with R=12rn-1. By Lemma 2.6, the right-hand side of (2.23) decays exponentially, say less than ce-α|xn|. By [5, Lemma 4.2], we have tn⋅∇a≥12∂a∂r for x∈Cn. Similarly by Lemma 2.1 (5), we have tn⋅∇xFλ(x,un)≤0 . Hence the left-hand side of (2.23) can be estimated as
1
p
∫
ℝ
N
t
n
⋅
∇
a
|
u
n
|
p
χ
𝑑
x
-
∫
ℝ
N
t
n
⋅
∇
x
F
λ
(
x
,
u
n
)
χ
𝑑
x
≥
1
p
∫
ℝ
N
t
n
⋅
∇
a
|
u
n
|
p
χ
𝑑
x
≥
1
2
p
inf
B
L
(
x
n
)
∂
a
∂
r
∫
B
L
(
x
n
)
|
u
n
|
p
𝑑
x
(2.24)
≥
m
4
p
inf
B
L
(
x
n
)
∂
a
∂
r
,
where
m
=
lim
n
→
∞
∫
B
L
(
x
n
)
|
u
n
|
p
d
x
=
lim
n
→
∞
∫
B
L
(
0
)
(
u
n
(
⋅
+
x
n
,
1
)
)
p
d
x
=
∫
B
L
(
0
)
U
1
p
d
x
>
0
for L large enough. Combining (2.23) and (2.24), we obtain
m
4
p
inf
B
L
(
x
n
)
∂
a
∂
r
≤
c
e
-
δ
|
x
n
|
,
which is impossible by (a3). Thus Λ=∅ and by the decomposition (2.1) we have already proved the following proposition.
Proposition 2.7.
Let un∈W1,p(RN) be a solution of ($P_{\lambda}$) with λ=λn≥0, n=1,2,…. Assume ∥un∥≤M, un⇀u in W1,p(RN). Then {un} converges in Lq(RN).
Proof of Theorem 1.1.
We use an indirect argument. Assume for any μn=1n that there exist un∈W1,p(ℝN), xn∈ℝN such that un is a solution of (Pλn), ∥un∥≤M. But there holds that
(2.25)
|
u
n
(
x
n
)
|
>
μ
n
e
-
μ
n
1
+
|
x
n
|
2
.
By Proposition 2.7, the sequence {un} converges in Lq(ℝN). By Remark 2.5, we have
|
u
n
(
x
)
|
=
v
n
(
x
)
≤
1
α
e
-
α
1
+
|
x
|
2
for some α>0, which contradicts (2.25), provided μn=1n<α. ∎
Corollary 2.8.
Let un∈W1,p(RN) be a solution of ($P_{\lambda}$) with λ=λn≥0, n=1,2,…. Assume Iλn(un)≤M, where
(2.26)
I
λ
(
u
)
=
1
p
∫
ℝ
N
(
|
∇
u
|
p
+
a
(
x
)
|
u
|
p
)
𝑑
x
-
∫
ℝ
N
F
λ
(
x
,
u
)
𝑑
x
.
Then, up to a subsequence, {un} converges in W1,p(RN) and there exists μ>0 such that
(2.27)
|
u
n
(
x
)
|
≤
1
μ
e
-
μ
1
+
|
x
|
2
.
Proof.
By Lemma 2.1 (4), we have
M
≥
I
λ
n
(
u
n
)
=
I
λ
n
(
u
n
)
-
1
r
〈
D
I
λ
n
(
u
n
)
,
u
n
〉
=
(
1
p
-
1
r
)
∫
ℝ
N
(
|
∇
u
n
|
p
+
a
(
x
)
|
u
n
|
p
)
𝑑
x
+
q
-
r
q
r
∫
ℝ
N
|
u
n
|
r
+
1
|
m
λ
(
x
,
u
n
)
|
q
-
r
-
1
b
λ
(
x
,
u
n
)
𝑑
x
≥
(
1
p
-
1
r
)
∫
ℝ
N
(
|
∇
u
n
|
p
+
a
(
x
)
|
u
n
|
p
)
𝑑
x
.
Hence {un} is bounded in W1,p(ℝN). By Theorem 1.1, inequality (2.27) holds. By Proposition 2.7, up to a subsequence, {un} converges in Lq(ℝN). Thus,
∫
ℝ
N
(
|
∇
u
n
|
p
-
2
∇
u
n
-
|
∇
u
m
|
p
-
2
∇
u
m
,
∇
u
n
-
∇
u
m
)
𝑑
x
+
∫
ℝ
N
a
(
x
)
(
|
u
n
|
p
-
2
u
n
-
|
u
m
|
p
-
2
u
m
)
(
u
n
-
u
m
)
𝑑
x
=
∫
ℝ
N
(
f
λ
n
(
x
,
u
n
)
-
f
λ
m
(
x
,
u
m
)
)
(
u
n
-
u
m
)
𝑑
x
≤
c
∫
ℝ
N
(
|
u
n
|
q
-
1
+
|
u
m
|
q
-
1
)
|
u
n
-
u
m
|
𝑑
x
→
0
as
n
,
m
→
∞
.
