1 Introduction
This paper is concerned with the existence of a positive solution to the problem
(1.1)
-
Δ
u
+
V
(
x
)
u
=
f
(
u
)
,
u
∈
D
1
,
2
(
ℝ
N
)
,
for N≥3, where the nonlinearity f is subcritical at infinity and supercritical near the origin, and the potential V vanishes at infinity. Our precise assumptions on V and f are stated below.
In their groundbreaking paper [10], Berestycki and Lions considered the case where V≡λ is constant and f has superlinear growth. They showed that if f is subcritical, then problem (1.1) has a solution for λ>0 and it does not have a solution for λ<0. They also studied the limiting case V≡0, which they called the zero mass case. They showed that if f is subcritical at infinity and supercritical near the origin, then the problem
(1.2)
-
Δ
u
=
f
(
u
)
,
u
∈
D
1
,
2
(
ℝ
N
)
,
has a ground state solution ω, which is positive, radially symmetric and decreasing in the radial direction.
The motivation for studying this type of equations came from some problems in particle physics related to the nonabelian gauge theory, which underlies strong interaction, called quantum chromodynamics (QCD). Their solutions give rise to some special solutions of the pure Yang–Mills equations via ’t Hooft’s ansatz, see [17].
For a radial potential V(|x|), Badiale and Rolando [4] established the existence of a positive radial solution to problem (1.1). On the other hand, under suitable hypotheses, but without assuming any symmetries on V, Benci, Grisanti and Micheletti showed in [7] that problem (1.1) has a positive least energy solution if V(x)≤0 for all x∈ℝN and V(x)<0 on a set of positive measure. They also showed that, if V(x)≥0 for all x∈ℝN and V(x)>0 on a set of positive measure, then this problem does not have a ground state solution, i.e., the corresponding variational functional does not attain its (least energy) mountain pass value. Other related results may be found in [2, 8, 9, 16].
The result that we present in this paper includes the existence of a positive bound state for positive or sign changing potentials which decay to 0 at infinity with a suitable velocity. More precisely, we assume that V and f have the following properties:
V
∈
L
N
/
2
(
ℝ
N
)
∩
L
r
(
ℝ
N
)
for some r>N/2 and ∫ℝN|V-|N/2<SN/2, where V-:=min{0,V} and S is the best constant for the embedding D1,2(ℝN)↪L2∗(ℝN), with 2∗:=2NN-2.
There exist constants A0>0 and κ>max{2,N-2} such that
V
(
x
)
≤
A
0
(
1
+
|
x
|
)
-
κ
for all
x
∈
ℝ
N
.
f
∈
𝒞
1
[
0
,
∞
)
and there exist constants A1>0 and 2<p<2∗<q such that, for m=-1,0,1,
|
f
(
m
)
(
s
)
|
≤
{
A
1
|
s
|
p
-
(
m
+
1
)
if
|
s
|
≥
1
,
A
1
|
s
|
q
-
(
m
+
1
)
if
|
s
|
≤
1
,
where f(-1):=F, f(0):=f, f(1):=f′ and F(s):=∫0sf(t)dt.
There exists a constant θ>2 such that 0≤θF(s)≤f(s)s<f′(s)s2 for all s>0.
The function g(s):=sf′(s)f(s) is a decreasing function of s>0 and
lim
s
→
∞
g
(
s
)
<
2
∗
-
1
<
lim
s
→
0
g
(
s
)
.
Our main result is the following one.
Theorem 1.1.
Assume that (V1)–(V2) and (f1)–(f3) hold true. Then problem (1.1) has a positive solution.
It is easy to see that the model nonlinearity
f
(
s
)
:=
s
q
-
1
1
+
s
q
-
p
satisfies assumptions (f1)–(f3).
We point out that assumptions (V1), (f1) and (f2) are quite natural and have been also considered in previous works, in particular, in [7]. Assumption (f3) guarantees that the limit problem (1.2) has a unique positive solution. This fact, together with some fine estimates, which involve assumption (V2), allows us to show the existence of a positive bound state for problem (1.1) when the ground state is not attained.
The positive mass case, in which the potential V tends to a positive constant at infinity, has been widely investigated. A brief account may be found in [13], where a result similar to Theorem 1.1 was obtained for subcritical nonlinearities. On the other hand, except for the case of the critical pure power nonlinearity, only few results are known for the zero mass case.
There are several delicate issues in dealing with the zero mass case. Already the variational formulation requires some care, because the energy space D1,2(ℝN) is only embedded in L2∗(ℝN). The growth assumptions (f1) on the nonlinearity, however, provide the basic interpolation and boundedness conditions that allow to establish the differentiability of the variational functional and to study its compactness properties. Benci and Fortunato, in [6], expressed these conditions in the framework of Orlicz spaces, which was also used and further developed in [2, 4, 7, 8, 9, 3, 16]. The crucial facts for our purposes are stated in Proposition 3.1 below.
Another sensitive issue is the lack of compactness. In the positive mass case, a fundamental tool for dealing with it is Lions’ vanishing lemma, whose proof relies deeply on the fact that the sequences involved are bounded in H1(ℝN). Once again, assumption (f1) allowed us to obtain a suitable version of this result for sequences which are only bounded in D1,2(ℝN) (see Lemma 3.5). This new version of Lions’ vanishing lemma plays a crucial role in the proof of the splitting lemma (Lemma 3.9), which describes the lack of compactness of the variational functional. When the ground state is not attained, due to the uniqueness of the positive solution to the limit problem (1.2), the splitting lemma provides an open interval of values at which the energy functional satisfies the Palais–Smale condition.
We give a topological argument to establish the existence of a critical value in this interval. This argument requires, on the one hand, some fine estimates which are based on the precise asymptotic decay for the solutions of the limit problem (1.2), obtained recently by Vétois in [18], and on a suitable deformation lemma for 𝒞1-manifolds that was proved by Bonnet in [11].
This paper is organized as follows. In Section 2 we collect the information that we need about the solutions of the limit problem. Section 3 is devoted to the study of the variational problem and, especially, to the compactness properties of the variational functional. In Section 4 we derive the estimates that we need, and we prove our main result.
3 The Variational Setting
For u,v∈D1,2(ℝN), we set
(3.1)
〈
u
,
v
〉
V
:=
∫
ℝ
N
∇
u
⋅
∇
v
+
V
(
x
)
u
v
,
∥
u
∥
V
2
:=
∫
ℝ
N
(
|
∇
u
|
2
+
V
(
x
)
u
2
)
.
By assumption (V1), these expressions are well defined and, using the Sobolev inequality, we conclude that ∥⋅∥V is a norm in D1,2(ℝN) which is equivalent to the standard one.
Let 2<p<2∗<q. The following proposition, combined with assumption (f1), provides the interpolation and boundedness properties that are needed to obtain a good variational problem.
Proposition 3.1.
Let α,β>0 and h∈C0(R). Assume αβ≤pq and β≤q, and that there exists M>0 such that
|
h
(
s
)
|
≤
M
min
{
|
s
|
α
,
|
s
|
β
}
for every
s
∈
ℝ
.
Then, for every t∈[qβ,pα], the map D1,2(RN)→Lt(RN) given by u↦h(u) is well defined, continuous and bounded.
Proof.
The decomposition u=u1Ωu+u1ℝN∖Ωu, where Ωu:={x∈ℝN:|u(x)|>1}, gives a continuous embedding of L2∗(ℝN), and hence of D1,2(ℝN), into the Orlicz space
L
p
(
ℝ
N
)
+
L
q
(
ℝ
N
)
:=
{
u
:
u
=
u
1
+
u
2
, with
u
1
∈
L
p
(
ℝ
N
)
,
u
2
∈
L
q
(
ℝ
N
)
}
,
whose norm is defined by
|
u
|
p
,
q
:=
inf
{
|
u
1
|
p
+
|
u
2
|
q
:
u
=
u
1
+
u
2
,
u
1
∈
L
p
(
ℝ
N
)
,
u
2
∈
L
q
(
ℝ
N
)
}
.
