1 Introduction
A very interesting area of nonlinear analysis lies in the study of elliptic equations involving fractional operators. Recently, great attention has been focused on these problems, both for the pure mathematical research and in view of concrete real-world applications. Indeed, this type of operators appears in a quite natural way in different contexts such as the description of several physical phenomena (see [23, 31, 45]).
In this paper, we study the existence of least energy sign-changing (or nodal) solutions for the nonlinear problem involving the fractional Laplacian
(1.1)
{
(
-
Δ
)
α
u
+
V
(
x
)
u
=
K
(
x
)
f
(
u
)
in
ℝ
N
,
u
∈
𝒟
α
,
2
(
ℝ
N
)
,
with α∈(0,1), N>2α and 𝒟α,2(ℝN) being defined as the completion of u∈Cc∞(ℝN) with respect to the Gagliardo semi-norm
[
u
]
:=
(
∬
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
)
1
/
2
,
where Cc∞(ℝN) is the space of smooth functions with compact support.
The operator (-Δ)α is the so-called fractional Laplacian which, up to a positive constant, may be defined through the kernel representation
(
-
Δ
)
α
u
(
x
)
:=
P
.
V
.
∫
ℝ
N
u
(
x
)
-
u
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
y
for every u:ℝN→ℝ sufficiently smooth. Here, the abbreviation P.V. stands for “in the principal value sense”.
For an elementary introduction to the fractional Laplacian and fractional Sobolev spaces, we refer the interested reader to [31, 45].
Equation (1.1) appears in the study of standing wave solutions ψ(x,t)=u(x)e-ıωt to the following fractional Schrödinger equation
ı
ℏ
∂
ψ
∂
t
=
ℏ
2
(
-
Δ
)
α
ψ
+
W
(
x
)
ψ
-
f
(
|
ψ
|
)
in
ℝ
N
,
where ℏ is the Planck constant, W:ℝN→ℝ is an external potential and f a suitable nonlinearity. The fractional Schrödinger equation is one of the most important objects of the fractional quantum mechanics because it appears in problems involving nonlinear optics, plasma physics and condensed matter physics. The previous equation has been introduced for the first time by Laskin [42, 43] as a result of expanding the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths. To be physically consistent, one of the main features of the Schrödinger equation is that, in the semi-classical limit in which the diffusion operator arises as a singular perturbation, the wave functions “concentrate” into “particles”; see [30, 29] for more details.
Subsequently, many papers appeared studying existence, multiplicity and behavior of solutions to fractional Schrödinger equations [1, 7, 8, 6, 9, 10, 11, 33, 34, 36, 38, 40, 21, 50]. Generally, nonlinear problems involving nonlocal operators have received a great interest from the mathematical community thanks to their intriguing structure and in view of several applications such as phase transition, optimization, obstacle problems, minimal surfaces and regularity theory; see, among others, the papers [24, 25, 27, 26, 41].
When α=1, the equation in (1.1) becomes the classical nonlinear Schrödinger equation
(1.2)
-
Δ
u
+
V
(
x
)
u
=
K
(
x
)
f
(
u
)
in
ℝ
N
,
which has been extensively studied in the last twenty years. We do not intend to review the huge bibliography of equations like (1.2), we just emphasize that the potential V:ℝn→ℝ has a crucial role concerning the existence and behavior of solutions. For instance, when V is a positive constant or V is radially symmetric, it is natural to look for radially symmetric solutions; see [51, 53]. On the other hand, after the seminal paper of Rabinowitz [49], where the potential V is assumed to be coercive, several different assumptions are adopted in order to obtain existence and multiplicity results, see [15, 16, 39]. An important class of problems associated with (1.2) is the zero mass case, which occurs when
lim
|
x
|
→
+
∞
V
(
x
)
=
0
.
To study these problems, many authors used several variational methods; see [2, 4, 5, 14, 19, 20, 22, 35] for problems set in ℝN, as well as [3, 17, 18, 28] for problems in bounded domain with homogeneous boundary conditions.
We notice that there is a huge literature on these classical topics, and, in recent years, also a lot of papers related to the study of fractional and nonlocal operators of elliptic type, through critical point theory, appeared. Indeed, a natural question is whether or not these techniques may be adapted in order to investigate the fractional analogue of the classical elliptic case. We refer to the recent books [32, 45], in which the analysis of some fractional elliptic problems, via classical variational methods and other novel approaches (see, for instance, [13, 37, 46, 47]), is performed.
In this spirit, the goal of the present paper is to study the nonlocal counterpart of (1.2), and to prove the existence of sign-changing solutions to problem (1.1).
Before stating our main result, we introduce the basic assumptions on V,K and f.
More precisely, we suppose that the functions V,K:ℝN→ℝ are continuous on ℝN, and we say that (V,K)∈𝒦 if the following conditions hold:
V
(
x
)
,
K
(
x
)
>
0
for all x∈ℝN and K∈L∞(ℝN).
If {An}n∈ℕ⊂ℝN is a sequence of Borel sets such that the Lebesgue measure m(An) is less than or equal to R for all n∈ℕ and some R>0, then
lim
r
→
+
∞
∫
A
n
∩
ℬ
r
c
(
0
)
K
(
x
)
𝑑
x
=
0
uniformly in n∈ℕ, where ℬrc(0):=ℝN∖ℬr(0) and
ℬ
r
(
0
)
:=
{
x
∈
ℝ
N
:
|
x
|
<
r
}
.
Furthermore, one of the following conditions occurs:
There exists m∈(2,2α*) such that
K
(
x
)
V
(
x
)
2
α
*
-
m
/
(
2
α
*
-
2
)
→
0
as
|
x
|
→
+
∞
,
where 2α*:=2NN-2α.
We recall that assumptions (h1)–(h4) were introduced for the first time by Alves and Souto in [2]. It is very important to observe that (h2) is weaker than any one of the conditions below used in the above-mentioned papers to study zero-mass problems:
There are r≥1 and ρ≥0 such that K∈Lr(ℝN∖ℬρ(0)).
K
=
H
1
+
H
2
, with H1 and H2 verifying (i) and (ii), respectively.
Now, we provide some examples of functions V and K satisfying (h1)–(h4). Let {Bn}n∈ℕ be a disjoint sequence of open balls in ℝN centered in ξn=(n,0,…,0) and consider a nonnegative function H3 such that
H
3
=
0
in
ℝ
N
∖
⋃
n
=
1
∞
ℬ
n
,
H
3
(
ξ
n
)
=
1
,
∫
ℬ
n
H
3
(
x
)
𝑑
x
=
2
-
n
.
Then the pairs (V,K) given by
K
(
x
)
=
V
(
x
)
=
H
3
(
x
)
+
1
log
(
2
+
|
x
|
)
and
K
(
x
)
=
H
3
(
x
)
+
1
log
(
2
+
|
x
|
)
and
V
(
x
)
=
H
3
(
x
)
+
(
1
log
(
2
+
|
x
|
)
)
2
α
*
-
2
2
α
*
-
m
for some m∈(2,2α*) belong to the class 𝒦.
For the nonlinearity f:ℝ→ℝ we assume that it is a C0-function and satisfies the following growth conditions in the origin and at infinity:
lim
|
t
|
→
0
f
(
t
)
|
t
|
=
0
if
(
h
3
)
holds
or
lim
|
t
|
→
0
f
(
t
)
|
t
|
m
-
1
<
+
∞
if
(
h
4
)
holds.
f has a quasicritical growth at infinity, namely
lim
|
t
|
→
+
∞
f
(
t
)
|
t
|
2
α
*
-
1
=
0
.
F has a superquadratic growth at infinity, that is,
lim
|
t
|
→
+
∞
F
(
t
)
|
t
|
2
=
+
∞
,
where, as usual, we set F(t):=∫0tf(τ)𝑑τ.
The map t↦f(t)|t| is strictly increasing for every t∈ℝ∖{0}.
As models for f we can take, for instance, the following nonlinearities:
f
(
t
)
=
(
t
+
)
m
and
f
(
t
)
=
{
log
2
(
t
+
)
m
if
t
≤
1
,
t
log
(
1
+
t
)
if
t
>
1
,
for some m∈(2,2α*).
Remark 1.1.
