1 Introduction
In this work, the groundbreaking paper [3] by V. Benci and P. H. Rabinowitz is revisited. The aim is to prove an abstract linking theorem for Cerami sequences [6], which will complement related works found in the literature and make possible to extend for many applications. Our interest in applications is twofold: On one hand, extending results for existence of solutions to nonlinear Schrödinger equations, elliptic systems or even Hamiltonian systems, with very general potentials which make the problems strongly indefinite. On the other hand working with nonlinear terms which do not satisfy any monotonicity condition such as those required to perform projections on the so called Nehari manifold as in, for instance, [20, 29]. In this purpose, spectral properties of self-adjoint operators are going to be exploited in order to get the geometry of a linking structure and then apply an abstract result to obtain a critical point to the functional associated to the nonlinear equation, namely a solution to the problem. Furthermore, a compactness structure given by Cerami sequences, (C)c sequences for short, is faced here since asymptotically linear problems at infinity are studied. Thus, inspired by the theory developed in [3], in this work a more general version of its main result [3, Theorem 1.29] is provided for (C)c sequences. To do so, a deformation lemma adapted for Cerami sequences is proved and then the abstract results obtained by V. Benci and P. Rabinowitz are extended.
Our main result, developed throughout this paper, is the following.
Theorem (Linking Theorem for Cerami Sequences).
Let E be a real Hilbert space with inner product (⋅,⋅), let E1 be a closed subspace of E and E2=E1⊥. Let I∈C1(E,R) satisfy the following conditions:
I
(
u
)
=
1
2
(
L
u
,
u
)
+
B
(
u
)
for all
u
∈
E
, where
u
=
u
1
+
u
2
∈
E
1
⊕
E
2
, Lu=L1u1+L2u2 and Li:Ei→Ei, i=1,2, is a bounded linear self-adjoint mapping.
B
is weakly continuous and uniformly differentiable on bounded subsets of
E.
There exist Hilbert manifolds
S
,
Q
⊂
E
such that
Q
is bounded and has boundary
∂
Q
, and constants
α
>
ω
and
v
∈
E
2
such that
S
⊂
v
+
E
1
and
I
≥
α
on
S,
If
I
(
u
n
)
, for a sequence
(
u
n
)
, is bounded and
(
1
+
∥
u
n
∥
)
∥
I
′
(
u
n
)
∥
→
0
as
n
→
+
∞
, then
(
u
n
)
is bounded.
Then I possesses a critical value c≥α.
It is important to highlight that (I2) implies that B′(u) maps weakly convergent to strongly convergent sequences, which gives a kind of partial compactness for I. Moreover, (I4) is a weakened version of the Cerami condition, (C)c condition for short, once the boundedness of any (C)c sequence will be enough to look for a nontrivial critical point without wondering whether it has a convergent subsequence (see the first paragraph of the proof of Theorem 2.4). Together, hypotheses (I2) and (I4) ensure the existence of a critical value c which can be characterized as a minimax level. Furthermore, hypotheses (I1) and (I3) produce a quite general linking geometry for the functional I so that both subspaces in the Hilbert decomposition are allowed to be infinite-dimensional. The conjunction of these hypotheses reproduces sufficient tools to obtain a nontrivial critical point for I in the desired applications scenario. In fact, our approach of constructing a linking structure by means of a sharp study on the spectral properties of the Schrödinger operator is a methodological novelty. This idea extremely enhances the current references since it lights up the fundamental relation between the asymptotic behavior of the nonlinear term and the spectrum features. Therefore, the purpose of revisiting the core of linking structures was precisely to understand this interaction and discard any unnecessary hypothesis.
The pioneering work in this direction is [1] by P. Bartolo, V. Benci and D. Fortunato, where a deformation lemma for Cerami sequences was developed assuming the (C)c condition, as a qualitative deformation lemma with the purpose of extending previous critical point results to non-super-quadratic problems. Thereafter, D. Costa and C. Magalhães [8] proved abstract linking results for strongly indefinite non-quadratic problems on bounded domains, making use of the deformation lemma introduced in [1] and proving that under their assumptions the associated functional satisfied the (C)c condition. Alternatively, as aforementioned, the same lines as in [3] are followed, and hence a deformation lemma without the (C)c condition is proved. Furthermore, our version of the linking theorem only requires the boundedness of Cerami sequences.
In the literature one also finds a paper by G. Li and C. Wang [18] which presented a similar argument, introducing a new kind of deformation lemma, without the (C)c condition, but subsequently used in a linking theorem under the (C)c condition. Moreover, in the abstract result they required that one of the subspaces in the linking decomposition is of finite dimension, while in our result both subspaces in the decomposition may be of infinite dimension. Their construction was inspired by ideas of M. Willem [31] for the quantitative deformation lemma. Although a kind of deformation lemma is developed, it is deeply different from theirs since nonstandard ideas in [3] are closely followed. In fact, the mapping η in deformation is in general determined by solving an appropriate initial value problem involving I′(η). However, this is not suitable for our purposes because is necessary to construct an η satisfying special properties, which will be fundamental in attaining the critical minimax level.
It is also important to mention the theory developed by W. Kryszewski and A. Szulkin in [16], where they solved a more general class of superlinear problems, with assumptions of periodicity. Developing a new degree theory and a weaker topology, they generalized abstract linking theorems introduced in [3] also working with Palais–Smale sequences. Following the same idea, G. Li and A. Szulkin [17] extended the results in [16] obtaining a (C)c sequence for the asymptotically linear case. Nevertheless, so as to get a nontrivial solution, without the (C)c condition, these authors required extra assumptions, including a monotonicity condition on the nonlinearity in an auxiliary problem solved in [30], which had been treated by adapting techniques in [16, 14].
Posteriorly, T. Bartsch and Y. Ding [2] complemented the results in [16] considering both, Palais–Smale and Cerami sequences, in order to apply their abstract results to a Dirac equation where the nonlinearity could be asymptotically linear or superlinear at infinity. Similarly, in [11], Y. Ding and B. Ruf worked with an asymptotically linear problem with a Dirac operator, but without periodicity conditions. Their operator satisfies that the essential spectrum is ℝ∖(-a,a) and that the discrete spectrum intersects the interval (0,q0) for some positive q0. Then they could make use of discrete and positive eigenvalues in the linking structure and apply a particular case of the result in [2], so as to obtain a Cerami sequence. Under their assumptions they were able to prove that their functional satisfied the (C)c condition, which yielded their results.
It is worth highlighting that [11] adapted assumptions and arguments introduced in another very inspiring paper [10], where L. Jeanjean and Y. Ding worked with Hamiltonian Systems, looking for homoclinic orbits, without any periodicity condition. These authors also applied the abstract critical point theory developed in [2], and in order to recover the desired compactness they imposed hypotheses controlling the size of the nonlinearity with respect to the behavior of the potential at infinity. Thus, their assumptions yielded the linking geometry and provided the (C)c condition. In contrast to the argument presented by these authors, our approach does not require the guarantee of the (C)c condition, the necessary compactness to solve the problem is embedded inside the four conditions assumed in the abstract result.
Still referring to abstract results involving linking structure, it is well known that M. Schechter and W. Zou have developed many relevant papers in this spirit; see especially [24, 23, 26, 25] among other works by these authors. In our understanding, their results are different from ours in the sense that, roughly speaking, they usually work with weaker linking geometries in order to get either a Palais–Smale or a Cerami bounded sequence. Then they apply widely alternative arguments to find a solution to the proposed application. On the other side, our idea is to obtain a result which could ensure the existence of a nontrivial critical point directly, without stressing either on geometry or on compactness of the associated functional, separately. Notwithstanding, it is worth pointing out clever abstract results obtained in [25, Theorem 2.1] and [26, Theorem 2.1], where Schechter and Zou also made use of the “Monotonicity Trick” developed by L. Jeanjean in [14], for the purpose of getting critical points for a family of functionals, converging to a critical point the functional associated to the initial problem. These results have been applied to solve asymptotically linear problems with spectral properties similar to those presented in this paper; see, for instance, [7].
Here we present two applications for Schrödinger equations for our abstract result. Other applications can be found in [27], where M. Soares proves the existence of solutions to Hamiltonian and elliptic systems by applying this abstract result.
For our applications we consider problem (P),
-
Δ
u
+
V
(
x
)
u
=
g
(
x
,
u
)
in
ℝ
N
,
N
≥
3
,
in the case where g(x,s)=h(x)f(s), and h satisfies
h
∈
L
∞
(
ℝ
N
)
∩
L
q
(
ℝ
N
)
, q=2*2*-p for some p∈(2,2*) and h>0 almost everywhere.
Furthermore, f is asymptotically linear satisfying the following conditions:
f
∈
C
(
ℝ
,
ℝ
)
and lims→0f(s)s=0.
There exists a>0 such that lims→+∞F(s)s2=a2, where F(s):=∫0sf(t)𝑑t, and F(s)≥0.
Setting Q(s):=12f(s)s-F(s)>0 for all s∈ℝ/{0}, we have
lim
s
→
+
∞
Q
(
s
)
=
+
∞
.
Moreover, for the first application, we assume that V satisfies the following conditions:
V
∈
C
(
ℝ
N
,
ℝ
)
and lim|x|→+∞V(x)=V∞>0.
Setting A:=-Δ+V(x) as an operator of L2(ℝN) and denoting by σ(A) the spectrum of A, we have
sup
[
σ
(
A
)
∩
(
-
∞
,
0
)
]
=
σ
-
<
0
<
σ
+
=
inf
[
σ
(
A
)
∩
(
0
,
+
∞
)
]
.
This application was inspired by [19], where L. Maia, J. Oliveira Junior and R. Ruviaro solved problem (P) with potential V satisfying (V1)–(V2). In addition, they required that 0∉σ(A) and assumed some more hypotheses of decay and compactness on V. About the nonlinearity they set h≡1, f∈C3(ℝN,ℝ) with some growth hypotheses on its derivatives, and assumed f(s)/s being increasing as well. Since this kind of potential ensures that the subspace where A is negative definite is finite-dimensional, they could apply the aforementioned version of the linking theorem introduced by G. Li and C. Wang in [18] to get the (C)c sequence. They used the associated problem at infinity and a splitting lemma to compare the levels of both problems and get the necessary compactness. Trying to improve their result, our abstract linking theorem for (C)c sequences is applied and a nontrivial critical point is obtained straightway, avoiding such monotonicity assumptions on f.