Therefore, our claim is a special case of [3, Proposition 3.5]. ∎
Let F(u):=∫0uf(s)ds. Assumption (f1) implies that |F(s)|≤A1|s|2∗ and |f(s)|≤A1|s|2∗-1. Therefore, the functionals Φ,Ψ:D1,2(ℝN)→ℝ, given by
Φ
(
u
)
:=
∫
ℝ
N
F
(
u
)
,
Ψ
(
u
)
:=
∫
ℝ
N
f
(
u
)
u
,
are well defined. Using Proposition 3.1, it is easy to show that Φ is of class 𝒞2 and Ψ is of class 𝒞1, see [9, Lemma 2.6] or [3, Proposition 3.8]. Hence, the functional IV:D1,2(ℝN)→ℝ, given by
I
V
(
u
)
:=
1
2
∫
ℝ
N
(
|
∇
u
|
2
+
V
(
x
)
u
2
)
-
∫
ℝ
N
F
(
u
)
,
is of class 𝒞2, with derivative
I
V
′
(
u
)
v
=
∫
ℝ
N
(
∇
u
⋅
∇
v
+
V
(
x
)
u
v
)
-
∫
ℝ
N
f
(
u
)
v
,
u
,
v
∈
D
1
,
2
(
ℝ
N
)
,
and the functional JV:D1,2(ℝN)→ℝ, defined by
J
V
(
u
)
:=
I
V
′
(
u
)
u
=
∫
ℝ
N
(
|
∇
u
|
2
+
V
(
x
)
u
2
)
-
∫
ℝ
N
f
(
u
)
u
,
is of class 𝒞1.
The solutions to problem (1.1) are the critical points of the functional IV. The nontrivial ones lie on the set
𝒩
V
:=
{
u
∈
D
1
,
2
(
ℝ
N
)
:
u
≠
0
,
J
V
(
u
)
=
0
}
.
We define
c
V
:=
inf
u
∈
𝒩
V
I
V
(
u
)
and we write I0, J0, 𝒩0 and c0 for the previous expressions with V=0.
The proofs of the next two lemmas use well-known arguments. We include them for the sake of completeness. Hereafter, C will denote a positive constant, not necessarily the same one.
Lemma 3.2.
There exists
ϱ
>
0
such that
∥
u
∥
V
≥
ϱ
for every
u
∈
𝒩
V
.
𝒩
V
is a closed
𝒞
1
-submanifold of
D
1
,
2
(
ℝ
N
)
and a natural constraint for the functional
I
V
.
If
u
∈
𝒩
V
, then the function
t
↦
I
V
(
t
u
)
is strictly increasing in
[
0
,
1
)
and strictly decreasing in
(
1
,
∞
)
. In particular,
I
V
(
u
)
=
max
t
>
0
I
V
(
t
u
)
.
Proof.
(a) Assumption (f1) implies that |f(s)s|≤A1|s|2∗. So, using Sobolev’s inequality, we get
J
V
(
u
)
≥
∥
u
∥
V
2
-
C
∫
ℝ
N
|
u
|
2
∗
≥
∥
u
∥
V
2
-
C
∥
u
∥
V
2
∗
for all
u
∈
D
1
,
2
(
ℝ
N
)
.
As 2∗>2, there exists ϱ>0 such that JV(u)>0 if 0<∥u∥V≤ϱ. This proves (a).
(b) It follows from (a) that 𝒩V is closed in D1,2(ℝN). Moreover, assumption (f2) yields
J
V
′
(
u
)
u
=
2
∥
u
∥
V
2
-
∫
ℝ
N
f
′
(
u
)
u
2
-
∫
ℝ
N
f
(
u
)
u
=
∫
ℝ
N
[
f
(
u
)
-
f
′
(
u
)
u
]
u
<
0
for every u∈𝒩V. This implies that 0 is a regular value of the restriction of JV to D1,2(ℝN)∖{0}, which is of class 𝒞1. Hence, 𝒩V is a 𝒞1-submanifold of D1,2(ℝN) and a natural constraint for IV.
(c) Let u∈𝒩V. From hypothesis (f2) and statement (a), we obtain that
I
V
(
u
)
=
I
V
(
u
)
-
1
θ
I
V
′
(
u
)
u
=
(
1
2
-
1
θ
)
∥
u
∥
V
2
+
∫
ℝ
N
(
1
θ
f
(
u
)
u
-
F
(
u
)
)
≥
(
1
2
-
1
θ
)
∥
u
∥
V
2
≥
(
1
2
-
1
θ
)
ϱ
2
.
Hence, cV>0.
(d) Let u∈𝒩V. Then
d
d
t
I
V
(
t
u
)
=
1
t
J
V
(
t
u
)
=
t
∥
u
∥
V
2
-
∫
ℝ
N
f
(
t
u
)
u
=
t
∫
ℝ
N
(
f
(
u
)
-
f
(
t
u
)
t
)
u
=
t
[
∫
u
>
0
(
f
(
u
)
u
-
f
(
t
u
)
t
u
)
u
2
+
∫
u
<
0
(
f
(
u
)
u
-
f
(
t
u
)
t
u
)
u
2
]
.
Property (f2) implies that f(s)s is strictly increasing for s>0 and strictly decreasing for s<0. Therefore, ddtIV(tu)>0 if t∈(0,1) and ddtIV(tu)<0 if t∈(1,∞). This proves (d). ∎
Lemma 3.3.
If u is a solution of (1.1), with IV(u)∈[cV,2cV), then u does not change sign.
Proof.
If u is a solution of (1.1), then 0=IV′(u)u±=JV(u±), where u+:=max{u,0} and u-:=min{u,0}. Thus, if u+≠0 and u-≠0, then u±∈𝒩V and
I
V
(
u
)
=
I
V
(
u
+
)
+
I
V
(
u
-
)
≥
2
c
V
,
contradicting our assumption. ∎
Lemma 3.4.
The limit problem (1.2) does not have a solution u with I0(u)∈(c0,2c0).
Proof.
If u is a solution of (1.2) such that I0(u)∈[c0,2c0), then, by Lemma 3.3, u does not change sign. So, by Proposition 2.1, we have that u=±ω, up to a translation. Hence, I0(u)=c0. ∎
The following version of Lions’ vanishing lemma plays a crucial role in the proof of Lemma 3.6 and of the splitting lemma (Lemma 3.9). Its proof was inspired by that of [1, Lemma 2]. We write
B
R
(
y
)
:=
{
x
∈
ℝ
N
:
|
x
-
y
|
<
R
}
.
Lemma 3.5.
If (uk) is bounded in D1,2(RN) and there exists R>0 such that
lim
k
→
∞
(
sup
y
∈
ℝ
N
∫
B
R
(
y
)
|
u
k
|
2
)
=
0
,
then limk→∞∫RNf(uk)uk=0.
Proof.
Fix ε∈(0,1) and set η:=2∗2>1. For each k, consider the function
w
k
:=
{
|
u
k
|
if
|
u
k
|
≥
ε
,
ε
-
(
η
-
1
)
|
u
k
|
η
if
|
u
k
|
≤
ε
.
Observe that
|
∇
w
k
|
2
=
η
2
ε
-
2
(
η
-
1
)
|
u
k
|
2
(
η
-
1
)
|
∇
u
k
|
2
≤
η
2
|
∇
u
k
|
2
if
|
u
k
|
≤
ε
and
|
w
k
|
2
≤
ε
-
(
2
∗
-
2
)
|
u
k
|
2
∗
≤
|
u
k
|
2
if
|
u
k
|
≤
ε
,
|
w
k
|
2
≤
|
u
k
|
2
-
2
∗
|
u
k
|
2
∗
≤
ε
-
(
2
∗
-
2
)
|
u
k
|
2
∗
if
|
u
k
|
≥
ε
.
Using these inequalities, we obtain that
|
w
k
|
2
≤
|
u
k
|
2
,
|
w
k
|
2
≤
ε
-
(
2
∗
-
2
)
|
u
k
|
2
∗
,
|
∇
w
k
|
2
≤
η
2
|
∇
u
k
|
2
.
Therefore, as (uk) is bounded in D1,2(ℝN), we have that
∥
w
k
∥
H
1
(
ℝ
N
)
2
:=
∫
ℝ
N
|
∇
w
k
|
2
+
|
w
k
|
2
≤
∫
ℝ
N
η
2
|
∇
u
k
|
2
+
∫
ℝ
N
ε
-
(
2
∗
-
2
)
|
u
k
|
2
∗
≤
C
,
i.e., (wk) is bounded in H1(ℝN). Moreover,
lim
k
→
∞
(
sup
y
∈
ℝ
N
∫
B
R
(
y
)
|
w
k
|
2
)
=
0
.