We notice that by (f4) it follows that the real function
t
↦
1
2
f
(
t
)
t
-
F
(
t
)
is strictly increasing for every t>0 and strictly decreasing for every t<0.
Now, we are ready to state the main result of this paper.
Theorem 1.2.
Suppose that (V,K)∈K and f∈C0(R,R) verifies either (f1) or (f~1) and (f2)–(f4). Then problem (1.1) possesses a least energy nodal weak solution. In addition, if the nonlinear term f is odd, then problem (1.1) has infinitely many nontrivial weak solutions not necessarily nodals.
The proof of Theorem 1.2 is obtained by exploiting variational arguments. One of the main difficulties in the study of problem (1.1) is related to the presence of the fractional Laplacian (-Δ)α which is a nonlocal operator. Indeed, the Euler–Lagrange functional associated to problem (1.1), that is,
J
(
u
)
:=
1
2
(
[
u
]
2
+
∫
ℝ
N
V
(
x
)
u
2
𝑑
x
)
-
∫
ℝ
N
K
(
x
)
F
(
u
)
𝑑
x
,
does not satisfy the decompositions (see Section 4)
〈
J
′
(
u
)
,
u
±
〉
=
〈
J
′
(
u
±
)
,
u
±
〉
,
J
(
u
)
=
J
(
u
+
)
+
J
(
u
-
)
,
which were fundamental in the application of variational methods to study (1.2); see [14, 18, 52, 53].
Anyway, along the paper we prove that the geometry of the classical minimization theorem is respected in the nonlocal framework: more precisely, we develop a functional analytical setting that is inspired by (but not equivalent to) the fractional Sobolev spaces, in order to correctly encode the variational formulation of problem (1.1); see Section 2. Secondly, the nonlinearity f is only continuous, so to overcome the nondifferentiability of the Nehari manifold associated to J, we adapt in our framework some ideas developed in [52]. Of course, also the compactness properties (see Proposition 3.2) required by these abstract theorems are satisfied in the nonlocal case, thanks to our functional setting.
Then, in order to obtain nodal solutions, we look for critical points of J(tu++su-), and, due to the fact that f is only continuous, we do not apply Miranda’s theorem [44] as in [3, 14], but we use an iterative procedure and the properties of J to prove the existence of a sequence which converges to a critical point of J(tu++su-) (see Lemma 4.1).
Finally, we emphasize that Theorem 1.2 improves the recent result established in [12], in which the existence of a least energy nodal solution to problem (1.1) has been proved under the stronger assumption that f∈C1, and satisfies the Ambrosetti–Rabinowitz condition.
The paper is organized as follows: In Section 2, we present the variational framework of the problem and compactness results, which will be useful for the next sections. In Section 3, we obtain some preliminary results which are useful to overcome the lack of differentiability of the Nehari manifold in which we look for weak solutions to problem (1.1). In Section 4, we provide the proofs of some technical lemmas. Finally, in Section 5 we prove the existence of a least energy nodal weak solution by using minimization arguments and a variant of the Deformation Lemma.
2 Preliminary Results
This section is devoted to the notations used along the present paper. In order to give the weak formulation of problem (1.1), we need to work in a special functional space. Indeed, one of the difficulties in treating problem (1.1) is related to his variational formulation. With respect to this, the standard fractional Sobolev spaces are not sufficient in order to study the problem. We overcome this difficulty by working in a suitable functional space, whose definition and basic analytical properties are recalled here.
For α∈(0,1), we denote by 𝒟α,2(ℝN) the completion of Cc∞(ℝN) with respect to the so-called Gagliardo semi-norm
[
u
]
:=
(
∬
ℝ
2
N
|
u
(
x
)
-
u
(
y
)
|
2
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
)
1
/
2
,
and Hα(ℝN) denotes the standard fractional Sobolev space, defined as the set of u∈𝒟α,2(ℝN) satisfying u∈L2(ℝN) with the norm
∥
u
∥
H
α
(
ℝ
N
)
:=
(
[
u
]
2
+
∥
u
∥
L
2
(
ℝ
N
)
2
)
1
/
2
.
Let us introduce the following functional space:
𝕏
:=
{
u
∈
𝒟
α
,
2
(
ℝ
N
)
:
∫
ℝ
N
V
(
x
)
|
u
|
2
𝑑
x
<
+
∞
}
endowed with the norm
∥
u
∥
:=
(
[
u
]
2
+
∫
ℝ
N
V
(
x
)
|
u
|
2
𝑑
x
)
1
/
2
.
At this point, we define, for q∈ℝ with q≥1, the Lebesgue space LKq(ℝN) as
L
K
q
(
ℝ
N
)
:=
{
u
:
ℝ
N
→
ℝ
measurable and
∫
ℝ
N
K
(
x
)
|
u
|
q
𝑑
x
<
∞
}
endowed with the norm
∥
u
∥
L
K
q
(
ℝ
N
)
:=
(
∫
ℝ
N
K
(
x
)
|
u
|
q
𝑑
x
)
1
/
q
.
Finally, we recall the following useful results, which extend the ones in [2].
Lemma 2.1.
Assume that (V,K)∈K. Then X is continuously embedded in LKq(RN) for every q∈[2,2α*] if (h3) holds. Moreover, X is continuously embedded in LKm(RN) if (h4) holds.
Lemma 2.2.
Assume that (V,K)∈K. The following facts hold:
𝕏
is compactly embedded into
L
K
q
(
ℝ
N
)
for all
q
∈
(
2
,
2
α
*
)
if (h3) holds.
𝕏
is compactly embedded into
L
K
m
(
ℝ
N
)
if (h4) holds.
Lemma 2.3.
Assume that (V,K)∈K and f satisfies either (f1)–(f2) or (f~1)–(f2). Let {un}n∈N be a sequence such that un⇀u in X. Then, up to a subsequence, one has
lim
n
→
∞
∫
ℝ
N
K
(
x
)
F
(
u
n
)
𝑑
x
=
∫
ℝ
N
K
(
x
)
F
(
u
)
𝑑
x
and
lim
n
→
∞
∫
ℝ
N
K
(
x
)
f
(
u
n
)
u
n
𝑑
x
=
∫
ℝ
N
K
(
x
)
f
(
u
)
u
𝑑
x
.
For a proof of these lemmas, one can see [12].
3 Existence of a Least Energy Nodal Solution
In this section, we obtain some preliminary results which are useful to overcome the lack of differentiability of the Nehari manifold in which we look for weak solutions to problem (1.1).
In the following, we search a nodal or sign-changing weak solution of problem (1.1), that is, a function u=u++u-∈𝕏 such that u+:=max{u,0}≠0, u-:=min{u,0}≠0 in ℝN and
∬
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
(
φ
(
x
)
-
φ
(
y
)
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
+
∫
ℝ
N
V
(
x
)
u
(
x
)
φ
(
x
)
𝑑
x
=
∫
ℝ
N
K
(
x
)
f
(
u
)
φ
(
x
)
𝑑
x
for every φ∈𝕏.
The energy functional associated to problem (1.1) is given by
J
(
u
)
:=
1
2
∥
u
∥
2
-
∫
ℝ
N
K
(
x
)
F
(
u
)
𝑑
x
,
where F(t):=∫0tf(τ)𝑑τ. By the assumptions on f, it is clear that J∈C1(𝕏,ℝ) and that its differential J′:𝕏→𝕏′ is given by
〈
J
′
(
u
)
,
φ
〉
=
∬
ℝ
2
N
(
u
(
x
)
-
u
(
y
)
)
(
φ
(
x
)
-
φ
(
y
)
)
|
x
-
y
|
N
+
2
α
𝑑
x
+
∫
ℝ
N
V
(
x
)
u
(
x
)
φ
(
x
)
𝑑
x
-
∫
ℝ
N
K
(
x
)
f
(
u
)
φ
(
x
)
𝑑
x
for every u,φ∈𝕏. Then the critical points of J are the weak solutions of problem (1.1).
Let us also observe that J satisfies the following decompositions:
J
(
u
)
=
J
(
u
+
)
+
J
(
u
-
)
-
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
-
(
x
)
u
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
and
〈
J
′
(
u
)
,
u
+
〉
=
〈
J
′
(
u
+
)
,
u
+
〉
-
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
-
(
x
)
u
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
.
