Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 317139, 11 pages
http://dx.doi.org/10.1155/2013/317139
Research Article
Ground State Solutions for the Periodic Discrete Nonlinear
Schrödinger Equations with Superlinear Nonlinearities
Ali Mai1,2,3 and Zhan Zhou1,2
1
School of Mathematics and Information Science, Guangzhou University, Guangdong, Guangzhou 510006, China
Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University,
Guangdong, Guangzhou 510006, China
3
Department of Applied Mathematics, Yuncheng University, Shanxi, Yuncheng 044000, China
2
Correspondence should be addressed to Zhan Zhou; zzhou0321@hotmail.com
Received 31 December 2012; Revised 22 March 2013; Accepted 24 March 2013
Academic Editor: Yuming Chen
Copyright © 2013 A. Mai and Z. Zhou. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the periodic discrete nonlinear Schrödinger equations with the temporal frequency belonging to a spectral gap. By
using the generalized Nehari manifold approach developed by Szulkin and Weth, we prove the existence of ground state solutions
of the equations. We obtain infinitely many geometrically distinct solutions of the equations when specially the nonlinearity is odd.
The classical Ambrosetti-Rabinowitz superlinear condition is improved.
1. Introduction
The following discrete nonlinear Schrödinger equation
(DLNS):
𝑖𝜓̇𝑛 = −Δ𝜓𝑛 + 𝜀𝑛 𝜓𝑛 − 𝜎𝜒𝑛 𝑓𝑛 (𝜓𝑛 ) ,
𝑛 ∈ Z,
(1)
We are interested in the existence of solitons of (1), that
is, solutions which are spatially localized time-periodic and
decay to zero at infinity. Thus, 𝜓𝑛 has the form
𝜓𝑛 = 𝑢𝑛 𝑒−𝑖𝜔𝑡 ,
(4)
lim 𝜓𝑛 = 0,
|𝑛| → ∞
where 𝜎 = ±1 and
Δ𝜓𝑛 = 𝜓𝑛+1 + 𝜓𝑛−1 − 2𝜓𝑛
(2)
is the discrete Laplacian operator, appears in many physical
problems, like polarons, energy transfer in biological materials, nonlinear optics, and so forth (see [1]). The parameter 𝜎
characterizes the focusing properties of the equation: if 𝜎 = 1,
the equation is self-focusing, while 𝜎 = −1 corresponds to the
defocusing equation. The given sequences {𝜀𝑛 } and {𝜒𝑛 } are
assumed to be 𝑇-periodic in 𝑛, that is, 𝜀𝑛+𝑇 = 𝜀𝑛 and 𝜒𝑛+𝑇 =
𝜒𝑛 . Moreover, {𝜒𝑛 } is a positive sequence. Here, 𝑇 is a positive
integer. We assume that 𝑓𝑛 (0) = 0 and the nonlinearity 𝑓𝑛 (𝑢)
is gauge invariant, that is,
𝑓𝑛 (𝑒𝑖𝜃 𝑢) = 𝑒𝑖𝜃 𝑓𝑛 (𝑢) ,
𝜃 ∈ R.
(3)
where {𝑢𝑛 } is a real-valued sequence and 𝜔 ∈ R is the
temporal frequency. Then, (1) becomes
−Δ𝑢𝑛 + 𝜀𝑛 𝑢𝑛 − 𝜔𝑢𝑛 = 𝜎𝜒𝑛 𝑓𝑛 (𝑢𝑛 ) ,
𝑛 ∈ Z,
lim 𝑢𝑛 = 0
(5)
(6)
|𝑛| → ∞
holds. Naturally, if we look for solitons of (1), we just need to
get the solutions of (5) satisfying (6).
Actually, we consider a more general equation:
𝐿𝑢𝑛 − 𝜔𝑢𝑛 = 𝜎𝜒𝑛 𝑓𝑛 (𝑢𝑛 ) ,
𝑛 ∈ Z,
(7)
with the same boundary condition (6). Here, 𝐿 is a secondorder difference operator
𝐿𝑢𝑛 = 𝑎𝑛 𝑢𝑛+1 + 𝑎𝑛−1 𝑢𝑛−1 + 𝑏𝑛 𝑢𝑛 ,
(8)
2
Abstract and Applied Analysis
where {𝑎𝑛 } and {𝑏𝑛 } are real-valued 𝑇-periodic sequences.
When 𝑎𝑛 ≡ −1 and 𝑏𝑛 = 2 + 𝜀𝑛 , we obtain (5).
We consider (7) as a nonlinear equation in the space 𝑙2
of two-sided infinite sequences. Note that every element of 𝑙2
automatically satisfies (6).