Staring at our hypotheses, it is also possible to say that our second application complements the work by L. Jeanjean and K. Tanaka in [15]. In fact, they assumed V(x)≥α>0, and so they worked with Ekeland’s principle to get a (C)c sequence and due to the geometry of their functional they applied the Mountain Pass Theorem to get a critical point. They also worked with an asymptotically linear problem where f(s)/s is not necessarily increasing. In addition, they assumed h≡1 and f(s)/s→a>infσ(A)>0 as |s|→+∞. Differently, in our case V changes sign and infσ(A)<0, which implies a linking geometry and prevents us to use the same argument. However, by considering the positive spectrum, a similar hypothesis is assumed:
a
>
inf
u
1
∈
E
1
,
u
≠
0
∫
ℝ
N
(
|
∇
u
1
(
x
)
|
2
+
V
(
x
)
u
1
2
(
x
)
)
𝑑
x
∫
ℝ
N
h
(
x
)
u
1
2
(
x
)
𝑑
x
≥
1
h
∞
inf
[
σ
(
A
)
∩
(
0
,
+
∞
)
]
=
σ
+
h
∞
>
0
,
where ∥h∥L∞(ℝN):=h∞ and E1 is the subspace of H1(ℝN) on which the operator A is positive definite. This hypothesis allows to develop a linking structure.
On the other hand, the work [9] by D. Costa and H. Tehrani can be cited since the same V as theirs is presented here. More specifically, they required (V1), and our hypothesis (V2) implies that either 0 is an isolated point of σ(A) or it is in a gap of the spectrum, which is also required by them. However, they did not work with an asymptotically linear problem, but they assumed the well-known Ambrosetti–Rabinowitz condition and required a nondecreasing nonlinearity. In their assumptions, h=a is a sign-changing function in C1(ℝN,ℝ) and such that lim|x|→+∞a(x)=a∞<0, differently from lim|x|→+∞h(x)=0 in our case. Moreover, instead of using an abstract linking theorem, they applied a method of approximations to solve their problem.
It is also worth mentioning the paper [12] by A. Edelson and C. Stuart since assumptions close to theirs are considered here. However, they required f(s)/s strictly increasing, which is removed here. Moreover, they applied the method of sub- and super-solution and bifurcation to get a solution to their problem.
Finally, for the second application we keep all assumptions on h and f, but on V we assume (V2) and replace assumption (V1) by the following:
V
∈
C
(
ℝ
N
,
ℝ
)
is (2π)N-periodic in x∈ℝN.
This application is motivated by the fact that in virtue of (V1) the subspace in which the operator A is negative definite is finite-dimensional. Since this is irrelevant for applying our abstract result, we sought for a problem where both subspaces, in which the operator A is positive and negative definite, are infinite-dimensional. In fact, (V1′) combined with (V2) ensures the desired. Although all other hypotheses are kept, this replacement changes completely the spectral properties of A, which are fundamental to determine the linking geometry. It would be interesting to note that we only require V being a periodic function in order to explore spectral properties, we do not need a periodic nonlinearity. Hence unlike most of the works in the literature (see [16, 2]), we do not make use of periodicity to translate a (C)c sequence and ensure the existence of a critical point.
3 A Quantitative Deformation Lemma and Proof of the Main Result
Inspired by [3], we find it suitable to state a variant of the standard quantitative deformation lemma for Cerami sequences, without Cerami condition, which is necessary to prove Theorem 2.4.
Lemma 3.1 (Deformation Lemma).
Let I∈C1(E,R) satisfying (I1)–(I2) as in Theorem 2.4. Then for any R∈N, ϱ>0 and ε∈(0,110), if s:=(R+2)2, there exist k∈N and η∈Γ, such that the following assertions hold:
I
(
η
t
k
s
(
u
)
)
≤
I
(
u
)
+
ϱ
for all
u
∈
ℬ
R
+
2
and
t
∈
[
0
,
1
]
.
If
c
∈
ℝ
and
∥
I
′
(
w
)
∥
(
1
+
∥
w
∥
)
≥
2
ε
for all
w
∈
ℬ
R
+
1
∩
I
-
1
(
[
c
-
ε
,
c
+
ε
]
)
, then
I
(
η
k
s
(
u
)
)
≤
c
-
ε
2
whenever
u
∈
ℬ
R
/
2
∩
I
-
1
(
[
c
-
ε
,
c
+
ε
]
)
.
Remark 3.2.
The mapping η is usually determined by solving an appropriate differential equation involving I′(η). Such an approach seems to fail here since it does not give an η satisfying (Γ1)–(Γ3) which are crucial for the purpose of proving Theorem 2.4. Hence, in order to prove Lemma 3.1 it is necessary to argue similarly to the proof of [3, Theorem 1.5].
Proof.
First, choose χ∈C∞(ℝ,ℝ) such that
χ
(
t
)
=
1
for
s
≤
R
+
1
,
χ
(
t
)
=
0
for
s
≥
R
+
2
,
χ
′
(
t
)
<
0
for
t
∈
(
R
+
1
,
R
+
2
)
,
χ
(
t
)
≤
(
R
+
2
-
t
)
2
for
t
∈
[
R
+
3
2
,
R
+
2
]
.
For u=u1+u2∈E1⊕E2, set Vi(u):=χ(∥ui∥)PiI′(u), i=1,2, and V(u):=V1(u)+V2(u). Now note that V(u)≡I′(u) in ℬR+1. Moreover, by (I1)–(I2), there is a constant M=M(R) such that ∥I′(u)∥≤M for u∈ℬR+2. With this, set
(3.1)
ε
¯
:=
1
M
s
min
(
ϱ
,
ε
2
)
.
Since I is uniformly differentiable on bounded sets, there is a δ=δ(ε¯,R)>0 such that
(3.2)
|
I
(
u
+
v
)
-
I
(
u
)
-
I
′
(
u
)
v
|
≤
ε
¯
∥
v
∥
for u,u+v∈ℬR+2 and ∥v∥≤δ. Assume δ≤1 and choose k∈ℕ such that
(3.3)
1
k
<
min
(
δ
2
M
,
1
8
(
R
+
2
)
(
1
+
max
ℝ
|
χ
′
(
s
)
|
(
∥
L
1
∥
+
∥
L
2
∥
)
)
.
Now define ηt(u):=u-tkV(u).
By assuming this claim and postponing its proof, it is necessary to check that ℬR+2 is an invariant set for ηt for the purpose of proving (i)–(ii). In fact, for u=u1+u2∈ℬR+2, by the definitions of η and V it follows that
(3.4)
∥
P
i
η
t
(
u
)
-
u
i
∥
=
∥
(
u
i
-
t
k
V
i
(
u
)
)
-
u
i
∥
=
t
k
∥
V
i
(
u
)
∥
≤
M
k
χ
(
∥
u
i
∥
)
.
Note that, for ui∈BR+3/2, the choice of χ implies that
M
k
χ
(
∥
u
i
∥
)
≤
1
2
≤
R
+
2
-
∥
u
i
∥
via (3.3), while for ∥ui∥≥R+32 it follows that
1
2
≥
R
+
2
-
∥
u
i
∥
≥
(
R
+
2
-
∥
u
i
∥
)
2
≥
M
k
χ
(
∥
u
i
∥
)
by the choice of χ and k. Hence, the right-hand side of (3.4) does not exceed R+2-∥ui∥, i=1,2, which implies that ℬR+2 is invariant for ηt. Indeed, from (3.4) and the triangular inequality, this yields
(3.5)
∥
P
i
η
t
(
u
)
∥
≤
M
k
χ
(
∥
u
i
∥
)
+
∥
u
i
∥
≤
R
+
2
.
Since dist(ηt(u),∂ℬR+2), the distance from ηt(u) to ∂ℬR+2 satisfies
dist
(
η
t
(
u
)
,
∂
ℬ
R
+
2
)
=
min
i
=
1
,
2
(
R
+
2
-
∥
P
i
(
η
t
(
u
)
)
∥
)
.
Thus (3.5) implies that ηt(u)∈ℬR+2.
In order to prove (i), observe that by (3.3) and the definitions of η, M and k it follows that
(3.6)
∥
η
t
(
u
)
-
u
∥
=
∥
-
t
k
V
(
u
)
∥
=
t
k
∥
V
(
u
)
∥
≤
M
k
<
δ
for all u∈ℬR+2. Hence, fixing u∈ℬR+2 and using (3.2) with v=-tkV(u), we obtain
(3.7)
I
(
u
+
v
)
=
I
(
η
t
(
u
)
)
≤
I
(
u
)
-
t
k
I
′
(
u
)
V
(
u
)
+
ε
¯
t
k
∥
V
(
u
)
∥
.
Since E2=E1⊥, we obtain P1(I′(u))⊥P2(I′(u)), and by the definition of V it follows that
(3.8)
I
′
(
u
)
V
(
u
)
=
χ
(
∥
u
1
∥
)
∥
P
1
(
I
′
(
u
)
)
∥
2
+
χ
(
∥
u
2
∥
)
∥
P
2
(
I
′
(
u
)
)
∥
2
≥
0
.
Thus, by (3.1), (3.6)–(3.8) and the definition of ε¯ it follows that
(3.9)
I
(
η
t
(
u
)
)
≤
I
(
u
)
+
ε
¯
M
k
≤
I
(
u
)
+
ϱ
M
s
M
k
=
I
(
u
)
+
ϱ
k
s
.
Provided that ℬR+2 is invariant under ηt, (i) holds by iterating (3.9) ks times. In fact, iterating twice means using ηt(u) instead of u in (3.9), and after that using again (3.9), but for u it yields
I
(
η
t
2
(
u
)
)
=
I
(
η
t
(
η
t
(
u
)
)
)
≤
I
(
η
t
(
u
)
)
+
ϱ
k
s
≤
I
(
u
)
+
2
ϱ
k
s
.
Then, after ks iterations, it yields
I
(
η
t
k
s
(
u
)
)
≤
I
(
u
)
+
k
s
ϱ
k
s
≤
I
(
u
)
+
ϱ
,
and (i) is proved.
In order to prove (ii), take u∈ℬR/2∩I-1([c-ε,c+ε]). Three cases are considered.
Case I: η1j(u)∈BR+1∩I-1([c-ε,c+ε]) for 1≤j≤ks. By definition, V(u)=I′(u) in ℬR+1. Then, by fixing j, 1≤j≤ks, the definition of η1 yields
η
1
j
(
u
)
-
η
1
j
-
1
(
u
)
=
-
1
k
V
(
η
1
j
-
1
(
u
)
)
=
-
1
k
I
′
(
η
1
j
-
1
(
u
)
)
.