The Nehari manifold associated to the functional J is given by
𝒩
:=
{
u
∈
𝕏
∖
{
0
}
:
〈
J
′
(
u
)
,
u
〉
=
0
}
.
Recalling that a nonzero critical point u of J is a least energy weak solution of problem (1.1) if
J
(
u
)
=
min
v
∈
𝒩
J
(
v
)
and since our purpose is to prove the existence of a least energy sign-changing weak solution of (1.1), we look for u∈ℳ such that
J
(
u
)
=
min
v
∈
ℳ
J
(
v
)
,
where ℳ is the subset of 𝒩 containing all sign-changing weak solutions of problem (1.1), that is,
ℳ
:=
{
w
∈
𝒩
:
w
+
≠
0
,
w
-
≠
0
,
〈
J
′
(
w
)
,
w
+
〉
=
〈
J
′
(
w
)
,
w
-
〉
=
0
}
.
Once f is only continuous, the following results are crucial since they allow us to overcome the non-differentiability of 𝒩. Below, we denote by 𝕊 the unit sphere on 𝕏.
Lemma 3.1.
Suppose that (V,K)∈K and f verifies conditions (f1)–(f4). Then the following facts hold true:
For each
u
∈
𝕏
∖
{
0
}
, let
h
u
:
ℝ
+
→
ℝ
be defined by
h
u
(
t
)
:=
J
(
t
u
)
. Then there is a unique
t
u
>
0
such that
h
u
′
(
t
)
>
0
in
(
0
,
t
u
)
,
h
u
′
(
t
)
<
0
in
(
t
u
,
+
∞
)
.
There is
τ
>
0
, independent of
u
, such that
t
u
≥
τ
for every
u
∈
𝕊
. Moreover, for each compact set
𝒲
⊂
𝕊
there is
C
𝒲
>
0
such that
t
u
≤
C
𝒲
for every
u
∈
𝒲
.
The map
η
^
:
𝕏
∖
{
0
}
→
𝒩
given by
η
^
(
u
)
:=
t
u
u
is continuous, and
η
:=
η
^
|
𝕊
is a homeomorphism between
𝕊
and
𝒩
. Moreover,
η
-
1
(
u
)
=
u
∥
u
∥
.
Proof.
(i) We distinguish two cases.
Let us assume that (h3) is verified. By using assumptions (f1) and (f2), given ε>0 there exists a positive constant Cε such that
|
F
(
t
)
|
≤
ε
t
2
+
C
ε
|
t
|
2
α
*
for every
t
∈
ℝ
.
The above inequality and the Sobolev embedding yield
J
(
t
u
)
=
t
2
2
[
u
]
2
+
t
2
2
∫
ℝ
N
V
(
x
)
|
u
|
2
𝑑
x
-
∫
ℝ
N
K
(
x
)
F
(
t
u
)
𝑑
x
≥
t
2
2
∥
u
∥
2
-
ε
∫
ℝ
N
K
(
x
)
t
2
u
2
𝑑
x
-
C
ε
∫
ℝ
N
K
(
x
)
t
2
α
*
|
u
|
2
α
*
𝑑
x
(3.1)
≥
t
2
2
∥
u
∥
2
-
ε
∥
K
/
V
∥
L
∞
(
ℝ
N
)
t
2
∥
u
∥
2
-
C
ε
C
′
∥
K
∥
L
∞
(
ℝ
N
)
t
2
α
*
∥
u
∥
2
α
*
.
Taking
0
<
ε
<
1
2
∥
K
/
V
∥
L
∞
(
ℝ
N
)
,
we get t0>0 sufficiently small such that
(3.2)
0
<
h
u
(
t
)
=
J
(
t
u
)
for all
t
<
t
0
.
On the other hand, suppose that (h4) is true. Then there exists a constant Cm>0 such that, for each ε∈(0,Cm), we obtain R>0 such that
(3.3)
∫
ℬ
R
c
(
0
)
K
(
x
)
|
u
|
m
𝑑
x
≤
ε
∫
ℬ
R
c
(
0
)
(
V
(
x
)
|
u
|
2
+
|
u
|
2
α
*
)
𝑑
x
for every u∈𝕏.
Now, by using (f~1) and (f2), the Sobolev embedding result, relation (3.3) and the Hölder inequality, we have that
J
(
t
u
)
≥
t
2
2
∥
u
∥
2
-
C
1
∫
ℝ
N
K
(
x
)
t
m
|
u
|
m
𝑑
x
-
C
2
∫
ℝ
N
K
(
x
)
t
2
α
*
|
u
|
2
α
*
𝑑
x
≥
t
2
2
∥
u
∥
2
-
C
1
t
m
ε
∫
ℬ
R
c
(
0
)
(
V
(
x
)
|
u
|
2
+
|
u
|
2
α
*
)
𝑑
x
-
C
1
t
m
∫
ℬ
R
(
0
)
K
(
x
)
|
u
|
m
𝑑
x
-
C
2
t
2
α
*
∥
K
∥
L
∞
(
ℝ
N
)
∫
ℝ
N
|
u
|
2
α
*
𝑑
x
≥
t
2
2
∥
u
∥
2
-
C
1
t
m
ε
∫
ℬ
R
c
(
0
)
(
V
(
x
)
|
u
|
2
+
|
u
|
2
α
*
)
𝑑
x
-
C
1
t
m
∥
K
∥
L
2
α
*
/
(
2
α
*
-
m
)
(
ℬ
R
(
0
)
)
(
∫
ℬ
R
(
0
)
K
(
x
)
|
u
|
m
𝑑
x
)
m
2
α
*
-
C
2
t
2
α
*
∥
K
∥
L
∞
(
ℝ
N
)
∫
ℝ
N
|
u
|
2
α
*
𝑑
x
(3.4)
≥
t
2
2
∥
u
∥
2
-
C
1
t
m
(
ε
∥
u
∥
2
+
ε
C
∥
u
∥
2
α
*
+
C
∥
K
∥
L
2
α
*
/
(
2
α
*
-
m
)
(
ℬ
R
(
0
)
)
∥
u
∥
m
)
-
C
2
C
t
2
α
*
∥
K
∥
L
∞
(
ℝ
N
)
∥
u
∥
2
α
*
.
This shows that condition (3.2) is verified also in this case.
Moreover, since F(t)≥0 for every t∈ℝ, we have
J
(
t
u
)
≤
t
2
2
∥
u
∥
2
-
∫
A
K
(
x
)
F
(
t
u
)
𝑑
x
,
where A⊂suppu is a measurable set with finite and positive measure. Hence,
lim sup
t
→
+
∞
J
(
t
u
)
∥
t
u
∥
2
≤
1
2
-
lim inf
t
→
∞
{
∫
A
K
(
x
)
[
F
(
t
u
)
(
t
u
)
2
]
(
u
∥
u
∥
)
2
𝑑
x
}
.
By (f3) and Fatou’s lemma, it follows that
(3.5)
lim sup
t
→
+
∞
J
(
t
u
)
∥
t
u
∥
2
≤
-
∞
.
Thus, there exists R>0 sufficiently large such that
h
u
(
R
)
=
J
(
R
u
)
<
0
.
By the continuity of hu and (f4), there is tu>0 which is a global maximum of hu with tuu∈𝒩.
Now, we aim to prove that tu is the unique critical point of hu. Arguing by contradiction, we assume that there are critical points t1,t2 of hu with t1>t2>0. Thus, we have hu′(t1)=hu′(t2)=0, or equivalently
∥
u
∥
2
-
∫
ℝ
N
K
(
x
)
f
(
t
1
u
)
u
t
1
𝑑
x
=
0
,
∥
u
∥
2
-
∫
ℝ
N
K
(
x
)
f
(
t
2
u
)
u
t
2
𝑑
x
=
0
.
Subtracting and taking into account (f4), we obtain
0
=
∫
ℝ
N
K
(
x
)
[
f
(
t
1
u
)
t
1
u
-
f
(
t
2
u
)
t
2
u
]
u
2
𝑑
x
>
0
,
which leads a contradiction.