As it is well known, the operator 𝐿 is a bounded and selfadjoint operator in 𝑙2 . The spectrum 𝜎(𝐿) is a union of a finite
number of closed intervals, and the complement R \ 𝜎(𝐿)
consists of a finite number of open intervals called spectral
gaps. Two of them are semi-infinite (see [2]). If 𝑇 = 1, then
finite gaps do not exist. However, in general, finite gaps exist,
and the most interesting case in (7) is when the frequency 𝜔
belongs to a finite spectral gap. Let us fix any spectral gap and
denote it by (𝛼, 𝛽).
DNLS equation is one of the most important inherently
discrete models. DNLS equation plays a crucial role in the
modeling of a great variety of phenomena, ranging from solid
state and condensed matter physics to biology (see [1, 3–6]
and references therein). In the past decade, solitons of the
periodic DNLS have become a hot topic. The existence of
solitons for the periodic DNLS equations with superlinear
nonlinearity [7–10] and with saturable nonlinearity [11–13]
has been studied, respectively. If 𝜔 is below or above the
spectrum of the difference operator −Δ + 𝜀𝑛 , solitons were
shown by using the Nehari manifold approach and a discrete
version of the concentration compactness principle in [14].
If 𝜔 is a lower edge of a finite spectral gap, the existence
of solitons was obtained by using variant generalized weak
linking theorem in [10]. If 𝜔 lies in a finite spectral gap, the
existence of solitons was proved by using periodic approximations in combination with the linking theorem in [8] and
the generalized Nehari manifold approach in [9], respectively.
The results were extended by Chen and Ma in [7]. In this
paper, we employ the generalized Nehari manifold approach
instead of periodic approximation technique to obtain the
existence of a kind of special solitons of (7), which called
ground state solutions, that is, nontrivial solutions with least
possible energy in 𝑙2 . We should emphasize that the results
are obtained under more general super nonlinearity than
the classical Ambrosetti-Rabinowitz superlinear condition
[8, 9, 15].
This paper is organized as follows. In Section 2, we first
establish the variational framework associated with (7) and
transfer the problem on the existence of solutions in 𝑙2
of (7) into that on the existence of critical points of the
corresponding functional. We then present the main results
of this paper and compare them with existing ones. Section 3
is devoted to the proofs of the main results.
2. Preliminaries and Main Results
(𝑉1 ) 𝜔 ∈ (𝛼, 𝛽),
(𝑓1 ) 𝑓𝑛 ∈ 𝐶(R, R) and 𝑓𝑛+𝑇 (𝑢) = 𝑓𝑛 (𝑢), and there exist
𝑎 > 0 and 𝑝 ∈ (2, ∞) such that
∀𝑛 ∈ Z, 𝑢 ∈ R,
(𝑓3 ) lim|𝑢| → ∞ 𝐹𝑛 (𝑢)/𝑢2 = ∞, where 𝐹𝑛 (𝑢) is the primitive
function of 𝑓𝑛 (𝑢), that is,
𝑢
𝐹𝑛 (𝑢) = ∫ 𝑓𝑛 (𝑠) 𝑑𝑠,
(10)
0
(𝑓4 ) 𝑢 󳨃→ 𝑓𝑛 (𝑢)/|𝑢| is strictly increasing on (−∞, 0) and
(0, ∞).
To state our results, we introduce some notations. Let
𝐸 = 𝑙2 (Z) .
𝐴 = 𝐿 − 𝜔,
(11)
Consider the functional 𝐽 defined on 𝐸 by
1
𝐽 (𝑢) = (𝐴𝑢, 𝑢)𝐸 − 𝜎 ∑ 𝜒𝑛 𝐹𝑛 (𝑢𝑛 ) ,
2
𝑛∈Z
(12)
where (⋅, ⋅)𝐸 is the inner product in 𝐸 and ‖ ⋅ ‖𝐸 is the
corresponding norm in 𝐸. The hypotheses on 𝑓𝑛 (𝑢) imply that
the functional 𝐽 ∈ 𝐶1 (𝐸, R) and (7) is easily recognized as the
corresponding Euler-Lagrange equation for 𝐽. Thus, to find
nontrivial solutions of (7), we need only to look for nonzero
critical points of 𝐽 in 𝐸.
For the derivative of 𝐽, we have the following formula:
(𝐽󸀠 (𝑢) , V) = (𝐴𝑢, V)𝐸 − 𝜎 ∑ 𝜒𝑛 𝑓𝑛 (𝑢𝑛 ) V𝑛 ,
∀V ∈ 𝐸.
𝑛∈Z
(13)
By (𝑉1 ), we have 𝜎(𝐴) ⊂ R\(𝛼−𝜔, 𝛽−𝜔). So, 𝐸 = 𝐸+ ⊕𝐸−
corresponds to the spectral decomposition of 𝐴 with respect
to the positive and negative parts of the spectrum, and
(𝐴𝑢, 𝑢)𝐸 ≥ (𝛽 − 𝜔) ‖𝑢‖2𝐸 ,
𝑢 ∈ 𝐸+ ,
(𝐴𝑢, 𝑢)𝐸 ≤ (𝛼 − 𝜔) ‖𝑢‖2𝐸 ,
𝑢 ∈ 𝐸− .