Then, by using (3.7) for η1j(u) and η1j-1(u) instead of u+v and u, by the definition of M, and due to (3.1), it yields
(3.10)
I
(
η
1
j
(
u
)
)
-
I
(
η
1
j
-
1
(
u
)
)
≤
-
1
k
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
2
+
1
k
ε
¯
M
≤
-
1
k
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
2
+
ε
2
k
s
.
Then, by the telescopic sum, and using (3.10) for all 1≤j≤ks, it follows that
(3.11)
I
(
η
1
k
s
(
u
)
)
-
I
(
u
)
=
∑
j
=
1
k
s
[
I
(
η
1
j
(
u
)
)
-
I
(
η
1
j
-
1
(
u
)
)
]
≤
∑
j
=
1
k
s
[
-
1
k
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
2
+
ε
2
k
s
]
.
Provided that
(
R
+
2
)
∥
I
′
(
u
)
∥
≥
(
1
+
∥
u
∥
)
∥
I
′
(
u
)
∥
≥
2
ε
from the assumption, by setting εs:=εs, it follows that ∥I′(u)∥2≥2εs for all u∈ℬR+1. Thus, (3.11) yields
(3.12)
I
(
η
1
k
s
(
u
)
)
-
I
(
u
)
≤
∑
j
=
1
k
s
[
-
2
ε
s
k
+
ε
s
2
k
]
=
-
3
ε
2
.
Therefore, since I(u)≤c+ε, inequality (3.12) implies that I(η1ks(u))≤I(u)-3ε2≤c-ε2, and the result holds for the first case.
Case II: η1j(u)∈BR+1∩I-1([c-ε,c+ε]) for 1≤j≤m-1 but η1m(u)∉I-1([c-ε,c+ε]) for some 1≤m≤ks. From (3.10) and since ∥I′(u)∥2≥2εs for all u∈ℬR+1, it follows that
I
(
η
1
j
(
u
)
)
-
I
(
η
1
j
-
1
(
u
)
)
≤
-
2
ε
s
k
+
ε
s
2
k
=
-
3
ε
s
2
k
,
and hence I(η1j(u))<I(η1j-1(u)) for 1≤j≤m. Then η1m(u)∉I-1([c-ε,c+ε]) and I(η1m(u))<I(η1m-1(u)) imply that I(η1m(u))<c-ε. Thus, by using again a telescopic sum, it follows that
(3.13)
I
(
η
1
k
s
(
u
)
)
=
I
(
η
1
m
(
u
)
)
+
∑
j
=
m
+
1
k
s
[
I
(
η
1
j
(
u
)
)
-
I
(
η
1
j
-
1
(
u
)
)
]
≤
c
-
ε
+
∑
j
=
m
+
1
k
s
[
I
(
η
1
j
(
u
)
)
-
I
(
η
1
j
-
1
(
u
)
)
]
.
By fixing j, m+1≤j≤ks, and using (3.7) as in (3.10), it yields
(3.14)
I
(
η
1
j
(
u
)
)
-
I
(
η
1
j
-
1
(
u
)
)
≤
-
1
k
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
2
+
ε
2
k
s
≤
ε
2
k
s
.
By replacing (3.14) in (3.13) for all m+1≤j≤ks, it follows that
I
(
η
1
k
s
(
u
)
)
≤
c
-
ε
+
∑
j
=
m
+
1
k
s
[
ε
2
k
s
]
=
c
-
ε
+
(
k
s
-
m
k
s
)
ε
2
≤
c
-
ε
2
.
Therefore, the result also holds in this case.
Case III: η1j(u)∈BR+1∩I-1([c-ε,c+ε]) for 1≤j≤m-1 but η1m(u)∉BR+1 for some 1≤m≤ks. Since u∈ℬR/2, it follows that
∥
u
∥
+
R
2
+
1
≤
R
+
1
≤
∥
η
1
m
(
u
)
∥
,
and hence, by the triangular inequality and the telescopic sum, it follows that
(3.15)
R
+
2
2
≤
∥
η
1
m
(
u
)
∥
-
∥
u
∥
≤
∥
η
1
m
(
u
)
-
u
∥
≤
∑
j
=
1
m
∥
η
1
j
(
u
)
-
η
1
j
-
1
(
u
)
∥
.
By the definition of η1 and since V(u)=I′(u) in ℬR+1, it follows that
(3.16)
∥
η
1
j
(
u
)
-
η
1
j
-
1
(
u
)
∥
=
1
k
∥
V
(
η
1
j
-
1
(
u
)
)
∥
=
1
k
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
for all 1≤j≤m because η1j(u)∈ℬR+1 for all 1≤j≤m-1. Hence, by replacing (3.16) in (3.15) and applying Hölder’s inequality for finite sums, it yields
(3.17)
R
+
2
2
≤
1
k
∑
j
=
1
m
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
≤
m
1
2
k
[
∑
j
=
1
m
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
2
]
1
2
.
From (3.10) it follows that
1
k
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
2
≤
-
[
I
(
η
1
j
(
u
)
)
-
I
(
η
1
j
-
1
(
u
)
)
]
+
ε
2
k
s
,
which yields
(3.18)
∥
I
′
(
η
1
j
-
1
(
u
)
)
∥
2
≤
k
(
I
(
η
1
j
-
1
(
u
)
)
-
I
(
η
1
j
(
u
)
)
)
+
ε
s
2
for all 1≤j≤m. Thus, by using (3.18) in (3.17), it yields
R
+
2
2
≤
m
1
2
k
[
∑
j
=
1
m
k
(
I
(
η
1
j
-
1
(
u
)
)
-
I
(
η
1
j
(
u
)
)
)
+
ε
s
2
]
1
2
=
m
1
2
k
[
k
(
I
(
u
)
-
I
(
η
1
m
(
u
)
)
)
+
m
ε
s
2
]
1
2
.
By squaring both sides, it follows that
s
4
=
(
R
+
2
2
)
2
≤
m
k
2
[
k
(
I
(
u
)
-
I
(
η
1
m
(
u
)
)
)
+
m
ε
s
2
]
=
m
k
[
I
(
u
)
-
I
(
η
1
m
(
u
)
)
+
m
ε
2
k
s
]
(3.19)
≤
m
k
[
c
+
ε
-
I
(
η
1
m
(
u
)
)
+
ε
2
]
.
By multiplying both sides of (3.19) by km, it yields
1
4
≤
k
s
4
m
≤
c
+
3
ε
2
-
I
(
η
1
m
(
u
)
)
,
which implies I(η1m(u))≤c+3ε2-10ε4=c-ε since 0<ε<110 by assumption. Now, as in Case II, by using a telescopic sum and (3.14), it yields
I
(
η
1
k
s
(
u
)
)
=
I
(
η
1
m
(
u
)
)
+
∑
j
=
m
+
1
k
s
[
I
(
η
1
j
(
u
)
)
-
I
(
η
1
j
-
1
(
u
)
)
]
≤
c
-
ε
+
∑
j
=
m
+
1
k
s
[
ε
2
k
s
]
=
c
-
ε
+
(
k
s
-
m
k
s
)
ε
2
≤
c
-
ε
2
.
Therefore, (ii) also holds for the third case.
Finally, to finish the proof of this lemma, it is left to prove the claim. Denoting by Id:E→E the identity map in E, since PiV(u)=Vi(u)=χ(∥ui∥)PiI′(u)=χ(∥ui∥)(Liui+PiB′(u)), we obtain that
P
i
η
t
(
u
)
=
P
i
(
u
-
t
k
V
(
u
)
)
=
(
Id
-
t
k
χ
(
∥
u
i
∥
)
L
i
)
u
i
-
t
k
χ
(
∥
u
i
∥
)
P
i
B
′
(
u
)
.
Hence, it is suitable to set
U
t
(
u
)
:=
∑
i
=
1
,
2
(
Id
-
t
k
χ
(
∥
u
i
∥
)
L
i
)
u
i
and
K
t
(
u
)
:=
-
t
k
∑
i
=
1
,
2
χ
(
∥
u
i
∥
)
P
i
B
′
(
u
)
.
In fact, observe that given un⇀u in E, (I2) and Lemma 2.3 imply that B′(un)→B′(u) in E′. Since Pi is continuous, it follows that Kt(un)→Kt(u). Thus Kt is completely continuous, and therefore is compact for all t∈[0,1]. Moreover, by the definitions of Ut and Kt, it is clear that η satisfies (Γ2)–(Γ3). Via (I1)–(I2) there is a constant C=C(r) such that ∥I′(u)∥≤C for all u∈Br. Hence by the definition of ηt,
∥
η
t
(
u
)
∥
≤
r
+
∥
I
′
(
u
)
∥
≤
r
+
C
,
and so ηt maps bounded sets to bounded sets, namely η satisfies (Γ4). Lastly, it suffices to show that Ut is a homeomorphism of E onto E, in order to complete the verification of (Γ1). By (Γ3), it suffices to show that PiUt is a homeomorphism of Ei onto Ei, i=1,2. Let u,v∈Ei. For any t∈[0,1], by the definition of χ, if ∥u∥,∥v∥≥R+2, then
t
k
∥
χ
(
∥
u
∥
)
L
i
u
-
χ
(
∥
v
∥
)
L
i
v
∥
=
0
≤
1
2
∥
u
-
v
∥
.
Hence, without loss of generality, suppose that ∥v∥≤R+2. By the definitions of χ and k, and via the Mean Value Theorem, for any t∈[0,1] it yields
t
k
∥
χ
(
∥
u
∥
)
L
i
u
-
χ
(
∥
v
∥
)
L
i
v
∥
≤
1
k
∥
L
i
∥
∥
u
-
v
∥
+
1
k
∥
L
i
∥
|
χ
(
∥
u
∥
)
-
χ
(
∥
v
∥
)
|
∥
v
∥
≤
1
k
∥
L
i
∥
(
R
+
2
)
[
1
+
max
ℝ
|
χ
′
(
s
)
|
]
∥
u
-
v
∥
(3.20)
≤
1
2
∥
u
-
v
∥
.
Thus, (3.20) holds for all u,v∈Ei. Now, for each w∈Ei fixed, set ℒw(u):Ei→Ei given by
ℒ
w
(
u
)
:=
t
k
χ
(
∥
u
∥
)
L
i
u
+
w
.