(ii) By (i), there exists tu>0 such that
(3.6)
t
u
2
∥
u
∥
2
=
∫
ℝ
N
K
(
x
)
f
(
t
u
u
)
t
u
u
𝑑
x
.
Then, estimating the right-hand side of (3.6) similarly to (3.1) and (3.4), we obtain that there exists τ>0, independent of u, such that tu≥τ. On the other hand, let 𝒲⊂𝕊 be a compact set. Assume by contradiction that there exists {un}n∈ℕ⊂𝒲 such that tn:=tun→∞. Therefore, there exists u∈𝒲 such that un→u in 𝕏. From (3.5) we have
(3.7)
J
(
t
n
u
n
)
→
-
∞
in
ℝ
.
Therefore, by using Remark 1.1, one has
(3.8)
J
(
v
)
=
J
(
v
)
-
1
2
〈
J
′
(
v
)
,
v
〉
=
∫
ℝ
N
K
(
x
)
[
1
2
f
(
v
)
v
-
F
(
v
)
]
𝑑
x
≥
0
for each v∈𝒩.
By taking into account that {tunun}n∈ℕ⊂𝒩, we conclude from (3.7) that (3.8) is not true, which is a contradiction.
(iii) Since J∈C1(𝕏,ℝ), J(0)=0 and since it satisfies (i) and (ii), the thesis follows by [52, Proposition 8]. The proof is now complete. ∎
Let us define the maps
ψ
^
:
𝕏
→
ℝ
and
ψ
:
𝕊
→
ℝ
by ψ^(u):=J(η^(u)) and ψ:=ψ^|𝕊.
The compactness condition assumed in the sequel is the well-known Palais–Smale condition at level d (briefly (PS)d), which in our framework reads as follows (see, for instance, [48, 53]):
J satisfies the Palais–Smale compactness condition at level d∈ℝ if any sequence {un}n∈ℕ in 𝕏 such that
J
(
u
n
)
→
d
and
sup
{
|
〈
J
′
(
u
n
)
,
φ
〉
|
:
φ
∈
𝕏
,
∥
φ
∥
=
1
}
→
0
as
n
→
∞
,
admits a strongly convergent subsequence in 𝕏.
The next result is a consequence of Lemma 3.1.
Proposition 3.2.
Suppose that (V,K)∈K and f verifies (f1)–(f4). Then one has the following assertions:
ψ
^
∈
C
1
(
𝕏
∖
{
0
}
,
ℝ
)
and
〈
ψ
^
′
(
u
)
,
v
〉
=
∥
η
^
(
u
)
∥
∥
u
∥
〈
J
′
(
η
^
(
u
)
)
,
v
〉
for every
u
∈
𝕏
∖
{
0
}
and
v
∈
𝕏
.
ψ
∈
C
1
(
𝕊
,
ℝ
)
and
〈
ψ
′
(
u
)
,
v
〉
=
∥
η
(
u
)
∥
〈
J
′
(
η
(
u
)
)
,
v
〉
for every
v
∈
T
u
𝕊
:=
{
v
∈
𝕏
:
〈
v
,
u
〉
=
∫
ℝ
2
N
(
v
(
x
)
-
v
(
y
)
)
(
u
(
x
)
-
u
(
y
)
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
+
∫
ℝ
N
V
(
x
)
u
v
𝑑
x
=
0
}
.
If
{
u
n
}
n
∈
ℕ
is a
(
PS
)
d
sequence for ψ , then {η(un)}n∈ℕ is a (PS)d sequence for J. Moreover, if {un}n∈ℕ⊂𝒩 is a bounded (PS)d sequence for J, then {η-1(un)}n∈ℕ is a (PS)d sequence for the functional ψ.
u
is a critical point of ψ if and only if η(u) is a nontrivial critical point for J. Moreover, the corresponding critical values coincide and
inf
u
∈
𝕊
ψ
(
u
)
=
inf
u
∈
𝒩
J
(
u
)
.
Remark 3.3.
We notice that the following equalities hold:
(3.9)
d
∞
:=
inf
u
∈
𝒩
J
(
u
)
=
inf
u
∈
𝕏
∖
{
0
}
max
t
>
0
J
(
t
u
)
=
inf
u
∈
𝕊
max
t
>
0
J
(
t
u
)
.
In particular, relations (3.1), (3.5) and (3.9) imply that
(3.10)
d
∞
>
0
.
4 Technical Lemmas
The aim of this section is to prove some technical lemmas related to the existence of a least energy nodal solution.
For each u∈𝕏 with u±≢0, let us consider the function hu:[0,+∞)×[0,+∞)→ℝ given by
h
u
(
t
,
s
)
:=
J
(
t
u
+
+
s
u
-
)
.
Let us observe that its gradient Φu:[0,+∞)×[0,+∞)→ℝ2 is defined by
(4.1)
Φ
u
(
t
,
s
)
:=
(
Φ
1
u
(
t
,
s
)
,
Φ
2
u
(
t
,
s
)
)
=
(
∂
h
u
∂
t
(
t
,
s
)
,
∂
h
u
∂
s
(
t
,
s
)
)
=
(
〈
J
′
(
t
u
+
+
s
u
-
)
,
u
+
〉
,
〈
J
′
(
t
u
+
+
s
u
-
)
,
u
-
〉
)
.
Lemma 4.1.
Suppose that (V,K)∈K and f verifies (f1)–(f4). Then it follows that
The pair
(
t
,
s
)
is a critical point of
h
u
with
t
,
s
>
0
if and only if
t
u
+
+
s
u
-
∈
ℳ
.
The map
h
u
has a unique critical point
(
t
+
,
s
-
)
, with
t
+
=
t
+
(
u
)
>
0
and
s
-
=
s
-
(
u
)
>
0
, which is the unique global maximum point of
h
u
.
The maps
a
+
(
r
)
:=
Φ
1
u
(
r
,
s
-
)
r
and
a
-
(
r
)
:=
Φ
2
u
(
t
+
,
r
)
r
are such that
(4.2)
{
a
+
(
r
)
>
0
if
r
∈
(
0
,
t
+
)
𝑎𝑛𝑑
a
+
(
r
)
<
0
if
r
∈
(
t
+
,
+
∞
)
,
a
-
(
r
)
>
0
if
r
∈
(
0
,
s
-
)
𝑎𝑛𝑑
a
-
(
r
)
<
0
if
r
∈
(
s
-
,
+
∞
)
.
Proof.
(i) Let us observe that by (4.1) we have
Φ
u
(
t
,
s
)
=
(
1
t
〈
J
′
(
t
u
+
+
s
u
-
)
,
t
u
+
〉
,
1
s
〈
J
′
(
t
u
+
+
s
u
-
)
,
s
u
+
〉
)
for every t,s>0. Then Φu(t,s)=0 if and only if
〈
J
′
(
t
u
+
+
s
u
-
)
,
t
u
+
〉
=
0
and
〈
J
′
(
t
u
+
+
s
u
-
)
,
s
u
-
〉
=
0
,
and this implies that tu++su-∈ℳ.
(ii) Firstly, we show that hu has a critical point. For each u∈𝕏 such that u±≠0 and s0 fixed, we define the function h1:[0,+∞)→[0,+∞) by h1(t):=hu(t,s0). Following the lines of Lemma 3.1 (i), we can infer that h1 has a maximum positive point.
Moreover, there exists a unique t0=t0(u,s0)>0 such that
h
1
′
(
t
)
>
0
if
t
∈
(
0
,
t
0
)
,
h
1
′
(
t
0
)
=
0
,
h
1
′
(
t
)
<
0
if
t
∈
(
t
0
,
+
∞
)
.
Thus, the map ϕ1:[0,+∞)→[0,+∞) defined by ϕ1(s):=t(u,s), where t(u,s) satisfies the properties just mentioned with s in place of s0, is well defined.
By the definition of h1, we have
(4.3)
h
1
′
(
ϕ
1
(
s
)
)
=
Φ
1
u
(
ϕ
1
(
s
)
,
s
)
=
0
for all
s
≥
0
,
that is,
(4.4)
0
=
|
ϕ
1
(
s
)
|
2
∥
u
+
∥
2
-
s
ϕ
1
(
s
)
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
-
(
x
)
u
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
-
∫
ℝ
N
K
(
x
)
f
(
ϕ
1
(
s
)
u
+
)
ϕ
1
(
s
)
u
+
𝑑
x
.