By the elementary inequalities (2.5) and (2.6), the sequence {un} converges in W1,p(ℝN). ∎
3 Existence of Multiple Solutions
Problem ($P_{\lambda}$) has a variational structure, given by the functional Iλ defined on W1,p(ℝN) (see (2.26)).
Lemma 3.1.
I
λ
(
λ
≥
0
)
satisfies the Palais–Smale condition.
Proof.
Let {un}⊂W1,p(ℝN) be a Palais–Smale sequence. By Lemma 2.1 (4), we have
∫
ℝ
N
(
1
r
u
n
f
λ
(
x
,
u
n
)
-
F
λ
(
x
,
u
n
)
)
𝑑
x
≥
0
.
Then
I
λ
(
u
n
)
-
1
r
〈
D
I
λ
(
u
n
)
,
u
n
〉
=
(
1
p
-
1
r
)
∫
ℝ
N
(
|
∇
u
n
|
p
+
a
(
x
)
|
u
n
|
p
)
𝑑
x
+
∫
ℝ
N
(
1
r
u
n
f
λ
(
x
,
u
n
)
-
F
λ
(
x
,
u
n
)
)
𝑑
x
≥
(
1
p
-
1
r
)
∫
ℝ
N
(
|
∇
u
n
|
p
+
a
(
x
)
|
u
n
|
p
)
𝑑
x
.
Hence {un} is bounded in W1,p(ℝN). Assume un⇀u in W1,p(ℝN), un→u in Llocq(ℝN), 1≤q<p*. By Lemma 2.1, we obtain
∫
ℝ
N
(
|
∇
u
n
|
p
-
2
∇
u
n
-
|
∇
u
m
|
p
-
2
∇
u
m
,
∇
u
n
-
∇
u
m
)
𝑑
x
+
∫
ℝ
N
a
(
x
)
(
|
u
n
|
p
-
2
u
n
-
|
u
m
|
p
-
2
u
m
)
(
u
n
-
u
m
)
𝑑
x
=
∫
ℝ
N
(
f
λ
(
x
,
u
n
)
-
f
λ
(
x
,
u
m
)
)
(
u
n
-
u
m
)
𝑑
x
≤
∫
ℝ
N
(
2
λ
e
-
λ
1
+
|
x
|
2
)
q
-
r
(
|
u
n
|
r
-
1
+
|
u
m
|
r
-
1
)
|
u
n
-
u
m
|
𝑑
x
≤
c
e
-
(
q
-
r
)
λ
R
∫
ℝ
N
∖
B
R
(
|
u
n
|
r
+
|
u
m
|
r
)
𝑑
x
+
c
(
∫
B
R
(
|
u
n
|
r
+
|
u
m
|
r
)
𝑑
x
)
r
-
1
r
(
∫
B
R
|
u
n
-
u
m
|
r
𝑑
x
)
1
r
(3.1)
≤
c
e
-
(
q
-
r
)
λ
R
+
c
(
∫
B
R
|
u
n
-
u
m
|
r
𝑑
x
)
1
r
→
0
as
n
,
m
→
∞
.
By (3.1) and the elementary inequalities (2.5), (2.6), we have that {un} is a Cauchy sequence in W1,p(ℝN). ∎
Now we define a sequence of critical values of Iλ.
c
k
(
λ
)
=
inf
A
∈
Γ
k
sup
u
∈
A
I
λ
(
u
)
,
λ
≥
0
,
k
=
1
,
2
,
…
,
where
Γ
k
=
{
A
∣
A
⊂
W
1
,
p
(
ℝ
N
)
,
A
compact
,
-
A
=
A
,
γ
(
A
∩
σ
-
1
(
S
ρ
)
)
≥
k
for all
σ
∈
G
}
,
G
=
{
σ
∣
σ
∈
C
(
W
1
,
p
(
ℝ
N
)
,
W
1
,
p
(
ℝ
N
)
)
,
σ
(
-
u
)
=
-
σ
(
u
)
for all
u
∈
W
1
,
p
(
ℝ
N
)
,
σ
(
u
)
=
u
if
I
1
(
u
)
<
0
}
,
S
ρ
=
{
u
∣
u
∈
W
1
,
p
(
ℝ
N
)
,
∥
u
∥
=
ρ
}
for a suitable
ρ
>
0
.