Note that due to (3.20), ℒw(u) is a contraction on Ei. Then it follows from the contracting mapping theorem that ℒw(u) has a unique fixed point uw. Therefore, uw∈Ei is the unique function such that ℒw(uw)=uw, which implies that PiUt(uw)=w is an one-to-one correspondence, and hence PiUt is a bijection. Furthermore, (3.20) yields
∥
P
i
U
t
(
u
)
-
P
i
U
t
(
v
)
∥
=
∥
(
u
-
v
)
-
t
k
(
χ
(
∥
u
∥
)
L
i
u
-
χ
(
∥
v
∥
)
L
i
v
)
∥
≥
∥
u
-
v
∥
-
1
2
∥
u
-
v
∥
=
1
2
∥
u
-
v
∥
,
which implies that (PiUt)-1 is continuous. Since PiUt is continuous by definition, it ensures that PiUt is a homeomorphism of Ei onto Ei. Consequently, η satisfies (Γ1), and finally the claim is proved. ∎
The following lemma gives a significant information about the level c.
Lemma 3.3 (See [3, Proposition 1.17]).
If I satisfies (I3), then c≥α.
Proof.
For the sake of completeness, this proof is included here. In fact, it suffices to show that
(3.21)
h
1
(
Q
¯
)
∩
S
≠
∅
for all h∈Λ. In fact, if (3.21) holds, there is a y∈h1(Q¯)∩S, and hence
(3.22)
sup
u
∈
Q
¯
I
(
h
1
(
u
)
)
≥
I
(
y
)
≥
inf
w
∈
S
I
(
w
)
≥
α
due to (I3) (i). Since (3.22) holds for all h∈Λ, the definition of c in (I4) yields c≥α. The proof of (3.21) follows from the stronger assertion that
(3.23)
h
t
(
Q
¯
)
∩
S
≠
∅
for all h∈Λ and t∈[0,1]. Since S-v⊂E1 via (I3) (i), inequality (3.23) is equivalent to finding, for each t∈[0,1], a u∈Q¯ such that
(3.24)
{
P
1
h
t
(
u
)
∈
S
-
v
,
P
2
h
t
(
u
)
=
v
.
In order to solve (3.24), it is necessary to convert it into an equivalent problem to which the linking geometry hypotheses can be applied. Suppose first that h∈Λ with the corresponding m=1. Let u=u1+u2∈E1⊕E2 as usual. By (Γ1) and (Γ3), (3.24) becomes
(3.25)
{
P
1
h
t
(
u
)
∈
S
-
v
,
P
2
h
t
(
u
)
=
U
t
(
u
2
)
+
P
2
K
t
(
u
)
=
v
.
More generally, suppose the second equation of (3.25) replaced by
(3.26)
P
2
h
t
(
u
)
=
P
2
Z
t
(
u
)
,
where Zt(u) is an arbitrary compact operator with Z0(u)=v. Note that in (3.25), Zt(u)=v is the compact constant operator for all t∈[0,1]. Again via (Γ1) and (Γ3), equation (3.26) is equivalent to
U
t
(
u
2
)
=
-
P
2
(
K
t
(
u
)
-
Z
t
(
u
)
)
,
which is equivalent to
u
2
=
U
t
-
1
(
-
P
2
(
K
t
(
u
)
-
Z
t
(
u
)
)
)
≡
P
2
Y
t
(
u
)
,
where Yt is compact due to the compactness of Kt and Zt, and Y0(u)=v since
U
0
-
1
(
-
P
2
(
K
0
(
u
)
-
Z
0
(
u
)
)
)
=
-
P
2
(
-
Z
0
(
u
)
)
=
v
.
Now, suppose by induction that (3.26) is equivalent to
(3.27)
u
2
=
P
2
Y
t
(
u
)
,
with Yt compact and Y0(u)=v, whenever h∈Λ with the corresponding m=n-1. Then let h∈Λ with m=n so that h=h(1)∘⋯∘h(m), and let h^=h(2)∘⋯∘h(m). Hence h=h(1)∘h^ and, again by (Γ1) and (Γ3), the equation
v
=
P
2
h
t
(
u
)
=
P
2
(
U
t
(
1
)
+
K
t
(
1
)
)
h
^
t
(
u
)
=
U
t
(
1
)
P
2
h
^
t
(
u
)
+
P
2
K
t
(
1
)
h
^
t
(
u
)
is equivalent to
(3.28)
P
2
h
^
t
(
u
)
=
(
U
t
(
1
)
)
-
1
(
-
P
2
K
t
(
1
)
(
h
^
t
(
u
)
)
+
v
)
=
:
P
2
Z
^
t
(
u
)
,
where h(1)=U(1)+K(1). Since Kt(1) is compact, Z^t given by the right-hand side of (3.28) is compact and P2Z^0(u)=v since K0=0. Thus, by the induction hypothesis there is a compact Yt such that (3.28) is equivalent to solving (3.27).
Now set Φt(u)=P1ht(u)+u2-P2Yt(u)+v, and note that Φ∈Σ since
P
2
Φ
t
=
u
2
-
(
P
2
Y
t
-
v
)
and
Φ
0
(
u
)
=
P
1
h
0
(
u
)
+
u
2
-
P
2
Y
0
(
u
)
+
v
=
P
1
u
+
u
2
-
v
+
v
=
u
.
In addition, P1Φt=P1ht, and provided that P2ht=v is equivalent to (3.27), due to all remarks above, it follows that P2Φt=v is equivalent to P2ht=v by the definition of Φt. Therefore, Φt(u)∈S if and only if ht(u)∈S, and hence to obtain (3.23) and complete the proof it is only necessary to show that
(3.29)
Φ
t
(
Q
)
∩
S
≠
∅
for all t∈[0,1]. Since Φ∈Σ and via (I3) (iii), S and ∂Q link. Then (3.29) holds if
(3.30)
Φ
t
(
∂
Q
)
∩
S
=
∅
.
Suppose the contrary, so there is a u∈∂Q and t∈[0,1] such that Φt(u)∈S. Then ht(u)∈S, but
h
t
(
∂
Q
)
⊂
I
α
+
ω
2
-
β
since h∈Λ. On the other hand,
S
∩
I
α
+
ω
2
-
β
=
∅
due to (I3) (i), and provided that β∈(0,α-ω2), it hence yields a contradiction. Thus (3.30) is satisfied and the proof of the lemma is complete. ∎
Now, under the knowledge of all results in the previous sections, Theorem 2.4 can be finally proved.
Proof of Theorem 2.4.
First, since the identity map h(u)=u is in Λ, we have c<+∞ in view of (I2) and (I4). Moreover, c≥α by Lemma 3.3. Now suppose that c is not a critical value of I. Then I′(u)≠0, for all u∈I-1(c), and hence there exists ε>0 such that
(3.31)
(
1
+
∥
u
∥
)
∥
I
′
(
u
)
∥
≥
2
ε
for all
u
∈
I
-
1
(
[
c
-
ε
,
c
+
ε
]
)
.
If not, for all n∈ℕ there exists a sequence of positive εn→0 and un∈I-1([c-εn,c+εn]) such that
(
1
+
∥
u
n
∥
)
∥
I
′
(
u
n
)
∥
<
2
ε
n
.
From (I4) this sequence is bounded, and thus it possesses a weakly convergent subsequence, still denoted by (un), namely un⇀u as n→+∞ for some u∈E. By (I2) and Lemma 2.3, one has B′(un)→B′(u) along this subsequence and by assumption, I′(un)→0. Then it follows that Lun=I′(un)-B′(un)→-B′(u) as n→+∞. On the other hand, Lun also converges weakly to Lu along this subsequence. Hence Lu=-B′(u) and Lun→Lu strongly. Then I′(u)=Lu+B′(u)=0. Since I(un)→c, again by (I2) it follows that
I
(
u
n
)
=
1
2
(
L
u
n
,
u
n
)
+
B
(
u
n
)
→
I
(
u
)
=
c
.
But it means that c is a critical value of I, contrary to the assumption. Thus there exists an ε as desired in (3.31). Further, ε<110 can be assumed. By the definition of the infimum, choose an h∈Λ with corresponding β such that
(3.32)
c
≤
sup
u
∈
Q
¯
I
(
h
1
(
u
)
)
≤
c
+
ε
and
h
t
(
∂
Q
)
⊂
I
α
-
ω
2
-
β
.
Since h∈Λ, we have that ht maps bounded sets on bounded sets due to the definition of Γ, and hence h1(Q¯) is bounded. Therefore, there is an R∈ℕ such that h1(Q¯)⊂BR/2. By Lemma 3.1, with ϱ=12min(β,ε), there exist η∈Γ and k∈ℕ such that ηtks satisfies (i) and (ii) of that lemma. Let gt(u)=ηtks(ht(u)). Provided that
h
t
(
∂
Q
)
⊂
I
α
-
ω
2
-
β
,
in view of Lemma 3.1 (i),
g
t
(
∂
Q
)
⊂
I
α
-
ω
2
-
β
2
.
Hence g∈Λ and from (2.1) it follows that
(3.33)
c
≤
sup
u
∈
Q
¯
I
(
g
1
(
u
)
)
.
From (3.32), I(h1(u))≤c+ε for all u∈Q¯. Thus, if h1(u)∈I-1([c-ε,c+ε]), by (3.31) it is possible to apply Lemma 3.1 (ii) to conclude that g1(u)∈Ic-ε/2. On the other hand, if h1(u)∈Ic-ε, then Lemma 3.1 (i) yields
I
(
g
1
(
u
)
)
=
I
(
η
1
k
s
(
h
1
(
u
)
)
)
≤
I
(
h
1
(
u
)
)
+
ϱ
≤
c
-
ε
+
ε
2
=
c
-
ε
2
,
and thus g1(u)∈Ic-ε/2 by the choice of ϱ. Consequently, it follows that
sup
u
∈
Q
¯
I
(
g
1
(
u
)
)
≤
c
-
ε
2
,
which contradicts (3.33) and the theorem is proved. ∎
4 Application to Asymptotically Linear Schrödinger Equations in ℝN
This section introduce two applications for the abstract critical point theorem developed previously. The main difference between them is how to obtain the linking geometry, based on their spectra, which are very different to each other.
First, consider problem (P),
(4.1)
-
Δ
u
+
V
(
x
)
u
=
g
(
x
,
u
)
in
ℝ
N
for N≥3, where the potential V satisfies (V1)–(V2) as stated in Section 1.
Remark 4.1.