Now, we prove some properties of ϕ1.
(a) The map ϕ1 is continuous. Let sn→s0 as n→∞ in ℝ. We want to prove that {ϕ1(sn)}n∈ℕ is bounded. Assume by contradiction that there is a subsequence, again denoted by {sn}n∈ℕ, such that ϕ1(sn)→+∞ as n→∞. So, ϕ1(sn)≥sn for n large. By (4.4), we have
∥
u
+
∥
2
-
s
n
ϕ
1
(
s
n
)
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
-
(
x
)
u
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
ϕ
1
(
s
n
)
u
+
)
ϕ
1
(
s
n
)
u
+
(
u
+
)
2
𝑑
x
.
Taking into account that sn→s0, ϕ1(sn)→+∞ as n→∞, assumptions (f3)-(f4) and Fatou’s lemma yields
∥
u
+
∥
2
=
lim inf
n
→
∞
∫
ℝ
N
K
(
x
)
f
(
ϕ
1
(
s
)
u
+
)
ϕ
1
(
s
)
u
+
(
u
+
)
2
𝑑
x
≥
+
∞
.
Thus we have a contradiction. So the sequence {ϕ1(sn)}n∈ℕ is bounded.
Therefore, there exists t0≥0 such that ϕ1(sn)→t0. Consider (4.4) with s=sn, and by passing to the limit as n→∞, we have
t
0
2
∥
u
+
∥
2
-
s
0
t
0
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
-
(
x
)
u
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
ϕ
1
(
t
0
)
u
+
)
ϕ
1
(
t
0
)
u
+
𝑑
x
,
that is, h1′(t0)=Φ1u(t0,s0)=0. As a consequence, t0=ϕ1(s0), i.e. ϕ1 is continuous.
(b)ϕ1(0)>0. Assume that there exists a sequence {sn}n∈ℕ such that ϕ1(sn)→0+ and sn→0 as n→∞. By assumption (f1), we get
∥
u
+
∥
2
≤
∥
u
+
∥
2
-
s
n
ϕ
1
(
s
n
)
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
-
(
x
)
u
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
ϕ
1
(
s
n
)
u
+
)
ϕ
1
(
s
n
)
u
+
(
u
+
)
2
𝑑
x
→
0
as
n
→
∞
,
and this fact gives a contradiction. So we deduce that ϕ1(0)>0.
(c) Now we show that ϕ1(s)≤s for s large. As a matter of fact, proceeding as in the first part of the proof of (a), we can see that it is not possible to find any sequence {sn}n∈ℕ such that sn→+∞ and ϕ1(sn)≥sn for all n∈ℕ. This implies that ϕ1(s)≤s for s large.
Analogously, for every t0≥0 we define h2(s):=hu(t0,s) and, as a consequence, we can find a map ϕ2 such that
(4.5)
h
2
′
(
ϕ
2
(
t
)
)
=
Φ
2
u
(
t
,
ϕ
2
(
t
)
)
for all
t
≥
0
and satisfying (a), (b) and (c).
By (c), we can find a positive constant C1 such that ϕ1(s)≤s and ϕ2(t)≤t for every t,s≥C1.
Let
C
2
:=
max
{
max
s
∈
[
0
,
C
1
]
ϕ
1
(
s
)
,
max
t
∈
[
0
,
C
1
]
ϕ
2
(
t
)
}
and C:=max{C1,C2}.
We define T:[0,C]×[0,C]→ℝ2 by T(t,s):=(ϕ1(s),ϕ2(t)). Let us note that
T
(
[
0
,
C
]
×
[
0
,
C
]
)
⊂
[
0
,
C
]
×
[
0
,
C
]
.
Indeed, for every t∈[0,C], we have that
{
ϕ
2
(
t
)
≤
t
≤
C
if
t
≥
C
1
,
ϕ
2
(
t
)
≤
max
t
∈
[
0
,
C
1
]
ϕ
2
(
t
)
≤
C
2
if
t
≤
C
1
.
Similarly, we can see that ϕ1(s)≤C for all s∈[0,C]. Moreover, since ϕi are continuous for i=1,2, it is clear that T is a continuous map.
Then, by the Brouwer fixed point theorem, there exists (t+,s-)∈[0,C]×[0,C] such that
(
ϕ
1
(
s
-
)
,
ϕ
2
(
t
+
)
)
=
(
t
+
,
s
-
)
.
Owing to this fact and recalling that ϕi>0, we have t+>0 and s->0. By (4.3) and (4.5), we have
Φ
1
u
(
t
+
,
s
-
)
=
Φ
2
u
(
t
+
,
s
-
)
=
0
,
that is, (t+,s-) is a critical point of hu. Next we aim to prove the uniqueness of (t+,s-).
Assuming that w∈ℳ, we have
Φ
w
(
1
,
1
)
=
(
Φ
1
w
(
1
,
1
)
,
Φ
2
w
(
1
,
1
)
)
=
(
∂
h
w
∂
t
(
1
,
1
)
,
∂
h
w
∂
s
(
1
,
1
)
)
=
(
〈
J
′
(
w
+
+
w
-
)
,
w
+
〉
,
〈
J
′
(
w
+
+
w
-
)
,
w
-
〉
)
=
(
0
,
0
)
,
which implies that (1,1) is a critical point of hw. Now, assume that (t0,s0) is a critical point of hw with 0<t0≤s0. This means that
〈
J
′
(
t
0
w
+
+
s
0
w
-
)
,
t
0
w
+
〉
=
0
and
〈
J
′
(
t
0
w
+
+
s
0
w
-
)
,
s
0
w
-
〉
=
0
,
or equivalently
(4.6)
t
0
2
∥
w
+
∥
2
-
s
0
t
0
∬
ℝ
2
N
w
+
(
x
)
w
-
(
y
)
+
w
-
(
x
)
w
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
t
0
w
+
)
t
0
w
+
𝑑
x
,
(4.7)
s
0
2
∥
w
-
∥
2
-
s
0
t
0
∬
ℝ
2
N
w
+
(
x
)
w
-
(
y
)
+
w
-
(
x
)
w
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
s
0
w
-
)
s
0
w
-
𝑑
x
.
Dividing by s02>0 in (4.7), we have
∥
w
-
∥
2
-
t
0
s
0
∬
ℝ
2
N
w
+
(
x
)
w
-
(
y
)
+
w
-
(
x
)
w
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
s
0
w
-
)
s
0
w
-
(
w
-
)
2
𝑑
x
,
and by using the fact that 0<t0≤s0, we can see that
(4.8)
∥
w
-
∥
2
-
∬
ℝ
2
N
w
+
(
x
)
w
-
(
y
)
+
w
-
(
x
)
w
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
≥
∫
ℝ
N
K
(
x
)
f
(
s
0
w
-
)
s
0
w
-
(
w
-
)
2
𝑑
x
.
Since w∈ℳ, we also have
(4.9)
∥
w
-
∥
2
-
∬
ℝ
2
N
w
+
(
x
)
w
-
(
y
)
+
w
-
(
x
)
w
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
w
-
)
w
-
(
w
-
)
2
𝑑
x
.
Putting together (4.8) and (4.9), we get
0
≥
∫
ℝ
N
K
(
x
)
[
f
(
s
0
w
-
)
s
0
w
-
(
w
-
)
2
-
f
(
w
-
)
w
-
(
w
-
)
2
]
𝑑
x
.
The above relation and assumption (f4) ensures that 0<t0≤s0≤1.
Now we prove that t0≥1. Dividing by t02>0 in (4.6), we have
∥
w
+
∥
2
-
s
0
t
0
∬
ℝ
2
N
w
+
(
x
)
w
-
(
y
)
+
w
-
(
x
)
w
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
t
0
w
+
)
t
0
w
+
(
w
+
)
2
𝑑
x
,
and by using 0<t0≤s0, we deduce that
(4.10)
∥
w
+
∥
2
-
∬
ℝ
2
N
w
+
(
x
)
w
-
(
y
)
+
w
-
(
x
)
w
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
≤
∫
ℝ
N
K
(
x
)
f
(
t
0
w
+
)
t
0
w
+
(
w
+
)
2
𝑑
x
.