For u∈Sρ we have
I
1
(
u
)
=
1
p
∫
ℝ
N
(
|
∇
u
|
p
+
a
(
x
)
|
u
|
p
)
𝑑
x
-
∫
ℝ
N
F
1
(
x
,
u
)
𝑑
x
≥
1
p
∫
ℝ
N
(
|
∇
u
|
p
+
a
(
x
)
|
u
|
p
)
𝑑
x
-
1
q
∫
ℝ
N
|
u
|
q
𝑑
x
≥
c
0
(
ρ
p
-
ρ
q
)
.
Choosing ρq-p<12, we have I1(u)≥m=12c0ρp for u∈Sρ.
The following proposition is known (see [1, 2, 11]).
Proposition 3.2.
Assume 0<λ≤1. Then we have the following results:
c
k
(
λ
)
>
0
, k=1,2,…, are critical values of Iλ.
If
c
k
(
λ
)
=
c
k
+
1
(
λ
)
=
⋯
=
c
k
+
m
-
1
(
λ
)
=
c
, then
γ
(
K
c
(
I
λ
)
)
≥
m
, where
K
c
(
I
λ
)
=
{
u
∣
u
∈
W
1
,
p
(
ℝ
N
)
,
D
I
λ
(
u
)
=
0
,
I
λ
(
u
)
=
c
}
.
Assume
p
=
2
. Then there exists
u
∈
W
1
,
p
(
ℝ
N
)
such that
I
λ
(
u
)
=
c
k
(
λ
)
, DIλ(u)=0 and m*(u)≥k, where m*(⋅) is the augmented Morse index.
Proof of Theorem 1.2.
Given k∈N, by Corollary 2.8 there exists μk>0 such that if u∈W1,p(ℝN), DIλ(u)=0 and Iλ(u)≤dk:=ck(1), then
|
u
(
x
)
|
≤
1
μ
k
e
-
μ
k
1
+
|
x
|
2
for all
x
∈
ℝ
N
.
Choose 0<λk<min{1,μk}. Let u1(λk),…,uk(λk) be k solutions of problem ($P_{\lambda}$) with λ=λk, corresponding to the critical values c1(λk)≤⋯≤ck(λk). Since Iλ is increasing as λ, we have that c1(λk)≤⋯≤ck(λk)≤dk, u1(λk),…,uk(λk) satisfy the estimate
|
u
(
x
)
|
≤
1
μ
k
e
-
μ
k
1
+
|
x
|
2
≤
1
λ
k
e
-
λ
k
1
+
|
x
|
2
,
hence u1(λk),…,uk(λk) are solutions of the original problem ($P_{0}$), k is arbitrary and ($P_{0}$) has infinitely many solutions. ∎
Remark 3.3.
We have proved that problem ($P_{0}$) has infinitely many solutions. We can prove a little more.
Claim.
The functional I has an infinite sequence of critical values c1<c2<⋯<ck<⋯. Moreover, if p=2, then limk→∞ck=+∞.
We use an indirect argument. Assume I has only finite critical values c1,…,ck. Denote
K
=
{
u
∣
u
∈
W
1
,
p
(
ℝ
N
)
,
D
I
(
u
)
=
0
}
.
Then by Corollary 2.8, the set K is compact. Assume γ(K)=m<+∞. For 0<λ<1, the functional Iλ has critical values c1(λ)≤c2(λ)≤⋯≤ckm+1(λ). If λ is small enough, they will be critical values of the functional I. We claim c1(λ)<cm+1(λ)<⋯<ckm+1(λ). Otherwise suppose say c=c1(λ)=cm+1(λ). By Proposition 3.2, we have γ(Kc)≥m+1, where
K
c
=
{
u
∣
u
∈
W
1
,
p
(
ℝ
N
)
,
D
I
λ
(
u
)
=
0
,
I
λ
(
u
)
=
c
}
⊂
K
.
This is a contradiction.
Now assume p=2. As in the proof of Theorem 1.2, we have that uk(λk) is a solution of problem ($P_{0}$). But by Proposition 3.2, we can assume m*(uk(λk))≥k. If the critical values of I are bounded above, then K is compact by Corollary 2.8, and the augmented Morse index of the critical points will be bounded above. We arrive at a contradiction.
Remark 3.4.
In problem ($P_{0}$) the positive nonlinear term has growth p-1. If we replace this term by a(x)|u|l-2u with l∈(1,q), the main theorems remain true.
and
Junfang Zhao