From Lions’ vanishing lemma (namely, [19, Lemma 1.21]), it follows that
w
k
→
0
in
L
s
(
ℝ
N
)
for each
2
<
s
<
2
∗
.
Now, using (f1), we obtain
|
∫
ℝ
N
f
(
u
k
)
u
k
|
≤
A
1
(
∫
|
u
k
|
≥
1
|
u
k
|
p
+
∫
|
u
k
|
≤
1
|
u
k
|
q
)
≤
A
1
(
∫
|
u
k
|
≥
ε
|
u
k
|
p
+
∫
ε
≤
|
u
k
|
≤
1
|
u
k
|
q
+
∫
|
u
k
|
≤
ε
|
u
k
|
q
)
≤
2
A
1
∫
|
u
k
|
≥
ε
|
u
k
|
p
+
A
1
∫
|
u
k
|
≤
ε
|
u
k
|
q
≤
2
A
1
∫
|
u
k
|
≥
ε
|
w
k
|
p
+
A
1
∫
|
u
k
|
≤
ε
|
u
k
|
q
-
2
∗
|
u
k
|
2
∗
≤
2
A
1
∫
ℝ
N
|
w
k
|
p
+
A
1
ε
q
-
2
∗
∫
ℝ
N
|
u
k
|
2
∗
.
As (uk) is bounded in D1,2(ℝN) and wk→0 in Lp(ℝN), we conclude that
|
∫
ℝ
N
f
(
u
k
)
u
k
|
≤
C
ε
q
-
2
∗
.
Since ε∈(0,1) was arbitrarily chosen, the statement is proved. ∎
We write ∇IV(u) and ∇JV(u) for the gradients of IV and JV at u with respect to the scalar product (3.1).
Lemma 3.6.
Let (uk) be a sequence in D1,2(RN) such that IV(uk)→d>0 and JV(uk)→0. Then there exist a1>a0>0 such that, after passing to a subsequence,
a
0
≤
∥
u
k
∥
V
≤
a
1
,
a
0
≤
∥
∇
J
V
(
u
k
)
∥
V
≤
a
1
,
|
J
V
′
(
u
k
)
u
k
|
≥
a
0
.
Proof.
From assumption (f2), we get that
d
+
o
(
1
)
=
I
V
(
u
k
)
≤
I
V
(
u
k
)
+
∫
ℝ
N
F
(
u
k
)
=
1
2
∥
u
k
∥
V
2
,
and
d
+
o
(
1
)
=
I
V
(
u
k
)
-
1
θ
J
V
(
u
k
)
=
(
1
2
-
1
θ
)
∥
u
k
∥
V
2
+
∫
ℝ
N
[
1
θ
f
(
u
k
)
u
k
-
F
(
u
k
)
]
≥
(
1
2
-
1
θ
)
∥
u
k
∥
V
2
.
Hence, (uk) is bounded and bounded away from 0 in D1,2(ℝN).
By assumption (f1), for any v∈D1,2(ℝN), we have that
|
∫
ℝ
N
[
f
′
(
u
k
)
u
k
+
f
(
u
k
)
]
v
|
≤
C
∫
ℝ
N
|
u
k
|
2
∗
-
1
|
v
|
≤
C
|
u
k
|
2
∗
2
∗
-
1
|
v
|
2
∗
≤
C
∥
u
k
∥
2
∗
-
1
∥
v
∥
≤
C
∥
v
∥
V
.
Therefore,
|
〈
∇
J
V
(
u
k
)
,
v
〉
V
|
=
|
2
〈
u
k
,
v
〉
V
-
∫
ℝ
N
[
f
′
(
u
k
)
u
k
+
f
(
u
k
)
]
v
|
≤
C
∥
v
∥
V
.
This implies that (∇JV(uk)) is bounded in D1,2(ℝN). Hence, after passing to a subsequence, we have that |JV′(uk)uk|→a≥0. Next, we show that a>0.
As JV(uk)→0, we have that
0
<
a
0
2
≤
∥
u
k
∥
V
2
=
∫
ℝ
N
f
(
u
k
)
u
k
+
o
(
1
)
.
So, by Lemma 3.5, there exist δ>0 and a sequence (yk) in ℝN such that
(3.2)
∫
B
1
(
y
k
)
|
u
k
|
2
=
sup
y
∈
ℝ
N
∫
B
1
(
y
)
|
u
k
|
2
>
δ
.
Set u~k:=uk(⋅+yk). Replacing (u~k) by a subsequence, we have that u~k⇀u weakly in D1,2(ℝN) and u~k→u in Lloc2(ℝN). Inequality (3.2) implies that u≠0. Hence, there exists a subset Λ of positive measure such that u(x)≠0 for every x∈Λ. Assumption (f2) implies that f′(s)s2-f(s)s>0 if s≠0. So, using Fatou’s lemma, we conclude that
a
=
lim
k
→
∞
|
J
V
′
(
u
k
)
u
k
|
=
lim
k
→
∞
|
2
∥
u
k
∥
V
2
-
∫
ℝ
N
(
f
′
(
u
k
)
u
k
2
+
f
(
u
k
)
u
k
)
|
=
lim
k
→
∞
∫
ℝ
N
(
f
′
(
u
k
)
u
k
2
-
f
(
u
k
)
u
k
)
=
lim
k
→
∞
∫
ℝ
N
(
f
′
(
u
~
k
)
u
~
k
2
-
f
(
u
~
k
)
u
~
k
)
≥
lim inf
k
→
∞
∫
Λ
(
f
′
(
u
~
k
)
u
~
k
2
-
f
(
u
~
k
)
u
~
k
)
≥
∫
Λ
(
f
′
(
u
)
u
2
-
f
(
u
)
u
)
>
0
.
This proves that a>0 and hence that (JV′(uk)uk) is bounded away from 0 in ℝ. It follows that
∥
∇
J
V
(
u
k
)
∥
V
≥
|
J
V
′
(
u
k
)
u
k
|
∥
u
k
∥
V
≥
a
~
>
0
.
This completes the proof. ∎
For σ∈ℝ, we set ℳσ:=JV-1(σ) if σ≠0, and ℳ0:=𝒩V. If |σ| is small enough and u∈ℳσ, we write ∇ℳσIV(u) for the orthogonal projection of ∇IV(u) onto the tangent space ℳσ at u.
Recall that a sequence (uk) in D1,2(ℝN) is said to be a (PS)d-sequence forIV if IV(uk)→d and ∇IV(uk)→0.
Lemma 3.7.
Let σk∈R and uk∈Mσk be such that σk→0, IV(uk)→d>0 and ∇MσkIV(uk)→0. Then (uk) is a (PS)d-sequence for IV.
Proof.
Let tk∈ℝ be such that
(3.3)
∇
I
V
(
u
k
)
=
∇
ℳ
σ
k
I
V
(
u
k
)
+
t
k
∇
J
V
(
u
k
)
.
Taking the scalar product with uk, we get that
σ
k
=
J
V
(
u
k
)
=
I
V
′
(
u
k
)
u
k
=
〈
∇
ℳ
σ
k
I
V
(
u
k
)
,
u
k
〉
V
+
t
k
J
V
′
(
u
k
)
u
k
.
By Lemma 3.6, we have that (uk) is bounded and (JV′(uk)uk) is bounded away from 0. So, as σk→0 and ∇ℳσkIV(uk)→0, we conclude that tk→0. Moreover, as (∇JV(uk)) is bounded in D1,2(ℝN), from equation (3.3) we get that ∇IV(uk)→0, as claimed. ∎
Lemma 3.8.