Since w∈ℳ, we also have
(4.11)
∥
w
+
∥
2
-
∬
ℝ
2
N
w
+
(
x
)
w
-
(
y
)
+
w
-
(
x
)
w
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
w
+
)
w
+
(
w
+
)
2
𝑑
x
.
Putting together (4.10) and (4.11), we get
0
≥
∫
ℝ
N
K
(
x
)
[
f
(
w
+
)
w
+
(
w
+
)
2
-
f
(
t
0
w
+
)
t
0
w
+
(
w
+
)
2
]
𝑑
x
.
By (f4), it follows that t0≥1. Consequently, t0=s0=1, and this proves that (1,1) is the unique critical point of hw with positive coordinates.
Let u±∈𝕏 be such that u±≠0, and let (t1,s1),(t2,s2) be critical points of hu with positive coordinates. By (i), it follows that
w
1
=
t
1
u
+
+
s
1
u
-
∈
ℳ
and
w
2
=
t
2
u
+
+
s
2
u
-
∈
ℳ
.
We notice that w2 can be written as
w
2
=
(
t
2
t
1
)
t
1
u
+
+
(
s
2
s
1
)
s
1
u
-
=
t
2
t
1
w
1
+
+
s
2
s
1
w
1
-
∈
ℳ
.
Since w1∈𝕏 is such that w1±≠0, we have that (t2/t1,s2/s1) is a critical point for hw1 with positive coordinates.
On the other hand, since w1∈ℳ, we can conclude that t2/t1=s2/s1=1, which gives t1=t2 and s1=s2.
Finally, we prove that hu has a maximum global point (t¯,s¯)∈(0,+∞)×(0,+∞). Let A+⊂suppu+ and A-⊂suppu- positive with finite measure. By assumption (f3) and the fact that F(t)≥0 for every t∈ℝ, it follows that
h
u
(
t
,
s
)
≤
1
2
∥
t
u
+
+
s
u
-
∥
2
-
∫
A
+
K
(
x
)
F
(
t
u
+
)
𝑑
x
-
∫
A
-
K
(
x
)
F
(
s
u
-
)
𝑑
x
≤
t
2
2
∥
u
+
∥
2
+
s
2
2
∥
u
-
∥
2
-
s
t
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
+
(
y
)
u
-
(
x
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
-
∫
A
+
K
(
x
)
F
(
t
u
+
)
𝑑
x
-
∫
A
-
K
(
x
)
F
(
s
u
-
)
𝑑
x
.
Let us suppose that |t|≥|s|>0. Then, by using the fact that F(t)≥0 for every t∈ℝ, we can see that
h
u
(
t
,
s
)
≤
(
t
2
+
s
2
)
[
1
2
∥
u
+
∥
2
+
1
2
∥
u
-
∥
2
-
1
2
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
+
(
y
)
u
-
(
x
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
]
-
t
2
∫
A
+
K
(
x
)
F
(
t
u
+
)
(
t
u
+
)
2
(
u
+
)
2
𝑑
x
.
Condition (f3), Fatou’s lemma and the fact that 0<t2+s2≤2t2 ensure that
lim sup
|
(
t
,
s
)
|
→
∞
h
u
(
t
,
s
)
t
2
+
s
2
≤
C
(
u
+
,
u
-
)
-
1
2
lim inf
|
t
|
→
∞
∫
A
+
K
(
x
)
F
(
t
u
+
)
(
t
u
+
)
2
(
u
+
)
2
𝑑
x
=
-
∞
,
where C(u+,u-)>0 is a constant depending only on u+ and u-.
Therefore,
(4.12)
lim
|
(
t
,
s
)
|
→
∞
h
u
(
t
,
s
)
=
-
∞
.
By (4.12), and recalling that hu is a continuous function, we deduce that hu has a maximum global point (t¯,s¯)∈(0,+∞)×(0,+∞).
The linearity of F and the positivity of K yield
(4.13)
∫
ℝ
N
K
(
x
)
(
F
(
t
u
+
)
+
F
(
s
u
-
)
)
𝑑
x
=
∫
ℝ
N
K
(
x
)
F
(
t
u
+
+
s
u
-
)
𝑑
x
.
By (4.13), for all u∈𝕏 such that u±≠0 and for every t,s≥0, it follows that
J
(
t
u
+
)
+
J
(
s
u
-
)
≤
J
(
t
u
+
+
s
u
-
)
.
So, for every u∈𝕏 such that u±≠0 one has
h
u
(
t
,
0
)
+
h
u
(
0
,
s
)
≤
h
u
(
t
,
s
)
for every t,s≥0.
Then
max
t
≥
0
h
u
(
t
,
0
)
<
max
t
,
s
>
0
h
u
(
t
,
s
)
and
max
s
≥
0
h
u
(
0
,
s
)
<
max
t
,
s
>
0
h
u
(
t
,
s
)
,
and this proves that (t¯,s¯)∈(0,+∞)×(0,+∞).
(iii) By Lemma 3.1 (i), we easily have that
Φ
1
u
(
r
,
s
-
)
=
∂
h
u
∂
t
(
r
,
s
-
)
>
0
if
r
∈
(
0
,
t
+
)
,
Φ
1
u
(
t
+
,
s
-
)
=
∂
h
u
∂
t
(
t
+
,
s
-
)
=
0
,
Φ
1
u
(
r
,
s
-
)
=
∂
h
u
∂
t
(
r
,
s
-
)
>
0
if
r
∈
(
t
+
,
+
∞
)
.
Therefore, (4.2) holds true. The proof of Lemma 4.1 is now complete. ∎
Lemma 4.2.
If {un}n∈N⊂M and un⇀u in X, then u∈X and u±≠0.
Proof.
Let us observe that there is β>0 such that
(4.14)
β
≤
∥
v
±
∥
for all
v
∈
ℳ
.
Indeed, if v∈ℳ, then
∥
v
±
∥
2
≤
∫
ℝ
N
K
(
x
)
f
(
v
±
)
v
±
𝑑
x
.
Assume that (h3) holds true. Then, by using (f1), (f2) and the Sobolev inequality, we can see that given ε>0 there exists a positive constant Cε such that
∥
v
±
∥
2
≤
∫
ℝ
N
K
(
x
)
f
(
v
±
)
v
±
𝑑
x
≤
ε
∥
K
/
V
∥
L
∞
(
ℝ
N
)
∫
ℝ
N
V
(
x
)
(
v
±
)
2
𝑑
x
+
C
ε
C
*
∥
K
∥
L
∞
(
ℝ
N
)
∥
v
±
∥
2
α
*
≤
ε
∥
K
/
V
∥
L
∞
(
ℝ
N
)
∥
v
±
∥
2
+
C
ε
C
*
∥
K
∥
L
∞
(
ℝ
N
)
∥
v
±
∥
2
α
*
.
Choosing
ε
∈
(
0
,
1
∥
K
/
V
∥
L
∞
(
ℝ
N
)
)
,
there exists a positive constant β1 such that ∥v±∥>β1.
Analogously, by assuming that (h4) holds, conditions (f~1) and (f2), the Sobolev embedding result and the Hölder inequality ensure that
∥
v
±
∥
2
≤
∫
ℝ
N
K
(
x
)
f
(
v
±
)
v
±
𝑑
x
≤
C
1
ε
∥
v
±
∥
2
+
C
1
C
*
(
ε
+
C
2
∥
K
∥
L
∞
(
ℝ
N
)
)
∥
v
±
∥
2
α
*
+
∥
K
∥
L
2
α
*
/
(
2
α
*
-
m
)
(
ℬ
R
(
0
)
)
C
*
∥
v
±
∥
m
.
Since m∈(2,2α*), we can choose ε sufficiently small such that it is possible to find a positive constant β2 such that ∥v±∥>β2.
Hence, if we set β:=min{β1,β2}, inequality (4.14) immediately holds.
So, if {un}n∈ℕ⊂ℳ, we have
(4.15)
β
2
≤
∫
ℝ
N
K
(
x
)
f
(
u
n
±
)
u
n
±
𝑑
x
for all
n
∈
ℕ
.