In view of hypothesis (V1), V(x) is bounded and σess(A)=[V∞,+∞) (see [28, Theorem 3.15, p. 44]), and hence σ(A)∩(-∞,V∞)=σd(A)∩(-∞,V∞). Furthermore, hypothesis (V2) implies that either 0∉σ(A) or 0∈σd(A) since 0∈(σ-,σ+) is an isolated point and the essential spectrum [V∞,+∞) does not have isolated points. Hence 0∉σess(A)=[V∞,+∞), which implies that V∞>0, namely V∞ is positive, and therefore assumption V∞>0 in (V1) is redundant. With this in hand, it is possible to introduce by means of the operator A a norm ∥⋅∥ equivalent to the usual norm ∥⋅∥H1(ℝN) in H1(ℝN) (see [9, Lemma 1.2]). Thus, E=(H1(ℝN),∥⋅∥) will be the Hilbert space used in order to apply Theorem 2.4. Finally, from (V2) we have ∅≠σ(A)∩(-∞,0)=σd(A)∩(-∞,0), i.e., the operator A has negative eigenvalues. Furthermore, this set is finite (see [13, Theorem 30, p. 150]). An example satisfying (V1)–(V2) is given by a continuous V(x) such that
V
(
x
)
=
{
-
V
0
,
|
x
|
<
R
,
V
∞
,
|
x
|
>
2
R
,
where V0>λ1(1)R2>0 is a constant and λ1(1) is the first eigenvalue of the operator (-Δ,H01(B1(0))).
Henceforth, consider the case where g(x,s)=h(x)f(s) and h satisfies (h1). Furthermore, f is asymptotically linear satisfying (f1)–(f3).
Remark 4.2.
Setting ∥h∥L∞(ℝN):=h∞, we observe that assumption (h1) implies that 0<h∞<+∞. In addition, (f2) implies that lims→+∞F(s)=+∞ and lims→+∞f(s)s=a. Moreover, due to assumptions (f1)–(f2) there exists κ>0 such that |f(s)|≤κ|s| for all s∈ℝ and a≤κ. An example satisfying (f1)–(f3) but not with f(s)s increasing, is a continuous f(s) such that
f
(
s
)
=
{
s
7
-
3
2
s
5
+
2
s
3
1
+
s
6
,
|
s
|
<
5
,
s
3
1
+
s
2
,
|
s
|
>
10
.
Let I:E→ℝ be the energy functional associated with problem (P) in (4.1), which is given by
I
(
u
)
=
1
2
∫
ℝ
N
(
|
∇
u
|
2
+
V
(
x
)
u
2
)
-
∫
ℝ
N
h
(
x
)
F
(
u
)
𝑑
x
for all u∈E. As observed in Remark 4.1, the set of eigenvalues σd(A)∩(-∞,0) is finite. Then one can denote it by {λi}i=1j for some j∈ℕ, counting multiplicities, and in addition denote by φi∈E the eigenfunction associated with λi for i=1,…,j and then set E-:=span{φi}i=1j. Moreover, by setting E0:=ker(A), if 0∉σ(A), then E0={0}, if not, then 0∈σd(A). Hence E0 is finite-dimensional. Thus, E-⊕E0 is a finite-dimensional subspace of E and, by setting E+:=(E-⊕E0)⊥, it is the subspace of E in which the operator A is positive definite. With this, E=E+⊕E-⊕E0 and every function u∈E can be uniquely written as u=u++u-+u0, with u+∈E+, u0∈E0 and u-∈E-. Furthermore, as in [9], the operator A induces a norm ∥⋅∥ equivalent to the standard H1(ℝN)-norm and a corresponding inner product (⋅,⋅) in E given by
∥
u
∥
2
:=
(
A
u
+
,
u
+
)
2
-
(
A
u
-
,
u
-
)
2
+
∥
u
0
∥
2
2
and
(
u
,
v
)
=
{
∫
ℝ
N
(
∇
u
(
x
)
∇
v
(
x
)
+
V
(
x
)
u
(
x
)
v
(
x
)
)
𝑑
x
=
(
A
u
,
v
)
2
if
u
,
v
∈
E
+
,
-
∫
ℝ
N
(
∇
u
(
x
)
∇
v
(
x
)
+
V
(
x
)
u
(
x
)
v
(
x
)
)
𝑑
x
=
-
(
A
u
,
v
)
2
if
u
,
v
∈
E
-
,
(
u
,
v
)
2
if
u
,
v
∈
E
0
,
0
if
u
∈
E
j
,
v
∈
E
k
,
j
≠
k
,
for j,k∈{+,-,0}. Henceforth, the Hilbert space used in this application is E=(H1(ℝN),∥⋅∥) and in addition it is possible to write
I
(
u
)
=
1
2
(
∥
u
+
∥
2
-
∥
u
-
∥
2
)
-
∫
ℝ
N
h
(
x
)
F
(
u
(
x
)
)
𝑑
x
for all u=u++u-+u0∈E. Note that the orthogonality among E+, E-, E0 ensures that u+, u-, u0 are also orthogonal in L2(ℝℕ). As usual, ∥u∥22=(u,u)2 denotes the norm and inner product in L2(ℝℕ). Then (uj,uk)2=0, ≠k and j,k∈{+,-,0}.
Denote by {ℰ(λ)} the spectral family of the operator A. In view of the spectral theory (see [4, Supplement S1.1], see also [21, Chapter 3]) it is possible to define E2:=ℰ(0)E=E-⊕E0 and E1:=(I-ℰ(0))E. Furthermore, ℰ(0)=ℰ(λ) for all 0<λ<σ+ by the definition of σ+ in (V2). Then E1=(I-ℰ(λ))E for all 0<λ<σ+. Hence, by [4, Theorem 1.1’, p. 394] it follows that σ+∥u1∥22≤∥u1∥2 for all u1∈E1. Therefore,
inf
u
1
∈
E
1
,
u
1
≠
0
∥
u
1
∥
2
∥
u
1
∥
2
2
≥
σ
+
,
and thus, by setting
(4.2)
a
0
:=
inf
u
1
∈
E
1
,
u
1
≠
0
∥
u
1
∥
2
∥
h
1
2
u
1
∥
2
2
≥
1
h
∞
inf
u
1
∈
E
1
,
u
1
≠
0
∥
u
1
∥
2
∥
u
1
∥
2
2
≥
σ
+
h
∞
>
0
,
it follows that
∥
u
1
∥
2
≥
a
0
∫
ℝ
N
h
(
x
)
u
1
2
(
x
)
𝑑
x
for all u1∈E1.
Under all the previous assumptions and notations, it is possible to state the first main result of this section.
Theorem 4.3.
Assume V satisfying (V1)–(V2), h satisfying (h1) and f satisfying (f1)–(f3), with a>a0. Then problem (P) has a nontrivial weak solution u∈H1(RN).
In order to prove Theorem 4.3 it is necessary to check that I satisfies hypotheses (I1)–(I4) in Theorem 2.4, and then it is possible to ensure the result by applying this theorem. First of all, see that I∈C1(E,ℝ) due to the hypotheses assumed about h and f. Moreover, on one hand,
(
L
u
,
u
)
=
(
L
1
u
1
+
L
2
u
2
,
u
1
+
u
2
)
=
(
L
1
u
1
,
u
1
)
+
(
L
2
u
2
,
u
2
)
,
and on the other hand, denoting by I1:E1→E1 the identity operator in E1, and by P-:E2→E2 the projector operator of E2 on E-, we note that u2∈E2 is such that u2=u-+u0. Hence,
∥
u
+
∥
2
-
∥
u
-
∥
2
=
∥
u
1
∥
2
-
∥
u
2
-
u
0
∥
2
=
(
I
1
(
u
1
)
,
u
1
)
+
(
-
P
-
(
u
2
)
,
u
2
)
.
Thus, by setting L1:=I1 and L2:=-P-, it follows that Li:Ei→Ei are bounded, linear and self-adjoint operators for i=1,2. Therefore, I(u)=12(Lu,u)+B(u), where
(4.3)
B
(
u
)
=
-
∫
ℝ
N
h
(
x
)
F
(
u
(
x
)
)
𝑑
x
,
and this gives (I1).
In order to prove (I2) the following lemma is needed.
Lemma 4.4.
Assume that (h1) and (f1)–(f2) hold for I. Then B given in (4.3) is weakly continuous and uniformly differentiable on bounded subsets.
Proof.
Let un⇀u be a sequence in E. Then un(x)→u(x) almost everywhere in ℝN, and F(un)(x)→F(u)(x) almost everywhere in ℝN since F(s) is a continuous function. Moreover, (4.11) yields
(
|
F
(
u
n
)
|
2
*
p
)
∈
L
1
(
ℝ
N
)
since 2<22*p<2* and (un)⊂E↪Ls(ℝN) for 2≤s≤2*. Thus
(
F
(
u
n
(
⋅
)
)
)
⊂
L
2
*
p
(
ℝ
N
)
is bounded, provided that (un) is bounded in E, and hence it is bounded in L22*p(ℝN) and in L2*(ℝN). Since F(un)(x)→F(u)(x) almost everywhere in ℝN and
∥
F
(
u
n
)
∥
L
2
*
p
(
ℝ
N
)
≤
C
for all
n
∈
ℕ
,
by Brezis–Lieb’s Lemma in [5], F(un)⇀F(u) in L2*p(ℝN). Provided that (h1) implies that h∈Lq(ℝN), where q is the conjugate exponent of 2*p, then
∫
ℝ
N
h
(
x
)
F
(
u
n
(
x
)
)
𝑑
x
→
∫
ℝ
N
h
(
x
)
F
(
u
(
x
)
)
𝑑
x
as n→+∞. Therefore, B is weakly continuous.