If uk⇀u weakly in D1,2(RN), then the following statements hold true:
∥
u
k
∥
V
2
=
∥
u
k
-
u
∥
2
+
∥
u
∥
V
2
+
o
(
1
)
,
∫
ℝ
N
|
f
(
u
k
)
-
f
(
u
)
|
|
φ
|
=
o
(
1
)
for every
φ
∈
𝒞
c
∞
(
ℝ
N
)
,
∫
ℝ
N
F
(
u
k
)
=
∫
ℝ
N
F
(
u
k
-
u
)
+
∫
ℝ
N
F
(
u
)
+
o
(
1
)
,
f
(
u
k
)
-
f
(
u
k
-
u
)
⟶
f
(
u
)
in
(
D
1
,
2
(
ℝ
N
)
)
′
.
Proof.
(a) Set vk:=uk-u. Assumption (V1) states that V∈LN/2(ℝN)∩Lr(ℝN), with r>N/2. As η:=2rr-1<2∗, we have that vk→0 in Llocη(ℝN). Given ε>0, we fix R>0 such that ∫ℝN∖BR(0)|V|N/2≤εN/2. Then
∫
ℝ
N
|
V
|
v
k
2
=
∫
B
R
(
0
)
|
V
|
v
k
2
+
∫
ℝ
N
∖
B
R
(
0
)
|
V
|
v
k
2
≤
|
V
|
r
|
v
k
|
L
η
(
B
R
(
0
)
)
2
+
|
V
|
L
N
/
2
(
ℝ
N
∖
B
R
(
0
)
)
∥
v
k
∥
2
≤
C
ε
for k large enough. It follows that
∥
u
k
∥
V
2
=
∥
u
k
-
u
∥
V
2
+
∥
u
∥
V
2
+
o
(
1
)
=
∥
u
k
-
u
∥
2
+
∥
u
∥
V
2
+
o
(
1
)
.
(b) Let s,t∈ℝ. By the mean value theorem, there exists ζ∈(0,1) such that
|
f
(
s
+
t
)
-
f
(
s
)
|
=
|
f
′
(
s
+
ζ
t
)
|
|
t
|
≤
A
1
min
{
|
s
+
ζ
t
|
p
-
2
,
|
s
+
ζ
t
|
q
-
2
}
|
t
|
≤
A
1
min
{
(
|
s
|
+
|
t
|
)
p
-
2
,
(
|
s
|
+
|
t
|
)
q
-
2
}
|
t
|
(3.4)
=
h
(
|
s
|
+
|
t
|
)
|
t
|
,
where h(s):=A1min{|s|p-2,|s|q-2}. Applying Proposition 3.1 to this function, we get that {h(|u|+|uk-u|)} is bounded in Lp/(p-2)(ℝN). So, as uk→u in Llocp(ℝN), we conclude that
∫
ℝ
N
|
f
(
u
k
)
-
f
(
u
)
|
|
φ
|
≤
∫
ℝ
N
h
(
|
u
|
+
|
u
k
-
u
|
)
|
u
k
-
u
|
|
φ
|
≤
|
h
(
|
u
|
+
|
u
k
-
u
|
)
|
p
p
-
2
|
φ
|
p
(
∫
supp
(
φ
)
|
u
k
-
u
|
p
)
1
/
p
=
o
(
1
)
.
(c) Arguing as in (b), we have that
|
F
(
s
+
t
)
-
F
(
s
)
|
≤
H
(
|
s
|
+
|
t
|
)
|
t
|
for all
s
,
t
∈
ℝ
,
where H(s):=A1min{|s|p-1,|s|q-1}. Let ε>0 and set vk:=uk-u. Then, noting that |F(s)|≤A1|s|2∗ and using Proposition 3.1, we may choose R>1 such that
∫
|
x
|
>
R
|
F
(
u
k
)
-
F
(
v
k
)
-
F
(
u
)
|
≤
∫
|
x
|
>
R
|
F
(
u
k
)
-
F
(
v
k
)
|
+
∫
|
x
|
>
R
|
F
(
u
)
|
≤
∫
|
x
|
>
R
H
(
|
v
k
|
+
|
u
|
)
|
u
|
+
A
1
∫
|
x
|
>
R
|
u
|
2
∗
≤
|
H
(
|
v
k
|
+
|
u
|
)
|
2
∗
/
(
2
∗
-
1
)
(
∫
|
x
|
>
R
|
u
|
2
∗
)
1
/
2
∗
+
A
1
∫
|
x
|
>
R
|
u
|
2
∗
≤
C
(
∫
|
x
|
>
R
|
u
|
2
∗
)
1
/
2
∗
+
A
1
∫
|
x
|
>
R
|
u
|
2
∗
<
ε
.
On the other hand, as uk→u in Llocp(ℝN), we have that
∫
|
x
|
≤
R
|
F
(
u
k
)
-
F
(
v
k
)
-
F
(
u
)
|
≤
∫
|
x
|
≤
R
|
F
(
u
k
)
-
F
(
u
)
|
+
∫
1
≤
|
x
|
≤
R
|
F
(
v
k
)
|
+
∫
|
x
|
≤
1
|
F
(
v
k
)
|
≤
∫
|
x
|
≤
R
H
(
|
u
k
|
+
|
u
|
)
|
v
k
|
+
A
1
∫
1
≤
|
x
|
≤
R
|
v
k
|
p
+
A
1
∫
|
x
|
≤
1
|
v
k
|
q
≤
|
H
(
|
u
k
|
+
|
u
|
)
|
p
/
(
p
-
1
)
(
∫
|
x
|
≤
R
|
v
k
|
p
)
1
/
p
+
C
∫
|
x
|
≤
R
|
v
k
|
p
≤
C
(
∫
|
x
|
≤
R
|
v
k
|
p
)
1
/
p
+
C
∫
|
x
|
≤
R
|
v
k
|
p
<
ε
if k is large enough. This proves (c).
(d) Let φ∈𝒞c∞(ℝN) and R>0. Set h(s):=A1min{|s|p-2,|s|q-2}. From (3.4) and Proposition 3.1, we get that
∫
|
x
|
>
R
|
f
(
u
k
)
-
f
(
u
k
-
u
)
|
|
φ
|
≤
∫
|
x
|
>
R
h
(
|
u
k
|
+
|
u
|
)
|
u
|
|
φ
|
≤
|
h
(
|
u
k
|
+
|
u
|
)
|
2
∗
/
(
2
∗
-
2
)
(
∫
|
x
|
>
R
|
u
|
2
∗
)
1
/
2
∗
|
φ
|
2
∗
≤
C
(
∫
|
x
|
>
R
|
u
|
2
∗
)
1
/
2
∗
∥
φ
∥
.
Moreover, as |f(u)|≤A1|u|2∗-1, we have that
∫
|
x
|
>
R
|
f
(
u
)
|
|
φ
|
≤
C
(
∫
|
x
|
>
R
|
u
|
2
∗
)
(
2
∗
-
1
)
/
2
∗
∥
φ
∥
.
Thus, given ε>0, we may choose R>0 large enough so that
∫
|
x
|
>
R
|
f
(
u
k
)
-
f
(
u
k
-
u
)
-
f
(
u
)
|
|
φ
|
≤
ε
∥
φ
∥
.
Next, we fix δ∈(0,1) such that η:=2∗δ∈(p,2∗) and ν:=ηη-1-δ∈[qq-2,pp-2]. As uk→u strongly in Llocη(ℝN), from (3.4) and Proposition 3.1, we get that
∫
|
x
|
≤
R
|
f
(
u
k
)
-
f
(
u
)
|
|
φ
|
≤
|
h
(
|
u
|
+
|
u
k
-
u
|
)
|
ν
(
∫
|
x
|
≤
R
|
u
k
-
u
|
η
)
1
/
η
|
φ
|
2
∗
≤
ε
∥
φ
∥
for k large enough and, similarly,
∫
|
x
|
≤
R
|
f
(
u
k
-
u
)
|
|
φ
|
≤
|
h
(
|
u
k
-
u
|
)
|
ν
(
∫
|
x
|
≤
R
|
u
k
-
u
|
η
)
1
/
η
|
φ
|
2
∗
≤
ε
∥
φ
∥
.
Therefore,
|
∫
ℝ
N
(
f
(
u
k
)
-
f
(
u
k
-
u
)
-
f
(
u
)
)
φ
|
≤
ε
∥
φ
∥
for
k
large enough.
This proves the claim. ∎
The following lemma is stated in [7] but its proof contains a gap. This can be fixed with the help of Lemma 3.5. We give the details.