Since un⇀u in 𝕏, bearing in mind Lemma 2.2, we can pass to the limit in (4.15) as n→∞.
More precisely, by using Lemma 2.3, it follows that
0
<
β
2
≤
∫
ℝ
N
K
(
x
)
f
(
u
±
)
u
±
𝑑
x
.
Thus u∈𝕏 and u±≠0. The proof is now complete. ∎
Let us denote by c∞ the number
c
∞
:=
inf
u
∈
ℳ
J
(
u
)
.
Since ℳ⊂𝒩, we deduce
(4.16)
c
∞
≥
d
∞
>
0
.
5 Proof of Theorem 1.2
In this section, we prove the existence of energy nodal weak solutions by using minimization arguments and a variant of the Deformation Lemma. We start by proving the existence of a minimum point of the functional J in ℳ.
Let {un}n∈ℕ⊂ℳ be such that
(5.1)
J
(
u
n
)
→
c
∞
in
ℝ
.
Our aim is to prove that {un}n∈ℕ is bounded in 𝕏.
Indeed, assume by contradiction that there exists a subsequence, denoted again by {un}n∈ℕ, such that ∥un∥→+∞ as n→∞. Thus, let us define
v
n
:=
u
n
∥
u
n
∥
for every n∈ℕ. Since {vn}n∈ℕ is bounded in 𝕏, due to the reflexivity of 𝕏, there exists v∈𝕏 such that
(5.2)
v
n
⇀
v
in
𝕏
.
Moreover, in virtue of Lemma 2.1, it follows that
(5.3)
v
n
(
x
)
→
v
(
x
)
a.e. in
ℝ
N
.
By Lemma 4.1 (i) and {un}n∈ℕ⊂ℳ, we have that t+(vn)=s-(vn)=∥un∥ and
(5.4)
J
(
u
n
)
=
J
(
∥
u
n
∥
v
n
)
≥
J
(
t
v
n
)
=
t
2
2
∥
v
n
∥
2
-
∫
ℝ
N
K
(
x
)
F
(
t
v
n
)
𝑑
x
=
t
2
2
-
∫
ℝ
N
K
(
x
)
F
(
t
v
n
)
𝑑
x
for every t>0 and n∈ℕ.
Suppose that v=0. Taking into account (5.2) and Lemma 2.3, we get
(5.5)
∫
ℝ
N
K
(
x
)
F
(
t
v
n
)
→
0
for all
t
>
0
.
By passing to the limit in (5.4) as n→∞, and combining (5.1) and (5.5), we have
c
∞
≥
t
2
2
for all
t
>
0
,
which is a contradiction.
Hence, v≠0. Taking into account the definitions of J and {vn}n∈ℕ, we have
(5.6)
J
(
u
n
)
∥
u
n
∥
2
=
1
2
-
∫
ℝ
N
K
(
x
)
F
(
v
n
∥
u
n
∥
)
(
v
n
∥
u
n
∥
)
2
(
v
n
)
2
𝑑
x
.
Now, since v≠0 and ∥un∥→+∞, by using (5.3) in addition to (f3), Fatou’s lemma ensures that
(5.7)
∫
ℝ
N
K
(
x
)
F
(
v
n
∥
u
n
∥
)
(
v
n
∥
u
n
∥
)
2
v
n
2
𝑑
x
→
+
∞
.
Since (5.7) holds true, bearing in mind (5.1) and passing to the limit in (5.6) as n→∞, we have a contradiction.
Therefore, {un}n∈ℕ⊂𝕏 is a bounded subsequence. As a consequence, there exists u∈𝕏 such that
(5.8)
u
n
⇀
u
in
𝕏
.
By Lemma 4.2, it follows that u±≠0. Moreover, by Lemma 4.1, there are two constants t+,s->0 such that
(5.9)
t
+
u
+
+
s
-
u
-
∈
ℳ
.
Now, our aim is to prove that t+,s-∈(0,1]. By (5.8) and Lemma 2.3, we have
(5.10)
∫
ℝ
N
K
(
x
)
f
(
u
n
±
)
u
n
±
𝑑
x
→
∫
ℝ
N
K
(
x
)
f
(
u
±
)
u
±
𝑑
x
and
(5.11)
∫
ℝ
N
K
(
x
)
F
(
u
n
±
)
𝑑
x
→
∫
ℝ
N
K
(
x
)
F
(
u
±
)
𝑑
x
.
Recalling that {un}n∈ℕ⊂ℳ, by using (5.8) and (5.10), Fatou’s lemma ensures that
〈
J
′
(
u
)
,
u
±
〉
=
∥
u
±
∥
2
-
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
-
(
x
)
u
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
-
∫
ℝ
N
K
(
x
)
f
(
u
±
)
u
±
𝑑
x
(5.12)
≤
lim inf
n
→
∞
〈
J
′
(
u
n
)
,
u
n
±
〉
=
0
.
Let us assume 0<t+<s-. By (5.9), we deduce
s
-
2
∥
u
-
∥
2
-
t
+
s
-
∬
ℝ
2
N
u
+
(
x
)
u
-
(
y
)
+
u
-
(
x
)
u
+
(
y
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
=
∫
ℝ
N
K
(
x
)
f
(
s
-
u
-
)
s
-
u
-
𝑑
x
,
and by using t+<s-, we obtain
(5.13)
∥
u
-
∥
2
-
∬
ℝ
2
N
u
-
(
x
)
u
+
(
y
)
+
u
-
(
y
)
u
+
(
x
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
≥
∫
supp
u
-
K
(
x
)
f
(
s
-
u
-
)
s
-
u
-
(
u
-
)
2
𝑑
x
.
By (5.12), we have
(5.14)
∥
u
-
∥
2
-
∬
ℝ
2
N
u
-
(
x
)
u
+
(
y
)
+
u
-
(
y
)
u
+
(
x
)
|
x
-
y
|
N
+
2
α
𝑑
x
𝑑
y
≤
∫
supp
u
-
K
(
x
)
f
(
u
-
)
u
-
(
u
-
)
2
𝑑
x
.
Putting together (5.13) and (5.14), we can deduce that
0
≥
∫
supp
u
-
K
(
x
)
[
f
(
s
-
u
-
)
s
-
u
-
-
f
(
u
-
)
u
-
]
(
u
-
)
2
𝑑
x
,
which yields s-∈(0,1] in virtue of (f4). Similarly, we can show that t+∈(0,1].
Now, we prove that
(5.15)
J
(
t
+
u
+
+
s
-
u
-
)
=
c
∞
.
By using the definitions of c∞, t+,s-∈(0,1], exploiting condition (f4) and taking into account relations (5.9), (5.10) and (5.11), we get
c
∞
≤
J
(
t
+
u
+
+
s
-
u
-
)
=
J
(
t
+
u
+
+
s
-
u
-
)
-
1
2
〈
J
′
(
t
+
u
+
+
s
-
u
-
)
,
t
+
u
+
+
s
-
u
-
〉
=
∫
ℝ
N
K
(
x
)
[
1
2
f
(
t
+
u
+
+
s
-
u
-
)
(
t
+
u
+
+
s
-
u
-
)
-
F
(
t
+
u
+
+
s
-
u
-
)
]
𝑑
x
=
∫
ℝ
N
K
(
x
)
[
1
2
f
(
t
+
u
+
)
(
t
+
u
+
)
-
F
(
t
+
u
+
)
]
𝑑
x
+
∫
ℝ
N
K
(
x
)
[
1
2
f
(
s
-
u
-
)
(
s
-
u
-
)
-
F
(
s
-
u
-
)
]
𝑑
x
≤
∫
ℝ
N
K
(
x
)
[
1
2
f
(
u
+
)
(
u
+
)
-
F
(
u
+
)
]
𝑑
x
+
∫
ℝ
N
K
(
x
)
[
1
2
f
(
u
-
)
(
u
-
)
-
F
(
u
-
)
]
𝑑
x
=
∫
ℝ
N
K
(
x
)
[
1
2
f
(
u
)
u
-
F
(
u
)
]
𝑑
x
=
lim
n
→
∞
∫
ℝ
N
K
(
x
)
[
1
2
f
(
u
n
)
u
n
-
F
(
u
n
)
]
𝑑
x
=
lim
n
→
∞
[
J
(
u
n
)
-
1
2
〈
J
′
(
u
n
)
,
u
n
〉
]
=
c
∞
.