Showing that B is uniformly differentiable on bounded subsets of E means that given ε>0 and BR⊂E, there exists δ>0 such that
|
B
(
u
+
v
)
-
B
(
u
)
-
B
′
(
u
)
v
|
<
ε
∥
v
∥
for all u+v∈BR with ∥v∥<δ. First, note that B satisfies
(4.4)
|
B
(
u
+
v
)
-
B
(
u
)
-
B
′
(
u
)
v
|
≤
h
∞
1
2
∫
ℝ
N
(
h
(
x
)
)
1
2
|
f
(
z
(
x
)
)
-
f
(
u
(
x
)
)
|
|
v
(
x
)
|
𝑑
x
since for ψ(t):=F(u+tv) it yields ψ′(t)=f(u+tv)v. Hence, the Mean Value Theorem implies there exists some function θ(x) such that 0<θ(x)<1 almost everywhere in ℝN and, by writing z=u+θv, it follows that
F
(
u
+
v
)
-
F
(
u
)
=
ψ
(
1
)
-
ψ
(
0
)
=
ψ
′
(
θ
)
=
f
(
z
)
v
,
almost everywhere in ℝN. Moreover, provided that h∈L∞(ℝN), from (4.4) we have
(4.5)
|
B
(
u
+
v
)
-
B
(
u
)
-
B
′
(
u
)
v
|
≤
h
∞
1
2
∥
ξ
∥
L
2
(
ℝ
N
)
∥
v
∥
L
2
(
ℝ
N
)
≤
h
∞
1
2
C
2
∥
ξ
∥
L
2
(
ℝ
N
)
∥
v
∥
,
where C2>0 is the constant given by the continuous embedding E↪L2(ℝN) and ξ:=h12(⋅)|f(z(⋅))-f(u(⋅))| belongs to L2(ℝN). Indeed, since 2*p is the conjugate exponent of q, by applying Hölder’s inequality for q and 2*p, it follows that
∫
ℝ
N
|
ξ
|
2
𝑑
x
=
∫
ℝ
N
h
(
x
)
|
f
(
z
(
x
)
)
-
f
(
u
(
x
)
)
|
2
𝑑
x
≤
∥
h
∥
L
q
(
ℝ
N
)
∥
f
(
z
(
x
)
)
-
f
(
u
(
x
)
)
∥
L
2
2
*
p
(
ℝ
N
)
2
<
+
∞
since
2
<
2
2
*
p
<
2
*
and
|
f
(
u
)
|
2
2
*
p
≤
κ
2
2
*
p
|
u
|
2
2
*
p
∈
L
1
(
ℝ
N
)
due to assumption (h1). Observe that by (4.5) it is sufficient to show that, given ε>0 and BR⊂E, there exists δ>0 such that
∥
ξ
∥
L
2
(
ℝ
N
)
≤
ε
h
∞
1
/
2
C
2
for all u+v∈BR with ∥v∥<δ.
In order to prove this indirectly, suppose there exist ε0>0 and BR0⊂E fixed such that for all δ>0 it is possible to obtain uδ+vδ∈BR0 with
∥
v
δ
∥
<
δ
and
∥
ξ
δ
∥
L
2
(
ℝ
N
)
>
ε
0
h
∞
1
/
2
C
2
,
where
ξ
δ
=
h
1
2
(
⋅
)
|
f
(
z
δ
(
⋅
)
)
-
f
(
u
δ
(
⋅
)
)
|
and
z
δ
=
u
δ
+
θ
v
δ
.
Choosing δn=1n, for each n∈ℕ there exist un+vn∈BR0 such that ∥vn∥≤1n and
∥
ξ
n
∥
L
2
(
ℝ
N
)
>
ε
0
h
∞
1
/
2
C
2
.
Hence, vn→0 in E and un⇀u in E, up to subsequences as n→+∞. In addition, zn⇀u in E and zn(x),un(x)→u(x) almost everywhere in ℝN, up to subsequences. Thus,
|
f
(
z
n
(
x
)
)
-
f
(
u
n
(
x
)
)
|
2
→
0
almost everywhere in ℝN as n→+∞. Moreover, (zn) and (un) are bounded in E and the Sobolev embedding L22*p(ℝN)↪E holds. Then
∥
|
f
(
z
n
)
-
f
(
u
n
)
|
2
∥
L
2
*
p
(
ℝ
N
)
2
*
p
≤
C
(
∥
f
(
z
n
)
∥
L
2
2
*
p
(
ℝ
N
)
2
2
*
p
+
∥
f
(
u
n
)
∥
L
2
2
*
p
(
ℝ
N
)
2
2
*
p
)
≤
C
κ
2
2
*
p
(
∥
z
n
∥
L
2
2
*
p
(
ℝ
N
)
2
2
*
p
+
∥
u
n
∥
L
2
2
*
p
(
ℝ
N
)
2
2
*
p
)
≤
2
C
(
C
2
2
*
p
κ
R
0
)
2
2
*
p
,
where C22*p>0 is the constant given by the continuous embedding E↪L22*p(ℝN). Therefore, the sequence (|f(zn)-f(un)|2) is bounded in L2*p(ℝN). Applying Brezis–Lieb’s Lemma again, it yields
|
f
(
z
n
)
-
f
(
u
n
)
|
2
⇀
0
in L2*p(ℝN) as n→+∞. Since h∈Lq(ℝN), which is the dual space of L2*p(ℝN), by weak convergence it yields
∥
ξ
n
∥
L
2
(
ℝ
N
)
2
=
∫
ℝ
N
h
(
x
)
|
f
(
z
n
(
x
)
)
-
f
(
u
n
(
x
)
)
|
2
𝑑
x
→
0
as n→+∞, which contradicts
∥
ξ
n
∥
L
2
(
ℝ
N
)
>
ε
0
h
∞
1
/
2
C
2
and completes the proof. ∎
In order to prove (I3), choose Q={re:r∈[0,r1]}⊕(E2∩Br2) and S=∂Bρ∩E1, where 0<ρ<r1<r2 are constants and e∈E1, ∥e∥=1, must be a suitable vector. Hence, observe that if a as in (f2) is such that a>a0, then for ε>0 small enough and aε:=a-ε it follows that a>aε>a0, and by the definition of a0 in (4.2), there exists some e0∈E1 such that
a
0
∫
ℝ
N
h
(
x
)
e
0
2
(
x
)
𝑑
x
≤
∥
e
0
∥
2
≤
a
ε
∫
ℝ
N
h
(
x
)
e
0
2
(
x
)
𝑑
x
.
By normalizing e0 it follows that e=e0∥e0∥∈E1 is such that
(4.6)
1
=
∥
e
∥
2
=
∫
ℝ
N
(
|
∇
e
(
x
)
|
2
+
V
(
x
)
e
2
(
x
)
)
𝑑
x
≤
a
ε
∫
ℝ
N
h
(
x
)
e
2
(
x
)
𝑑
x
.
Therefore, choose such e for the structure of Q. Furthermore, by Lemma 2.2, S and ∂Q “link”, where ∂Q can be written as ∂Q=Q1∩Q2∩Q3, with Q1={0}⊕(E2∩Br2), Q2={re:r∈[0,r1]}⊕(E2∩∂Br2) and Q3={r1e}⊕(E2∩Br2). The following lemma shows that I satisfies (I3) (i)–(ii) in Theorem 2.4 for some α>0, ω=0 and arbitrary v∈E2.
Lemma 4.5.
Assume that (V1)–(V2), (h1) and (f1)–(f2) hold for I. For Q and S as above, and for sufficiently large r1>0, the inequalities I|S≥α>0 and I|∂Q≤0 hold for some α>0.
Proof.
By definition, S⊂E1, and hence for all u1∈S it yields
(4.7)
I
(
u
1
)
≥
1
2
ρ
2
-
h
∞
∫
ℝ
N
(
ε
2
|
u
1
(
x
)
|
2
+
C
ε
p
|
u
1
(
x
)
|
p
)
𝑑
x
=
ρ
2
(
1
2
(
1
-
ε
h
∞
C
2
2
)
-
C
ε
p
h
∞
C
p
p
ρ
p
-
2
)
.
Thus, if ε,ρ are sufficiently small, from (4.7) we obtain I(u1)≥α>0.
Now, for the purpose of checking that I|∂Q≤0<α, consider the three cases as follows.
Case I: u∈Q1⊂E2. Thus
I
(
u
)
=
-
1
2
∥
u
∥
2
-
∫
ℝ
N
h
(
x
)
F
(
u
(
x
)
)
𝑑
x
≤
0
since h(x)F(u(x))≥0 for all x∈ℝN.
Case II: u∈Q2. Thus u=u1+u2, where u1=re, with 0≤∥u1∥=r≤r1 and ∥u2∥=r2>r1, and therefore
I
(
u
)
=
1
2
(
∥
u
1
∥
2
-
r
2
2
)
-
∫
ℝ
N
h
(
x
)
F
(
u
(
x
)
)
𝑑
x
≤
1
2
(
r
1
2
-
r
2
2
)
<
0
.
Case III: u∈Q3. Thus u=r1e+u2, where 0≤∥u2∥≤r2. If r1≤∥u2∥≤r2, then
I
(
u
)
=
1
2
(
r
1
2
-
∥
u
2
∥
2
)
-
∫
ℝ
N
h
(
x
)
F
(
u
(
x
)
)
𝑑
x
≤
1
2
(
r
1
2
-
r
1
2
)
≤
0
.
If 0≤∥u2∥<r1, put u2=r1v2, where v2∈B1∩E2. Thus,
(4.8)
I
(
u
)
=
1
2
r
1
2
(
1
-
∥
v
2
∥
2
)
-
∫
ℝ
N
h
(
x
)
F
(
u
(
x
)
)
𝑑
x
≤
1
2
r
1
2
(
1
-
∫
ℝ
N
2
h
(
x
)
F
(
r
1
(
e
(
x
)
+
v
2
(
x
)
)
)
r
1
2
𝑑
x
)
.
Claim.
The limit
lim
r
1
→
+
∞
∫
ℝ
N
2
h
(
x
)
F
(
r
1
(
e
(
x
)
+
v
2
(
x
)
)
)
r
1
2
𝑑
x
=
a
∫
ℝ
N
h
(
x
)
[
e
(
x
)
+
v
2
(
x
)
]
2
𝑑
x
,
is uniform for v2∈B1∩E2.
Assume this claim postponing its proof in order to conclude Case III. From uniform convergence in B1∩E2, for each ε>0 there exists r0>0 such that, for all r1≥r0,
∫
ℝ
N
[
a
-
2
F
(
r
1
(
e
(
x
)
+
v
2
(
x
)
)
)
r
1
2
(
e
(
x
)
+
v
2
(
x
)
)
2
]
h
(
x
)
(
e
(
x
)
+
v
2
(
x
)
)
2
𝑑
x
<
ε
∫
ℝ
N
h
(
x
)
(
e
(
x
)
+
v
2
(
x
)
)
2
𝑑
x
for all v2∈B1∩E2. Thus,
1
2
r
1
2
(
1
-
∫
ℝ
N
2
h
(
x
)
F
(
r
1
(
e
(
x
)
+
v
2
(
x
)
)
)
r
1
2
𝑑
x
)
<
1
2
r
1
2
(
1
-
a
ε
∫
ℝ
N
h
(
x
)
e
2
(
x
)
𝑑
x
)
,
where aε=(a-ε). Substituting in (4.8), we obtain
(4.9)
I
(
u
)
<
1
2
r
1
2
(
1
-
a
ε
∫
ℝ
N
h
(
x
)
e
2
(
x
)
𝑑
x
)
.
Therefore, from (4.6) and (4.9) it follows that I(u)<0, and the result holds.