Lemma 3.9 (Splitting lemma).
Let (uk) be a bounded (PS)d-sequence for IV. Then, after replacing (uk) by a subsequence, there exists a solution u of problem (1.1), a number m∈N∪{0}, m nontrivial solutions w1,…,wm to the limit problem (1.2), and m sequences of points (yj,k)∈RN, 1≤j≤m, satisfying
|
y
j
,
k
|
→
∞
and
|
y
j
,
k
-
y
i
,
k
|
→
∞
if
i
≠
j
,
u
k
-
∑
i
=
1
m
w
i
(
⋅
-
y
i
,
k
)
→
u
in
D
1
,
2
(
ℝ
N
)
,
d
=
I
V
(
u
)
+
∑
i
=
1
m
I
0
(
w
i
)
.
Proof.
Passing to a subsequence, we have that uk⇀u weakly in D1,2(ℝN). It follows from Lemma 3.8 that
o
(
1
)
=
I
V
′
(
u
k
)
φ
=
〈
u
k
,
φ
〉
V
-
∫
ℝ
N
f
(
u
k
)
φ
=
〈
u
,
φ
〉
V
-
∫
ℝ
N
f
(
u
)
φ
+
o
(
1
)
=
I
V
′
(
u
)
φ
+
o
(
1
)
for every φ∈𝒞c∞(ℝN). Hence, u solves (1.1). Set u1,k:=uk-u. Then u1,k⇀0 weakly in D1,2(ℝN). Moreover, Lemma 3.8 implies that
(3.5)
{
I
V
(
u
k
)
=
I
0
(
u
1
,
k
)
+
I
V
(
u
)
+
o
(
1
)
,
o
(
1
)
=
I
V
′
(
u
k
)
=
I
0
′
(
u
1
,
k
)
+
o
(
1
)
in
(
D
1
,
2
(
ℝ
N
)
)
′
.
If u1,k→0 strongly in D1,2(ℝN), the proof is completed. So assume it does not. Then, as IV′(u1,k)u1,k→0, after passing to a subsequence, we have that
0
<
C
≤
∥
u
1
,
k
∥
V
2
=
∫
ℝ
N
f
(
u
1
,
k
)
u
1
,
k
+
o
(
1
)
.
So, by Lemma 3.5, there exist δ>0 and a sequence (y1,k) in ℝN such that
(3.6)
∫
B
1
(
y
1
,
k
)
|
u
1
,
k
|
2
=
sup
y
∈
ℝ
N
∫
B
1
(
y
)
|
u
1
,
k
|
2
>
δ
.
Set vk:=u1,k(⋅+y1,k). Passing to a subsequence, we have that vk⇀w1 weakly in D1,2(ℝN) and vk→w1 in Lloc2(ℝN). Inequality (3.6) implies that w1≠0 and, as u1,k⇀0 weakly in D1,2(ℝN), we conclude that |y1,k|→∞. Next, we shall show that w1 is a solution to the limit problem (1.2). Let φ∈𝒞c∞(ℝN) and set φk:=φ(⋅-y1,k). Using Lemma 3.8 and performing a change of variable, we obtain that
I
0
′
(
w
1
)
φ
+
o
(
1
)
=
I
0
′
(
v
k
)
φ
=
I
0
′
(
u
1
,
k
)
φ
k
=
o
(
1
)
.
This proves that w1 solves problem (1.2). Moreover, Lemma 3.8 implies that
I
0
(
v
k
)
=
I
0
(
v
k
-
w
1
)
+
I
0
(
w
1
)
+
o
(
1
)
,
o
(
1
)
=
I
V
′
(
v
k
)
=
I
0
′
(
v
k
-
w
1
)
+
o
(
1
)
in
(
D
1
,
2
(
ℝ
N
)
)
′
.
Set u2,k:=u1,k-w1(⋅-y1,k)=uk-u-w1(⋅-y1,k). Then u2,k⇀0 weakly in D1,2(ℝN) and, after a change of variable, from identities (3.5) and (3.7), we obtain that
(3.7)
{
I
0
(
u
2
,
k
)
=
I
0
(
u
1
,
k
)
-
I
0
(
w
1
)
+
o
(
1
)
=
I
V
(
u
k
)
-
I
V
(
u
)
-
I
0
(
w
1
)
+
o
(
1
)
,
I
0
′
(
u
2
,
k
)
=
I
0
′
(
u
1
,
k
)
+
o
(
1
)
=
I
V
′
(
u
k
)
+
o
(
1
)
=
o
(
1
)
in
(
D
1
,
2
(
ℝ
N
)
)
′
.
If u2,k→0 strongly in D1,2(ℝN), the proof is completed. If not, we repeat the argument. After a finite number of steps, we will arrive to a sequence (um+1,k) which converges strongly to 0 in D1,2(ℝN). This completes the proof. ∎
Corollary 3.10 (Compactness).
If cV is not attained by IV on NV, then the following statements hold true:
If
σ
k
∈
ℝ
and
u
k
∈
ℳ
σ
k
are such that
σ
k
→
0
, IV(uk)→d∈(c0,2c0) and ∇ℳσkIV(uk)→0, then (uk) contains a convergent subsequence.
Proof.
(a) Let (uk) be a minimizing sequence for IV on 𝒩V. By Ekeland’s variational principle and Lemma 3.6, we may assume that (uk) is a bounded (PS)cV-sequence for IV. As cV is not attained, the splitting lemma implies that cV≥c0.
(b) By Lemmas 3.6 and 3.7, we have that (uk) is a bounded (PS)d-sequence for IV. Arguing by contradiction, assume that (uk) does not contain a convergent subsequence. Then, the splitting lemma yields a solution w of the limit problem (1.2) with d=I0(w), contradicting Lemma 3.4. This proves our claim. ∎
4 Existence of a Positive Solution
The proof of our main result requires some delicate estimates. The following lemma will help us obtain them.
Lemma 4.1.
If
y
0
,
y
∈
ℝ
N
, y0≠y and α , and β are positive constants such that α+β>N, then there exists C1=C1(α,β,|y-y0|)>0 such that
∫
ℝ
N
d
x
(
1
+
|
x
-
R
y
0
|
)
α
(
1
+
|
x
-
R
y
|
)
β
≤
C
1
R
-
μ
for all
R
≥
1
, where
μ
:=
min
{
α
,
β
,
α
+
β
-
N
}
.
If
y
0
,
y
∈
ℝ
N
∖
{
0
}
, and κ and ϑ are positive constants such that κ+2ϑ>N, then there exists C2=C2(κ,ϑ,|y0|,|y|)>0 such that
∫
ℝ
N
d
x
(
1
+
|
x
|
)
κ
(
1
+
|
x
-
R
y
0
|
)
ϑ
(
1
+
|
x
-
R
y
|
)
ϑ
≤
C
2
R
-
τ
for all
R
≥
1
, where
τ
:=
min
{
κ
,
2
ϑ
,
κ
+
2
ϑ
-
N
}
.
Proof.
(a) After a suitable translation, we may assume that y=-y0. Let 2ρ:=|y-y0|>0. In the following, C will denote different positive constants which depend on α, β and ρ. If |x-Ry0|≤ρR, then |x-Ry|≥ρR. Hence,
∫
B
ρ
R
(
R
y
0
)
d
x
(
1
+
|
x
-
R
y
0
|
)
α
(
1
+
|
x
-
R
y
|
)
β
=
∫
B
ρ
R
(
R
y
0
)
d
x
(
1
+
|
x
-
R
y
0
|
)
α
(
ρ
R
)
β
=
C
R
-
β
∫
B
ρ
R
(
0
)
d
x
(
1
+
|
x
|
)
α
≤
C
(
R
-
β
+
R
N
-
(
α
+
β
)
)
≤
C
R
-
μ
.
Similarly,
∫
B
ρ
R
(
R
y
)
d
x
(
1
+
|
x
-
R
y
0
|
)
α
(
1
+
|
x
-
R
y
|
)
β
≤
C
(
R
-
α
+
R
N
-
(
α
+
β
)
)
≤
C
R
-
μ
.