(14)
For any 𝑢, V ∈ 𝐸, letting 𝑢 = 𝑢+ + 𝑢− with 𝑢± ∈ 𝐸± and
V = V+ + V− with V± ∈ 𝐸± , we can define an equivalent inner
product (⋅, ⋅) and the corresponding norm ‖ ⋅ ‖ on 𝐸 by
(𝑢, V) = (𝐴𝑢+ , V+ )𝐸 − (𝐴𝑢− , V− )𝐸 ,
‖𝑢‖ = (𝑢, 𝑢)1/2 , (15)
respectively. So, 𝐽 can be rewritten as
1 󵄩 󵄩2
𝐽 (𝑢) = 󵄩󵄩󵄩𝑢+ 󵄩󵄩󵄩 −
2
1 󵄩󵄩 − 󵄩󵄩2
󵄩𝑢 󵄩 − 𝜎 ∑ 𝜒𝑛 𝐹𝑛 (𝑢𝑛 ) .
2󵄩 󵄩
𝑛∈Z
(16)
We define for 𝑢 ∈ 𝐸 \ 𝐸− , the subspace
𝐸 (𝑢) := R𝑢 + 𝐸− = R𝑢+ ⊕ 𝐸− ,
(17)
and the convex subset
The following are the basic hypotheses to establish the main
results of this paper:
󵄨
󵄨󵄨
𝑝−1
󵄨󵄨𝑓𝑛 (𝑢)󵄨󵄨󵄨 ≤ 𝑎 (1 + |𝑢| )
(𝑓2 ) 𝑓𝑛 (𝑢) = 𝑜(|𝑢|) as 𝑢 → 0,
(9)
𝐸̂ (𝑢) := R+ 𝑢 + 𝐸− = R+ 𝑢+ ⊕ 𝐸− ,
(18)
of 𝐸, where, as usual, R+ = [0, ∞). Let
M = {𝑢 ∈ 𝐸 \ 𝐸− : 𝐽󸀠 (𝑢) 𝑢 = 0, 𝐽󸀠 (𝑢) V = 0 ∀V ∈ 𝐸− } ,
(19)
𝑐 = inf 𝐽 (𝑢) .
𝑢∈M
(20)
Abstract and Applied Analysis
3
In this paper, we also consider the multiplicity of solutions
of (7).
For each ℓ ∈ Z, let
ℓ ∗ 𝑢 = (𝑢𝑛+ℓ𝑇 )𝑛∈Z ,
∀𝑢 = (𝑢𝑛 )𝑛∈Z ,
(21)
which defines a Z-action on 𝐸. By the periodicity of the
coefficients, we know that both 𝐽 and 𝐽󸀠 are Z-invariants.
Therefore, if 𝑢 ∈ 𝐸 is a critical point of 𝐽, so is ℓ∗𝑢. Two critical
points 𝑢1 , 𝑢2 ∈ 𝐸 of 𝐽 are said to be geometrically distinct if
𝑢1 ≠ ℓ ∗ 𝑢2 for all ℓ ∈ Z.
Now, we are ready to state the main results.
Theorem 1. Suppose that conditions (𝑉1 ), (𝑓1 )–(𝑓4 ) are satisfied. Then, one has the following conclusions.
(1) If either 𝜎 = 1 and 𝛽 ≠ ∞ or 𝜎 = −1 and 𝛼 ≠ − ∞, then
(7) has at least a nontrivial ground state solution.
(2) If either 𝜎 = 1 and 𝛽 = ∞ or 𝜎 = −1 and 𝛼 = −∞,
then (7) has no nontrivial solution.
Theorem 2. Suppose that conditions (𝑉1 ), (𝑓1 )–(𝑓4 ) are satisfied and 𝑓𝑛 is odd in 𝑢. If either 𝜎 = 1 and 𝛽 ≠ ∞ or 𝜎 = −1
and 𝛼 ≠ −∞, then (7) has infinitely many pairs of geometrically
distinct solutions.
In what follows, we always assume that 𝜎 = 1. The other
case can be reduced to 𝜎 = 1 by switching 𝐿 to −𝐿 and 𝜔 to
−𝜔.