Now, we prove the claim. In order to do so, define for all n∈ℕ the functional Jn:B1∩E2→ℝ given by
J
n
(
v
2
)
:=
∫
ℝ
N
[
a
-
2
F
(
n
(
e
(
x
)
+
v
2
(
x
)
)
)
n
2
(
e
(
x
)
+
v
2
(
x
)
)
2
]
h
(
x
)
(
e
(
x
)
+
v
2
(
x
)
)
2
𝑑
x
.
The continuity of F implies that Jn is continuous for all n∈ℕ. From (f2) and by the equivalence of the H1(ℝN) and E norms,
0
≤
J
n
(
v
2
)
≤
a
h
∞
(
∥
e
∥
2
2
+
∥
v
2
∥
2
2
)
≤
2
C
2
2
a
h
∞
for all v2∈B1∩E2. Then, by seeing that E2 is finite-dimensional, B1∩E2 is compact, and since Jn is continuous in B1∩E2, it attains a maximum value, denoted by un∈B1∩E2. By considering this sequence of maximums (un), and provided that ∥un∥≤1 for all n∈ℕ, the sequence is bounded. Again, provided that E2 is finite-dimensional, such a sequence converges, up to subsequences, in B1∩E2, namely, un→u in the E-norm. Moreover, for all v2∈B1∩E2 and all n∈ℕ, the inequality 0≤Jn(v2)≤Jn(un) holds, that is,
0
≤
∫
ℝ
N
[
a
-
2
F
(
n
(
e
(
x
)
+
v
2
(
x
)
)
)
n
2
(
e
(
x
)
+
v
2
(
x
)
)
2
]
h
(
x
)
(
e
(
x
)
+
v
2
(
x
)
)
2
𝑑
x
(4.10)
≤
∫
ℝ
N
[
a
-
2
F
(
n
(
e
(
x
)
+
u
n
(
x
)
)
)
n
2
(
e
(
x
)
+
u
n
(
x
)
)
2
]
h
(
x
)
(
e
(
x
)
+
u
n
(
x
)
)
2
𝑑
x
.
Now, note that un(x)→u(x) almost everywhere in ℝN. Then by (f2),
[
a
-
2
F
(
n
(
e
(
x
)
+
u
n
(
x
)
)
)
n
2
(
e
(
x
)
+
u
n
(
x
)
)
2
]
h
(
x
)
(
e
(
x
)
+
u
n
(
x
)
)
2
→
0
almost everywhere in ℝN as n→+∞. More than this, since un→u in E, we have un→u in L2(ℝN), and so by the Lebesgue Dominated Convergence Theorem it follows
lim
n
→
+
∞
∫
ℝ
N
[
a
-
2
F
(
n
(
e
(
x
)
+
u
n
(
x
)
)
)
n
2
(
e
(
x
)
+
u
n
(
x
)
)
2
]
h
(
x
)
(
e
(
x
)
+
u
n
(
x
)
)
2
𝑑
x
=
0
.
Now, applying in (4.10) the limit as n→+∞, the Sandwich Theorem yields
lim
n
→
+
∞
∫
ℝ
N
[
a
-
2
F
(
n
(
e
(
x
)
+
v
2
(
x
)
)
)
n
2
(
e
(
x
)
+
v
2
(
x
)
)
2
]
h
(
x
)
(
e
(
x
)
+
v
2
(
x
)
)
2
𝑑
x
=
0
uniformly for all v2∈B1∩E2, and the claim is proved. ∎
For now, observe that by (f1) and (f2), given ε>0 and 2<p<2*, there exists a constant Cε>0 such that
(4.11)
|
F
(
s
)
|
≤
ε
2
|
s
|
2
+
C
ε
p
|
s
|
p
and
|
f
(
s
)
|
≤
ε
|
s
|
+
C
ε
|
s
|
p
-
1
for all s∈ℝ.
In order to verify (I4) it is necessary to ensure the boundedness of Cerami sequences for I. The next lemma gives this result.
Lemma 4.6.
Assume that (V1)-(V2),(h1) and (f1)–(f3) hold for I and let (un)⊂E be a Cerami sequence for I on an arbitrary level c∈R. Then (un) is bounded.
Proof.
Suppose by contradiction that ∥un∥→+∞ as n→+∞, up to subsequences. By defining vn:=un∥un∥, it follows that (vn) is a bounded sequence in E. Then vn⇀v as n→+∞, up to subsequences. Let us show that neither v=0 nor v≠0 can occur.
First, suppose that v≠0, which means that there exists Ω⊂ℝN such that |Ω|>0 and v(x)≠0 for all x∈Ω. Since vn(x)→v(x) almost everywhere in Ω, one concludes that |un(x)|→+∞ almost everywhere for x∈Ω. Hence, in view of (f3) one arrives at h(x)Q(un(x))→+∞ as n→+∞ almost everywhere in Ω. Applying Fatou’s lemma, one obtains
(4.12)
lim inf
n
→
+
∞
∫
Ω
h
(
x
)
Q
(
u
n
(
x
)
)
𝑑
x
≥
∫
Ω
lim inf
n
→
+
∞
h
(
x
)
Q
(
u
n
(
x
)
)
d
x
=
+
∞
.
Provided that (un) is a Cerami sequence on level c, it follows that
(4.13)
c
+
o
n
(
1
)
=
I
(
u
n
)
-
1
2
I
′
(
u
n
)
u
n
=
∫
ℝ
N
h
(
x
)
Q
(
u
n
(
x
)
)
𝑑
x
≥
∫
Ω
h
(
x
)
Q
(
u
n
(
x
)
)
𝑑
x
.
Combining (4.12) and (4.13) yields a contradiction. Therefore, one must have v=0.
Setting vn=v+,n+v-,n+v0,n, where vi,n∈Ej, j=+,-,0, up to subsequences it yields
(4.14)
{
v
n
⇀
v
=
v
+
+
v
-
+
v
0
in
E
=
E
+
+
E
-
+
E
0
,
v
n
→
v
in
L
loc
2
(
ℝ
N
)
.
Since v=0, we have v+=v-=v0=0, and from (4.14) one has vj,n(x)→0 almost everywhere in ℝN. Moreover, v0,n→0 in E, provided that E0 is finite-dimensional. In addition, (un) is a Cerami sequence, and hence I′(un)u+,n→0 and I′(un)u-,n→0 as n→+∞. Therefore,
o
n
(
1
)
=
I
′
(
u
n
)
u
+
,
n
∥
u
n
∥
2
-
I
′
(
u
n
)
u
-
,
n
∥
u
n
∥
2
=
∥
v
+
,
n
∥
2
+
∥
v
-
,
n
∥
2
-
∫
ℝ
N
h
(
x
)
[
f
(
u
n
(
x
)
)
u
n
(
x
)
v
n
(
x
)
(
v
+
,
n
(
x
)
-
v
-
,
n
(
x
)
)
]
𝑑
x
=
1
-
∥
v
0
,
n
∥
2
-
∫
ℝ
N
h
(
x
)
[
f
(
u
n
(
x
)
)
u
n
(
x
)
(
v
+
,
n
2
(
x
)
-
v
-
,
n
2
(
x
)
)
]
𝑑
x
,
which implies that,
(4.15)
∫
ℝ
N
h
(
x
)
[
f
(
u
n
(
x
)
)
u
n
(
x
)
(
v
+
,
n
2
(
x
)
-
v
-
,
n
2
(
x
)
)
]
𝑑
x
→
1
-
∥
v
0
∥
2
=
1
as n→+∞. However, since |f(s)s|≤κ for all s∈ℝ∖{0} and provided that h∈Lq(ℝN), with q=2*2*-p and 2<22*p<2*, Hölder’s inequality ensures that
ϕ
n
(
x
)
:=
f
(
u
n
(
x
)
)
u
n
(
x
)
(
v
+
,
n
2
(
x
)
-
v
-
,
n
2
(
x
)
)
∈
L
2
*
p
(
ℝ
N
)
for all n and (ϕn) is a bounded sequence in L2*p(ℝN) since (vn) is bounded in E. Furthermore, in view of (4.14) one has ϕn(x)→0 almost everywhere in ℝN. Thus, it yields that (ϕn) converges weakly to 0 in L2*p(ℝN), up to subsequences.
Recalling that h∈Lq(ℝN), with q=2*2*-p, the weak convergence implies that
(4.16)
∫
ℝ
N
h
(
x
)
[
f
(
u
n
(
x
)
)
u
n
(
x
)
(
v
+
,
n
2
(
x
)
-
v
-
,
n
2
(
x
)
)
]
𝑑
x
→
0
as n→+∞. By looking at (4.15) and (4.16), one arrives at a contradiction. ∎
In view of the last result, to obtain (I4) we fix b>0 and take (un) such that
I
(
u
n
)
⊂
[
c
-
b
,
c
+
b
]
and
∥
I
′
(
u
n
)
∥
(
1
+
∥
u
n
∥
)
→
0
as n→+∞. Supposing that (un) is unbounded, we take (unk)⊂(un) such that ∥unk∥→+∞ as k→∞. By seeing that I(unk)⊂[c-b,c+b] is bounded in ℝ, it implies that I(unk)→d, up to subsequences. Then (unk) is a Cerami sequence on level d, up to subsequences. Hence (unk) is bounded, up to subsequences, by Lemma 4.6. However, it yields a contradiction since ∥unk∥→+∞ as k→+∞. Thus, (un) is bounded.
Now, after all theses results, one is finally ready to prove Theorem 4.3.
Proof of Theorem 4.3.
In view of all assumptions, I satisfies (I1)–(I4) in Theorem 2.4, so it is possible to apply this theorem for I. Theorem 2.4 provides a critical value c≥α>ω=0 of I. Therefore, there exists u∈E such that I(u)=c>0 and I′(u)=0, hence u≠0. Provided that I∈C1(E,ℝ), it follows that u is a nontrivial weak solution to (P) in H1(ℝN). ∎
Remark 4.7.
Note that problem (P) in (4.1) has just been solved with conditions on the potential V, which ensures a spectrum with negative and positive parts. Such conditions imply that the subspace E-, corresponding to the negative spectrum, was finite-dimensional (see Remark 4.1)). Although the fact of E2=E-⊕E0 being finite-dimensional was not necessary to apply Theorem 4.3, this information was used to obtain the linking geometry (see the proof of the claim in Lemma 4.5). However, with minor changes it is possible to prove the linking geometry indirectly, without the assumption that dimE2 is finite.
Since Theorem 2.4 does not require that any subspace in the linking decomposition needs to be finite-dimensional, the main goal now is to work with the same problem, but assuming conditions on V which give both subspaces in the linking decomposition to be infinite-dimensional. Henceforth, consider problem (P), but replacing condition (V1) on V by condition (V1′), and also assuming condition (V2). The assumptions on h and f are the same as before.