Let
H
+
:=
{
z
∈
ℝ
N
:
|
z
-
R
y
|
≥
|
z
-
R
y
0
|
}
and
H
-
:=
{
z
∈
ℝ
N
:
|
z
-
R
y
|
≤
|
z
-
R
y
0
|
}
.
Setting x=Rz, we obtain
∫
H
+
∖
B
ρ
R
(
R
y
0
)
d
x
(
1
+
|
x
-
R
y
0
|
)
α
(
1
+
|
x
-
R
y
|
)
β
≤
∫
H
+
∖
B
ρ
R
(
R
y
0
)
d
x
(
1
+
|
x
-
R
y
0
|
)
α
+
β
≤
∫
H
+
∖
B
ρ
(
y
0
)
R
N
d
x
(
R
|
z
-
y
0
|
)
α
+
β
=
C
R
N
-
(
α
+
β
)
≤
C
R
-
μ
.
Similarly,
∫
H
-
∖
B
ρ
R
(
R
y
)
d
x
(
1
+
|
x
-
R
y
0
|
)
α
(
1
+
|
x
-
R
y
|
)
β
≤
C
R
-
μ
.
Since ℝN∖[BρR(Ry0)∪BρR(Ry)]=[H+∖BρR(Ry0)]∪[H-∖BρR(Ry)], the previous estimates yield (a).
(b) From Hölder’s inequality and inequality (a), we obtain
∫
ℝ
N
d
x
(
1
+
|
x
|
)
κ
(
1
+
|
x
-
R
y
0
|
)
ϑ
(
1
+
|
x
-
R
y
|
)
ϑ
≤
(
∫
ℝ
N
d
x
(
1
+
|
x
|
)
κ
(
1
+
|
x
-
R
y
0
|
)
2
ϑ
)
1
/
2
(
∫
ℝ
N
d
x
(
1
+
|
x
|
)
κ
(
1
+
|
x
-
R
y
|
)
2
ϑ
)
1
/
2
≤
C
2
R
-
τ
,
as claimed. ∎
Let ω be the positive radial ground state of the limit problem (1.2). Fix y0∈ℝN, with |y0|=1, and let B2(y0):={x∈ℝN:|x-y0|≤2}. For R≥1 and each y∈∂B2(y0), we define
ω
0
R
:=
ω
(
⋅
-
R
y
0
)
,
ω
y
R
:=
ω
(
⋅
-
R
y
)
,
and we set
ε
R
:=
∫
ℝ
N
f
(
ω
0
R
)
ω
y
R
=
∫
ℝ
N
f
(
ω
(
x
-
R
y
0
)
)
ω
(
x
-
R
y
)
d
x
.
As before, C will denote a positive constant, not necessarily the same one.
Lemma 4.2.
There exists a constant C3>0 such that
ε
R
≤
C
3
R
-
(
N
-
2
)
for all y∈∂B2(y0) and all R≥1.
Proof.
By assumption (f1), we have that |f(s)|≤A1|s|2∗-1. On the other hand, from estimates (2.1) and Lemma 4.1 (a), we obtain
(4.1)
∫
ℝ
N
(
ω
0
R
)
2
∗
-
1
ω
y
R
≤
C
∫
ℝ
N
d
x
(
1
+
|
x
-
R
y
0
|
)
N
+
2
(
1
+
|
x
-
R
y
|
)
N
-
2
≤
C
R
-
(
N
-
2
)
.
Therefore,
∫
ℝ
N
f
(
ω
0
R
)
ω
y
R
≤
A
1
∫
ℝ
N
(
ω
0
R
)
2
∗
-
1
ω
y
R
≤
C
R
-
(
N
-
2
)
for all y∈∂B2(y0) and all R≥1, as claimed. ∎
Lemma 4.3.
There exists a constant C4>0 such that
∫
ℝ
N
f
(
s
ω
0
R
)
t
ω
y
R
≥
C
4
R
-
(
N
-
2
)
for all s,t≥12, y∈∂B2(y0) and R≥1.
Proof.
Note that if |x|<1, then, for every y∈∂B2(y0) and R≥1,
1
+
|
x
-
R
(
y
-
y
0
)
|
<
1
+
|
x
|
+
R
|
y
-
y
0
|
<
4
R
.
Assumption (f3) implies that f(s)s is strictly increasing for s>0; hence, so is f. Performing a change of variable and using estimate (2.1), we obtain
∫
ℝ
N
f
(
s
ω
0
R
)
t
ω
y
R
≥
t
∫
ℝ
N
f
(
1
2
ω
0
R
)
ω
y
R
≥
1
2
∫
B
1
(
R
y
0
)
f
(
1
2
ω
0
R
)
ω
y
R
≥
1
4
[
min
x
∈
B
1
(
0
)
f
(
1
2
ω
(
x
)
)
]
∫
B
1
(
0
)
ω
(
x
-
R
(
y
-
y
0
)
)
d
x
≥
C
∫
B
1
(
0
)
(
1
+
|
x
-
R
(
y
-
y
0
)
|
)
-
(
N
-
2
)
d
x
≥
C
R
-
(
N
-
2
)
for all s,t≥12, y∈∂B2(y0) and R≥1, as claimed. ∎
Note that Lemmas 4.2 and 4.3 yield
(4.2)
C
4
R
-
(
N
-
2
)
≤
ε
R
:=
∫
ℝ
N
f
(
ω
0
R
)
ω
y
R
≤
C
3
R
-
(
N
-
2
)
for all y∈∂B2(y0) and R≥1.
Lemma 4.4.
For each b>1, there exists a constant Cb>0 such that
|
∫
ℝ
N
(
s
f
(
ω
0
R
)
-
f
(
s
ω
0
R
)
)
ω
y
R
|
≤
C
b
|
s
-
1
|
ε
R
for all s∈[0,b], y∈∂B2(y0) and R≥1.
Proof.
Fix t∈ℝ and set g(s):=sf(t)-f(st). By the mean value theorem, there exists ζ between 1 and s such that
|
s
f
(
t
)
-
f
(
s
t
)
|
=
|
g
(
s
)
-
g
(
1
)
|
=
|
f
(
t
)
-
f
′
(
ζ
t
)
t
|
|
s
-
1
|
≤
(
|
f
(
t
)
|
+
|
f
′
(
ζ
t
)
t
|
)
|
s
-
1
|
≤
A
1
(
|
t
|
2
∗
-
1
+
|
ζ
|
2
∗
-
2
|
t
|
2
∗
-
1
)
|
s
-
1
|
≤
A
1
(
1
+
b
2
∗
-
2
)
|
t
|
2
∗
-
1
|
s
-
1
|
for all
s
∈
[
0
,
b
]
,
where the second-to-last inequality follows from assumption (f1). So, from inequalities (4.1) and (4.2), we obtain that
∫
ℝ
N
(
s
f
(
ω
0
R
)
-
f
(
s
ω
0
R
)
)
ω
y
R
≤
C
|
s
-
1
|
∫
ℝ
N
(
ω
0
R
)
2
∗
-
1
ω
y
R
≤
C
|
s
-
1
|
R
-
(
N
-
2
)
≤
C
|
s
-
1
|
ε
R
for all s∈[0,b], y∈∂B2(0) and R≥1, as claimed. ∎
Lemma 4.5.
There exists τ>N-2 such that
∫
ℝ
N
V
+
(
ω
0
R
+
ω
y
R
)
2
≤
C
R
-
τ
for every y∈∂B2(y0) and R≥1.
Proof.
From assumption (V2), estimates (2.1) and Lemma 4.1 (b), we immediately obtain that
∫
ℝ
N
V
+
(
ω
0
R
+
ω
y
R
)
2
=
∫
ℝ
N
V
+
(
ω
0
R
)
2
+
2
∫
ℝ
N
V
+
ω
0
R
ω
y
R
+
∫
ℝ
N
V
+
(
ω
y
R
)
2
≤
C
R
-
τ
,
with τ:=min{κ,2(N-2),κ+N-4}>N-2. ∎
For each R≥1, y∈∂B2(y0) and λ∈[0,1], we define
z
λ
,
y
R
:=
λ
ω
0
R
+
(
1
-
λ
)
ω
y
R
.
Lemma 4.6.
For each R≥1, y∈∂B2(y0) and λ∈[0,1], there exists a unique Tλ,yR>0 such that
T
λ
,
y
R
z
λ
,
y
R
∈
𝒩
V
.