Hence (5.15) holds true. Furthermore, the above calculation implies that t+=s-=1.
Now, we prove that u=u++u- is a critical point of the functional J, arguing by contradiction. Thus, let us suppose that J′(u)≠0. By continuity, there exist δ,μ>0 such that
μ
≤
|
J
′
(
v
)
|
since
∥
v
-
u
∥
≤
3
δ
.
Define D:=[12,32]×[12,32] and g:D→𝕏± by
g
(
t
,
s
)
:=
t
u
+
+
s
u
-
,
where 𝕏±:={u∈𝕏:u±≠0}.
By Lemma 4.1, we deduce
J
(
g
(
1
,
1
)
)
=
c
∞
,
J
(
g
(
t
,
s
)
)
<
c
∞
in
D
∖
{
(
1
,
1
)
}
.
Thus, we have
(5.16)
β
:=
max
(
t
,
s
)
∈
∂
D
J
(
g
(
t
,
s
)
)
<
c
∞
.
Now, we apply [53, Theorem 2.3] with
𝒮
~
:=
{
v
∈
𝕏
:
∥
v
-
u
∥
≤
δ
}
and c:=c∞.
Choosing
ε
:=
min
{
c
∞
-
β
4
,
μ
δ
8
}
,
we deduce that there exists a deformation η∈C([0,1]×𝕏,𝕏) such that the following assertions hold:
η
(
t
,
v
)
=
v
if v∈J-1([c∞-2ε,c∞+2ε]).
J
(
η
(
1
,
v
)
)
≤
c
∞
-
ε
for each v∈𝕏 with ∥v-u∥≤δ and J(v)≤c∞+ε.
J
(
η
(
1
,
v
)
)
≤
J
(
v
)
for all u∈𝕏.
By (ii) and (iii), we conclude that
(5.17)
max
(
t
,
s
)
∈
∂
D
J
(
η
(
1
,
g
(
t
,
s
)
)
)
<
c
∞
.
To complete the proof it suffices to prove that
(5.18)
η
(
1
,
g
(
D
)
)
∩
ℳ
≠
∅
.
Indeed, the definition of c∞ and (5.18) contradict (5.17).
Hence, let us define the maps
h
(
t
,
s
)
:=
η
(
1
,
g
(
t
,
s
)
)
,
ψ
0
(
t
,
s
)
:=
(
J
′
(
g
(
t
,
1
)
)
t
u
+
,
J
′
(
g
(
1
,
s
)
)
s
u
-
)
,
ψ
1
(
t
,
s
)
:=
(
1
t
J
′
(
h
(
t
,
1
)
)
h
(
t
,
1
)
+
,
1
s
J
′
(
h
(
1
,
s
)
)
h
(
1
,
s
)
-
)
.
By Lemma 4.1 (iii), the C1-function γ+(t)=hu(t,1) has a unique global maximum point t=1 (note that tγ+′(t)=〈J′(g(t,1)),tu+〉). By density, given ε>0 small enough, there is
γ
+
,
ε
∈
C
∞
(
[
1
2
,
3
2
]
)
such that
∥
γ
+
-
γ
+
,
ε
∥
C
1
(
[
1
2
,
3
2
]
)
<
ε
,
with t+ being the unique maximum global point of γ+,ε in [12,32]. Therefore,
∥
γ
+
′
-
γ
+
,
ε
′
∥
C
(
[
1
2
,
3
2
]
)
<
ε
,
γ
+
,
ε
′
(
1
)
=
0
,
γ
+
,
ε
′′
(
1
)
<
0
.
Analogously, there exists γ-,ε∈C∞([12,32]) such that
∥
γ
-
′
-
γ
-
,
ε
′
∥
C
(
[
1
2
,
3
2
]
)
<
ε
,
γ
+
,
ε
′
(
1
)
=
0
,
γ
+
,
ε
′′
(
1
)
<
0
,
where γ-(s)=hu(1,s).
Let us define ψε∈C∞(D) by ψε(t,s):=(tγ+,ε′(t),sγ-,ε′(s)) and note that
∥
ψ
ε
-
ψ
0
∥
C
(
D
)
<
3
2
2
ε
,
(
0
,
0
)
∉
ψ
ε
(
∂
D
)
,
and (0,0) is a regular value of ψε in D. On the other hand, (1,1) is the unique solution of ψε(t,s)=(0,0) in D. By the definition of Brouwer’s degree, we conclude that
deg
(
ψ
0
,
D
,
(
0
,
0
)
)
=
deg
(
ψ
ε
,
D
,
(
0
,
0
)
)
=
sgn
Jac
(
ψ
ε
)
(
1
,
1
)
for ε small enough.
Since
Jac
(
ψ
ε
)
(
1
,
1
)
=
[
γ
+
,
ε
′
(
1
)
+
γ
+
,
ε
′′
(
1
)
]
×
[
γ
-
,
ε
′
(
1
)
+
γ
-
,
ε
′′
(
1
)
]
=
γ
+
,
ε
′′
(
1
)
×
γ
-
,
ε
′′
(
1
)
>
0
,
we obtain that
deg
(
ψ
0
,
D
,
(
0
,
0
)
)
=
sgn
[
γ
+
,
ε
′′
(
1
)
×
γ
-
,
ε
′′
(
1
)
]
=
1
,
where Jac(ψε) is the Jacobian determinant of ψε and sign denotes the sign function.
On the other hand, by (5.16) we have
(5.19)
J
(
g
(
t
,
s
)
)
≤
β
<
β
+
c
∞
2
=
c
∞
-
2
(
c
∞
-
β
4
)
≤
c
∞
-
2
ε
for all
(
t
,
s
)
∈
∂
D
.
By (5.19) and (i), it follows that g=h on ∂D. Therefore, ψ1=ψ0 on ∂D, and consequently
deg
(
ψ
1
,
D
,
(
0
,
0
)
)
=
deg
(
ψ
0
,
D
,
(
0
,
0
)
)
=
1
,
which shows that ψ1(t,s)=(0,0) for some (t,s)∈D.
Now, in order to verify that (5.18) holds true, we prove that
(5.20)
ψ
1
(
1
,
1
)
=
(
J
′
(
h
(
t
,
1
)
)
h
(
1
,
1
)
+
,
J
′
(
h
(
1
,
1
)
)
h
(
1
,
1
)
-
)
=
0
.
As a matter of fact, (5.20) and the fact that (1,1)∈D yield h(1,1)=η(1,g(1,1))∈ℳ.
We argue as follows: If the zero (t,s) of ψ1 obtained above is equal to (1,1), there is nothing to do. On the other hand, if (t,s)≠(1,1), we take 0<δ1<min{|t-1|,|s-1|} and consider
D
1
:=
[
1
-
δ
1
2
,
1
+
δ
1
2
]
×
[
1
-
δ
1
2
,
1
+
δ
1
2
]
.
Then (t,s)∈D∖D1. Hence, we can repeat for D1 the same argument used for D, so that we can find a couple (t1,s1)∈D1 such that ψ1(t1,s1)=0. If (t1,s1)=(1,1), there is nothing to prove. Otherwise, we can continue with this procedure and find in the n-th step that (5.20) holds, or produce a sequence (tn,sn)∈Dn-1∖Dn which converges to (1,1) and such that
(5.21)
ψ
1
(
t
n
,
s
n
)
=
0
for every
n
∈
ℕ
.
Thus, taking the limit as n→∞ in (5.21) and using the continuity of ψ1, we get (5.20). Therefore, u:=u++u- is a critical point of J.
Finally, we consider the case when f is odd. Clearly, the functional ψ is even. From (3.10) and (4.16) we have that ψ is bounded from below in 𝕊. Taking into account Lemma 2.2 and Lemma 2.3, we can infer that ψ satisfies the Palais–Smale condition on 𝕊. Then, by Proposition 3.2 and [48], we can conclude that the functional J has infinitely many critical points.
Acknowledgements
The authors warmly thank the anonymous referee for her/his useful and nice comments on the paper.
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Giovany M. Figueiredo