Remark 4.8.
In view of (V1′), we have that V is periodic and continuous, and hence bounded. In addition, by [22, Theorem XIII.100, p. 309] the spectral theory asserts that the operator A has a pure absolutely continuous spectrum which is bounded from below and consists of closed disjoint intervals. Namely, σ(A)=σess(A)=σac(A)=⋃[αi,βi]. In view of (V2), the operator A has nonempty negative and positive spectra, and 0 is in the gap between them. Indeed, if 0∈σ(A), it would be an isolated point of the spectrum, which contradicts the fact that the operator A has a pure continuous spectrum. Moreover, since σ(A)=σess(A), the negative and positive spectrum are both part of the essential spectrum. Thus, if {ℰ(λ)} is the spectral family of the operator A, by spectral theory and the definition of the essential spectrum (see [4, 22]), the subspaces associated with the negative and positive spectrum, namely ℰ(0)(H1(ℝN)) and (I-ℰ(0))(H1(ℝN)) are both infinite-dimensional.
By setting E1:=(I-ℰ(0))(H1(ℝN)) and E2:=ℰ(0)(H1(ℝN)), these subspaces are such that the operator A is positive definite in the former and negative definite in the latter. Indeed, by the spectral family definition (see [4, Theorem 1.1’, p. 394], see also [21, Chapter 3]), for all u1∈E1 and for all u2∈E2,
σ
+
∥
u
1
∥
2
2
≤
∫
ℝ
N
(
|
∇
u
1
(
x
)
|
2
+
V
(
x
)
u
1
2
(
x
)
)
𝑑
x
and
-
σ
-
∥
u
2
∥
2
2
≤
-
∫
ℝ
N
(
|
∇
u
2
(
x
)
|
2
+
V
(
x
)
u
2
2
(
x
)
)
𝑑
x
.
Because of this, it is possible to proceed similarly to before and consider the norm induced by the operator A:
∥
u
∥
2
=
{
∫
ℝ
N
(
|
∇
u
|
2
+
V
(
x
)
u
2
)
𝑑
x
=
(
A
u
,
u
)
2
if
u
∈
E
1
,
-
∫
ℝ
N
(
|
∇
u
|
2
+
V
(
x
)
u
2
)
𝑑
x
=
-
(
A
u
,
u
)
2
if
u
∈
E
2
,
which is equivalent to the usual norm in H1(ℝN). Set E=(H1(ℝN),∥⋅∥). Then E=E1⊕E2, namely every u∈E can be uniquely written as u=u1+u2, with ui∈Ei and
∥
u
∥
2
=
∥
u
1
∥
2
+
∥
u
2
∥
2
=
(
A
u
1
,
u
1
)
-
(
A
u
2
,
u
2
)
.
Here the conclusions from Remark 4.2 also hold, and again the functional I:E→ℝ associated with (P) is written as
I
(
u
)
=
1
2
(
∥
u
1
∥
2
-
∥
u
2
∥
2
)
-
∫
ℝ
N
h
(
x
)
F
(
u
(
x
)
)
𝑑
x
for all u=u1+u2∈E.
Now it is possible to state the second main result of this section.
Theorem 4.9.
Consider problem (P) with V satisfying (V1′)–(V2), h satisfying (h1) and f satisfying (f1)–(f3), with a>a0. Then (P) has a nontrivial weak solution u∈H1(RN).
For the purpose of applying Theorem 2.4 to solve this problem, proceeding as before, it is necessary to show that this problem satisfies all required assumptions of the abstract linking theorem.
Observe that, as before, I∈C1(E,ℝ) due to the hypotheses assumed about h and f. Moreover, on one hand,
(
L
u
,
u
)
=
(
L
1
u
1
+
L
2
u
2
,
u
1
+
u
2
)
=
(
L
1
u
1
,
u
1
)
+
(
L
2
u
2
,
u
2
)
,
and on the other hand, denoting by Ii:Ei→Ei the identity operator in Ei for i=1,2, we note that
∥
u
1
∥
2
-
∥
u
2
∥
2
=
(
u
1
,
u
1
)
-
(
u
2
,
u
2
)
=
(
I
1
(
u
1
)
,
u
1
)
+
(
-
I
2
(
u
2
)
,
u
2
)
.
Thus, by setting Li:=Ii for i=1,2, it follows that Li:Ei→Ei are bounded, linear and self-adjoint operators for i=1,2. Thus, I satisfies (I1) in Theorem 2.4 as before.
Furthermore, all assumptions are kept on h and f. Then (I2) and (I4) also hold here with the same proofs as before. Therefore, it is only necessary to show the linking geometry in (I3), which will have a different proof in this case, provided that E2 is infinite-dimensional. First of all, set S:=∂Bρ∩E1 and Q={re+u2:r≥0,u2∈E2,∥re+u2∥≤r1}, where 0<ρ<r1 are constants and e∈E1, ∥e∥=1, is chosen as before. In fact, by assuming a>a0 and by the definition of a0, there exists some unitary e∈E1 such that
(4.17)
a
0
∫
ℝ
N
h
(
x
)
e
2
(
x
)
𝑑
x
≤
∥
e
∥
2
=
1
<
a
∫
ℝ
N
h
(
x
)
e
2
(
x
)
𝑑
x
.
Such e makes the following lemma true. Moreover, it is possible to show that S and Q “link”, following the same lines as in Lemma 2.2. The next lemma shows the linking geometry (I3) (i)–(ii) of Theorem 2.4 for some α>0, ω=0 and arbitrary v∈E2.
Lemma 4.10.
Assume that (V1′)–(V2), (h1) and (f1)–(f2) hold for I. For Q and S as above, and for sufficiently large r1>0, it follows that I|S≥α>0 and I|∂Q≤0 for some α>0.
Proof.
The proof that I|S≥α>0 for some α>0 is the same as in Lemma 4.5, and thus it is not repeated here. In purpose of proving that I|∂Q≤0, observe that I(u2)≤0, for all u2∈E2 Then it suffices to show that I(re+u)≤0 for r>0,u∈E2 and ∥re+u∥≥r1 for some r1>0 large enough. By arguing indirectly, assume that for some sequence (rne+un)⊂ℝ+e⊕E2 such that ∥rne+un∥→+∞, the inequality I(rne+un)>0 holds for all n∈ℕ. Then, by seeking a contradiction, the desired result holds. Firstly, set
v
n
:=
r
n
e
+
u
n
∥
r
n
e
+
u
n
∥
=
s
n
e
+
w
n
,
where sn∈ℝ+, wn∈E2 and ∥vn∥=1. Provided that (vn) is bounded, up to subsequences it follows that vn⇀v=se+w in E. Then vn(x)→v(x) almost everywhere in ℝN. By seeing that
1
=
∥
s
n
e
+
w
n
∥
2
=
s
n
2
+
∥
w
n
∥
2
,
it ensures that 0≤sn2≤1, wn⇀w in E and sn→s in ℝ+. Then it yields
(4.18)
I
(
r
n
e
+
u
n
)
∥
r
n
e
+
u
n
∥
2
=
s
n
2
-
1
2
-
∫
ℝ
N
h
(
x
)
F
(
r
n
e
(
x
)
+
u
n
(
x
)
)
∥
r
n
e
+
u
n
∥
2
𝑑
x
>
0
,
and hence 0<s≤1. Moreover, from (4.17) there exists a bounded domain Ω0⊂ℝN such that
1
<
a
∫
Ω
0
h
(
x
)
e
2
(
x
)
𝑑
x
.
Hence, supp(e)∩Ω0≠∅, and it follows that
(4.19)
0
>
s
2
-
s
2
a
∫
Ω
0
h
(
x
)
e
2
(
x
)
𝑑
x
≥
s
2
(
2
-
a
∫
Ω
0
h
(
x
)
e
2
(
x
)
𝑑
x
)
-
1
-
a
∫
Ω
0
h
(
x
)
w
2
(
x
)
𝑑
x
.
On the other hand, since vn⇀v in E, it converges strongly in L2(Ω0), and since ∥rne+un∥→+∞ as n→+∞, in view of (f2) it follows that
h
(
x
)
F
(
r
n
e
(
x
)
+
u
n
(
x
)
)
∥
r
n
e
+
u
n
∥
2
=
h
(
x
)
F
(
v
n
(
x
)
∥
r
n
e
(
x
)
+
u
n
(
x
)
∥
)
v
n
2
(
x
)
v
n
(
x
)
2
∥
r
n
e
+
u
n
∥
2
→
h
(
x
)
a
2
v
2
(
x
)
almost everywhere in Ω0∩supp(v), which is not empty since v=se+w, (e,w)2=0 and supp(e)∩Ω0≠∅. Thus, by the Lebesgue Dominated Convergence Theorem,
∫
Ω
0
h
(
x
)
F
(
r
n
e
(
x
)
+
u
n
(
x
)
)
∥
r
n
e
+
u
n
∥
2
𝑑
x
→
a
2
∫
Ω
0
h
(
x
)
(
s
2
e
2
(
x
)
+
w
2
(
x
)
)
𝑑
x
as n→+∞. From (4.18),
(4.20)
0
<
2
s
n
2
-
1
-
2
∫
Ω
0
h
(
x
)
F
(
r
n
e
(
x
)
+
u
n
(
x
)
)
∥
r
n
e
+
u
n
∥
2
𝑑
x
.
By passing to the limit in (4.20) as n→+∞, it yields
0
≤
2
s
2
-
1
-
a
∫
Ω
0
h
(
x
)
(
s
2
e
2
(
x
)
+
w
2
(
x
)
)
𝑑
x
=
s
2
(
2
-
a
∫
Ω
0
h
(
x
)
e
2
(
x
)
𝑑
x
)
-
1
-
a
∫
Ω
0
h
(
x
)
w
2
(
x
)
𝑑
x
,
which is contrary to (4.19). Therefore, the lemma is proved. ∎
By Lemma 4.10, the functional I satisfies (I3) of Theorem 2.4. Now, Theorem 4.9 can be proved.
Proof of Theorem 4.9.
Due to all hypotheses on I, it satisfies (I1)–(I4) in Theorem 2.4. Then it is possible to apply this theorem for I. By Theorem 2.4, c≥α>0 is a critical value of I. Therefore, there exists u∈E, such that I(u)=c>0 and I′(u)=0, provided that I(u)>0, and thus u≠0. Since I∈C1(E,ℝ), we obtain that u is a nontrivial weak solution to (P) in H1(ℝN). ∎
and