Moreover, there exist R0≥1 and T0>2 such that Tλ,yR∈[0,T0) for all R≥R0, y∈∂B2(y0) and λ∈[0,1], and Tλ,yR is a continuous function of the variables λ,y and R.
Proof.
The proof is the same as that of [13, Lemma 3.2], with obvious changes. ∎
Lemma 4.7.
For λ=12, we have that Tλ,yR→2 as R→∞ uniformly in y∈∂B2(y0).
Proof.
Since ω solves (1.2), we have that
J
0
(
ω
0
R
+
ω
y
R
)
=
J
0
(
ω
0
R
)
+
J
0
(
ω
y
R
)
+
2
∫
ℝ
N
∇
ω
0
R
⋅
∇
ω
y
R
-
∫
ℝ
N
[
f
(
ω
0
R
+
ω
y
R
)
-
f
(
ω
0
R
)
]
ω
0
R
-
∫
ℝ
N
[
f
(
ω
0
R
+
ω
y
R
)
-
f
(
ω
y
R
)
]
ω
y
R
=
2
∫
ℝ
N
∇
ω
⋅
∇
ω
y
-
y
0
R
-
∫
ℝ
N
[
f
(
ω
0
R
+
ω
y
R
)
-
f
(
ω
0
R
)
]
ω
0
R
-
∫
ℝ
N
[
f
(
ω
0
R
+
ω
y
R
)
-
f
(
ω
y
R
)
]
ω
y
R
.
Set h(s):=A1min{|s|p-2,|s|q-2}≤|s|2∗-2. Using inequality (3.4), we get that
∫
ℝ
N
|
f
(
ω
0
R
+
ω
y
R
)
-
f
(
ω
0
R
)
|
ω
0
R
≤
∫
ℝ
N
h
(
ω
0
R
+
ω
y
R
)
ω
y
R
ω
0
R
≤
|
ω
0
R
+
ω
y
R
|
2
∗
2
∗
-
2
(
∫
ℝ
N
(
ω
y
R
)
2
∗
/
2
(
ω
0
R
)
2
∗
/
2
)
2
/
2
∗
≤
C
(
∫
ℝ
N
(
ω
y
-
y
0
R
)
2
∗
/
2
ω
2
∗
/
2
)
2
/
2
∗
and, similarly,
∫
ℝ
N
|
f
(
ω
0
R
+
ω
y
R
)
-
f
(
ω
y
R
)
|
ω
y
R
≤
C
(
∫
ℝ
N
ω
2
∗
/
2
(
ω
y
-
y
0
R
)
2
∗
/
2
)
2
/
2
∗
.
From the above inequalities, we conclude that J0(ω0R+ωyR)=oR(1), where oR(1)→0 as R→∞ uniformly in y∈∂B2(y0). Hence, for λ=12, we get that
J
V
(
2
z
λ
,
y
R
)
=
J
V
(
ω
0
R
+
ω
y
R
)
=
J
0
(
ω
0
R
+
ω
y
R
)
+
∫
ℝ
N
V
(
ω
0
R
+
ω
y
R
)
2
=
o
R
(
1
)
.
This yields the claim. ∎
The proof of the next result follows that of [15, Lemma 2.1].
Lemma 4.8.
For each a>0, there exists Ca≥0 such that
F
(
s
+
t
)
-
F
(
s
)
-
F
(
t
)
-
f
(
s
)
t
-
f
(
t
)
s
≥
-
C
a
(
s
t
)
1
+
ν
/
2
for all s,t∈[0,a] and ν∈(0,q-2).
Proof.
The inequality is clearly satisfied if s=0 or t=0. Assumption (f3) implies that f is increasing for s>0. Therefore,
F
(
s
+
t
)
-
F
(
s
)
=
∫
s
s
+
t
f
(
ζ
)
d
ζ
≥
f
(
s
)
t
for all s,t>0. Moreover, by (f2), we have that f(s)=o(|s|1+ν) for any ν∈(0,q-2). Hence, there exists Ma>0 such that f(s)≤Mas1+ν for all s∈[0,a]. It follows that
F
(
s
+
t
)
-
F
(
s
)
-
F
(
t
)
-
f
(
s
)
t
-
f
(
t
)
s
≥
-
F
(
t
)
-
f
(
t
)
s
≥
-
M
a
∫
0
t
ζ
1
+
ν
d
ζ
-
M
a
s
t
1
+
ν
=
-
M
a
(
t
2
+
ν
2
+
ν
+
s
t
1
+
ν
)
for all s,t∈[0,a]. So, if t≤s, we get that
F
(
s
+
t
)
-
F
(
s
)
-
F
(
t
)
-
f
(
s
)
t
-
f
(
t
)
s
≥
-
3
2
M
a
(
s
t
)
1
+
ν
/
2
.
As this expression is symmetric in s and t, it holds true also when s≤t, and the proof is complete. ∎
With the previous lemmas on hand, we now prove the following estimate.
Proposition 4.9.
There exists R1≥1 and, for each R>R1, a number ηR>0 such that
I
V
(
T
λ
,
y
R
z
λ
,
y
R
)
≤
2
c
0
-
η
R
for all λ∈[0,1] and all y∈∂B2(y0).
Proof.
To simplify the notation, let us set
s
:=
T
λ
,
y
R
λ
,
t
:=
T
λ
,
y
R
(
1
-
λ
)
,
ω
0
:=
ω
0
R
,
ω
y
:=
ω
y
R
.
Then s,t∈[0,T0) if R≥R0, with R0≥1 and T0>2, as in Lemma 4.6, and
I
V
(
T
λ
,
y
R
z
λ
,
y
R
)
=
I
V
(
s
ω
0
+
t
ω
y
)
=
s
2
2
∫
ℝ
N
|
∇
ω
0
|
2
+
s
2
2
∫
ℝ
N
V
ω
0
2
+
t
2
2
∫
ℝ
N
|
∇
ω
y
|
2
+
t
2
2
∫
ℝ
N
V
ω
y
2
+
s
t
∫
ℝ
N
∇
ω
0
⋅
∇
ω
y
+
s
t
∫
ℝ
N
V
ω
0
ω
y
-
∫
ℝ
N
F
(
s
ω
0
+
t
ω
y
)
=
I
1
+
I
2
+
I
3
+
I
4
+
I
5
,
where
I
1
=
s
2
2
∫
ℝ
N
|
∇
ω
0
|
2
-
∫
ℝ
N
F
(
s
ω
0
)
+
t
2
2
∫
ℝ
N
|
∇
ω
y
|
2
-
∫
ℝ
N
F
(
t
ω
y
)
,
I
2
=
s
2
2
∫
ℝ
N
V
ω
0
2
+
t
2
2
∫
ℝ
N
V
ω
y
2
+
s
t
∫
ℝ
N
V
ω
0
ω
y
,
I
3
=
s
t
∫
ℝ
N
∇
ω
0
⋅
∇
ω
y
,
I
4
=
-
∫
ℝ
N
[
F
(
s
ω
0
+
t
ω
y
)
-
F
(
s
ω
0
)
-
F
(
t
ω
y
)
-
f
(
s
ω
0
)
t
ω
y
-
f
(
t
ω
y
)
s
ω
0
]
,
I
5
=
-
∫
ℝ
N
f
(
s
ω
0
)
t
ω
y
-
∫
ℝ
N
f
(
t
ω
y
)
s
ω
0
.
Next, we estimate each Ii, with i=1,…,5. As ω0 and ωy are ground states of the limit problem (1.2), Lemma 3.2 (d) yields
I
1
=
I
0
(
s
ω
0
)
+
I
0
(
t
ω
y
)
≤
I
0
(
ω
0
)
+
I
0
(
ω
y
)
=
2
c
0
.
From Lemma 4.5 and estimates (4.2), we get that
I
2
≤
C
R
-
τ
=
o
(
ε
R
)
.
Lemma 4.8, with ν∈(2N-2,q-2), and Lemma 4.1 (a), with α=β=(1+ν2)(N-2), imply, for some μ>N-2, that
I
4
≤
C
|
s
t
|
1
+
ν
2
∫
ℝ
N
(
ω
0
ω
y
)
1
+
ν
2
≤
C
R
-
μ
=
o
(
ε
R
)
.
We write the sum of the remaining terms as
I
3
+
I
5
=
t
2
∫
ℝ
N
[
s
f
(
ω
0
)
-
f
(
s
ω
0
)
]
ω
y
+
s
2
∫
ℝ
N
[
t
f
(
ω
y
)
-
f
(
t
ω
y
)
]
ω
0
-
1
2
∫
ℝ
N
f
(
s
ω
0
)
t
ω
y
-
1
2
∫
ℝ
N
f
(
t
ω
y
)
s
ω
0
.
By Lemma 4.4, there exists a constant C>0 such that
t
2
∫
ℝ
N
|
s
f
(
ω
0
)
-
f
(
s
ω
0
)
|
ω
y
+
s
2
∫
ℝ
N
|
t
f
(
ω
y
)
-
f
(
t
ω
y
)
|
ω
0
≤
C
(
|
s
-
1
|
+
|
t
-
1
|
)
ε
R
for all s,t∈[0,T0], y∈∂B2(y0) and R≥R0. Moreover, Lemma 4.3 yields a constant C0>0 such that
1
2
∫
ℝ
N
f
(
s
ω
0
)
t
ω
y
+
1
2
∫
ℝ
N
f
(
t
ω
y
)
s
ω
0
≥
C
0
ε
R
for all s,t≥12, y∈∂B2(y0) and R≥R0. By Lemma 4.7, if λ=12, then s,t→1 as R→∞. Therefore, there exist R1≥R0 and δ∈(0,12) such that
I
3
+
I
5
≤
-
C
0
2
ε
R
for all λ∈[12-δ,12+δ], y∈∂B2(y0) and R≥R1. Summing up, we have shown that
(4.3)
I
V
(
s
ω
0
+
t
ω
y
)
≤
2
c
0
-
C
0
2
ε
R
+
o
(
ε
R
)
for all λ∈[12-δ,12+δ], y∈∂B2(y0), R≥R1.
On the other hand, by Lemma 3.2 (d), there exists ϑ∈(0,c0) such that
I
1
=
I
0
(
s
ω
0
)
+
I
0
(
t
ω
y
)
≤
2
c
0
-
ϑ
for all λ∈[0,12-δ]∪[12+δ,1], y∈∂B2(y0) and R sufficiently large. Since I2+⋯+I5≤O(εR), we conclude that
(4.4)
I
V
(
s
ω
0
+
t
ω
y
)
≤
2
c
0
-
2
ϑ
+
O
(
ε
R
)
for all λ∈[0,12-δ]∪[12+δ,1], y∈∂B2(y0) and R sufficiently large.
Inequalities (4.3) and (4.4) yield the statement of the proposition. ∎
Lemma 4.10.
For any δ>0, there exists R2>0 such that
I
V
(
T
λ
,
y
R
z
λ
,
y
R
)
<
c
0
+
δ
for λ=1, every y∈∂B2(y0) and R≥R2. In particular, cV≤c0.
Proof.
By Lemma 4.6, Tλ,yR is bounded uniformly in λ,y and R. So, from Lemmas 3.2 (d) and 4.5, we obtain that
I
V
(
T
1
,
y
R
z
1
,
y
R
)
=
I
0
(
T
1
,
y
R
ω
y
R
)
+
(
T
1
,
y
R
)
2
∫
ℝ
N
V
(
ω
y
R
)
2
≤
c
0
+
o
R
(
1
)
,
where oR(1)→0 as R→∞ uniformly in y∈∂B2(y0), and the claim is proved. ∎
For c∈ℝ, set
I
V
c
:=
{
u
∈
D
1
,
2
(
ℝ
N
)
:
I
V
(
u
)
≤
c
}
.
Lemma 4.11 (Deformation).
If cV is not attained by IV on NV, then cV=c0. If, moreover, IV does not have a critical value in (c0,2c0), then, for any given δ,η∈(0,c04), there exists a continuous function
π
:
𝒩
V
∩
I
V
2
c
0
-
η
→
𝒩
V
∩
I
V
c
0
+
δ
such that π(u)=u for all u∈NV∩IVc0+δ.
Proof.
If cV is not attained, Corollary 3.10 (a) and Lemma 4.10 imply that cV=c0.
Recall that 𝒩V is a 𝒞1-manifold. From Lemma 3.6 and Corollary 3.10 (b), we have that the following statement is true: If σk∈ℝ and uk∈ℳσk are such that σk→0, IV(uk)→d∈(c0,2c0) and either ∇ℳσkIV(uk)→0 or ∇JV(uk)→0, then (uk) contains a convergent subsequence. (In fact, Lemma 3.6 says that (∇JV(uk)) must be bounded away from 0). This allows us to apply [11, Theorem 2.5] to conclude that there exists ε^>0 such that for each ε∈(0,ε^), there exists a homeomorphism ϕ:𝒩V→𝒩V such that
ϕ
(
u
)
=
u
if IV(u)∉[d-ε,d+ε],
I
V
(
ϕ
(
u
)
)
≤
I
V
(
u
)
for all u∈𝒩V,
I
V
(
ϕ
(
u
)
)
≤
d
-
ε
for all u∈𝒩V, with IV(u)≤d+ε.
Our claim follows easily from this fact. ∎
Let 𝔟:L2∗(ℝN)∖{0}→ℝN be a barycenter map, i.e., a continuous map such that
𝔟
(
u
(
⋅
-
y
)
)
=
𝔟
(
u
)
+
y
and
𝔟
(
u
∘
Θ
-
1
)
=
Θ
(
𝔟
(
u
)
)
for all u∈L2∗(ℝN)∖{0} and y∈ℝN, and every linear isometry Θ of ℝN, see [5, 12]. Note that 𝔟(u)=0 if u is radial.
Lemma 4.12.
If cV is not attained by IV on NV, then there exists δ>0 such that
𝔟
(
u
)
≠
0
for all
u
∈
𝒩
V
∩
I
V
c
0
+
δ
.
Proof.
The proof is the same as that of [13, Lemma 3.11]. ∎
Proof of Theorem 1.1.
If cV is attained by IV at some u∈𝒩V, then u is a nontrivial solution of problem (1.1). So assume that cV is not attained. Then, by Lemma 4.11, cV=c0. We will show that IV has a critical value in (c0,2c0).
Lemma 4.12 allows us to choose δ∈(0,c04) such that
𝔟
(
u
)
≠
0
for all
u
∈
𝒩
V
∩
I
V
c
0
+
δ
and, by Proposition 4.9 and Lemma 4.10, we may choose R≥1 and η∈(0,c04) such that
I
V
(
T
λ
,
y
R
z
λ
,
y
R
)
≤
{
2
c
0
-
η
for all
λ
∈
[
0
,
1
]
and all
y
∈
∂
B
2
(
y
0
)
,
c
0
+
δ
for
λ
=
1
and all
y
∈
∂
B
2
(
y
0
)
.
Define ι:B2(y0)→𝒩V∩IV2c0-η by
ι
(
(
1
-
λ
)
y
0
+
λ
y
)
:=
T
λ
,
y
R
z
λ
,
y
R
,
with
λ
∈
[
0
,
1
]
,
y
∈
∂
B
2
(
y
0
)
.
Arguing by contradiction, assume that IV does not have a critical value in (c0,2c0). Then, by Lemma 4.11, there exists a continuous function
π
:
𝒩
V
∩
I
V
2
c
0
-
η
→
𝒩
V
∩
I
V
c
0
+
δ
such that π(u)=u for all u∈𝒩V∩IVc0+δ. The function ψ:B2(y0)→∂B2(y0), given by
ψ
(
x
)
:=
2
(
𝔟
∘
π
∘
ι
)
(
x
)
|
(
𝔟
∘
π
∘
ι
)
(
x
)
|
,
is well defined and continuous, and ψ(y)=y for every y∈∂B2(y0). This is a contradiction. Therefore, IV must have a critical point u∈𝒩V, with IV(u)∈(c0,2c0).
By Lemma 3.3, u does not change sign and, since f is odd, -u is also a solution of (1.1). This proves that problem (1.1) has a positive solution. ∎
and
Liliane A. Maia