Remark 3. In [8], the author considered (7) with 𝑓𝑛 defined
by
𝑓𝑛 (𝑢) = |𝑢|2 𝑢,
(22)
which obviously satisfies (𝑓1 )–(𝑓4 ); the author also discussed
the case where 𝑓 satisfies the Ambrosetti-Rabinowitz condition; that is, there exists 𝜇 > 2 such that
Remark 5. In [7], it is shown that (7) has at least a nontrivial
solution 𝑢 ∈ 𝑙2 if 𝑓 satisfies (𝑉1 ), (𝑓2 ), (𝑓3 ), and the following
conditions:
(𝐵1 ) 𝐹𝑛 (𝑢) ≥ 0 for any 𝑢 ∈ R and 𝐻𝑛 (𝑢) := (1/2)𝑓𝑛 (𝑢)𝑢 −
𝐹𝑛 (𝑢) > 0 if 𝑢 ≠ 0,
(𝐵2 ) 𝐻𝑛 (𝑢) → ∞ as |𝑢| → ∞, and there exist 𝑟0 > 0 and
𝛾 > 1 such that |𝑓𝑛 (𝑢)|𝛾 /|𝑢|𝛾 ≤ 𝑐0 𝐻𝑛 (𝑢) if |𝑢| ≥ 𝑟0 ,
where 𝑐0 is a positive constant,
In our paper, we use (9) and (𝑓4 ) instead of (𝐵1 ) and (𝐵2 ).
3. Proofs of Main Results
We assume that (𝑉1 ) and (𝑓1 )–(𝑓4 ) are satisfied from now on.
Lemma 6. 𝐹𝑛 (𝑢) > 0 and (1/2)𝑓𝑛 (𝑢)𝑢 > 𝐹𝑛 (𝑢) for all 𝑢 ≠ 0.
Proof. By (𝑓2 ) and (𝑓4 ), it is easy to get that
𝐹𝑛 (𝑢) > 0 ∀𝑢 ≠ 0.
(25)
Set 𝐻𝑛 (𝑢) = (1/2)𝑓𝑛 (𝑢)𝑢 − 𝐹𝑛 (𝑢). It follows from (𝑓4 ) that
𝐻𝑛 (𝑢) =
𝑢
𝑢
𝑓𝑛 (𝑢) − ∫ 𝑓𝑛 (𝑠) 𝑑𝑠
2
0
𝑓 (𝑢) 𝑢
𝑢
> 𝑓𝑛 (𝑢) − 𝑛
∫ 𝑠𝑑𝑠 = 0.
2
𝑢
0
(26)
So, (1/2)𝑓𝑛 (𝑢)𝑢 > 𝐹𝑛 (𝑢) for all 𝑢 ≠ 0.
(23)
To continue the discussion, we need the following proposition.
Clearly, (23) implies that 𝐹𝑛 (𝑢) ≥ 𝑐|𝑢| > 0 for |𝑢| ≥ 1. So, it
is a stronger condition than (𝑓3 ).
Proposition 7 (see [16, 17]). Let 𝑢, 𝑠, V ∈ R be numbers with
𝑠 ≥ −1 and 𝑤 := 𝑠𝑢 + V ≠ 0. Then,
Remark 4. In [9], the author assumed that 𝑓𝑛 satisfies the
following condition: there exists 𝜃 ∈ (0, 1) such that
𝑠
𝑓𝑛 (𝑢) [𝑠 ( + 1) 𝑢 + (1 + 𝑠) V] + 𝐹𝑛 (𝑢) − 𝐹𝑛 (𝑢 + 𝑤) < 0.
2
(27)
0 < 𝜇𝐹𝑛 (𝑢) ≤ 𝑓𝑛 (𝑢) 𝑢,
𝑢 ≠ 0.
𝜇
0 < 𝑢−1 𝑓𝑛 (𝑢) ≤ 𝜃𝑓𝑛󸀠 (𝑢) ,
𝑢 ≠ 0.
(24)
Obviously, (24) implies (23) with 𝜇 = 1 + (1/𝜃), so it is
a stronger condition than the Ambrosetti-Rabinowitz condition. In our paper, the nonlinearities satisfy more general
superlinear assumptions instead of (24) which also implies
(𝑓4 ). However, we do not assume that 𝑓𝑛 is differentiable and
satisfies (24), M is not a 𝐶1 manifold of 𝐸, and the minimizers
on M may not be critical points of 𝐽. Hence, the method of [9]
does not apply any more. Nevertheless, M is still a topological
manifold, naturally homeomorphic to the unit sphere in 𝐸+
(see in detail in Section 3). We use the generalized Nehari
manifold approach developed by Szulkin and Weth which is
based on reducing the strongly indefinite variational problem
to a definite one and prove that the minimizers of 𝐽 on M are
indeed critical points of 𝐽.
Lemma 8. If 𝑢 ∈ M, then
𝐽 (𝑢 + 𝑤) < 𝐽 (𝑢)
𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑤 ∈ 𝑈
:= {𝑠𝑢 + V : 𝑠 ≥ −1, V ∈ 𝐸− } ,
𝑤 ≠ 0.
(28)
Hence, 𝑢 is the unique global maximum of 𝐽|𝐸(𝑢)
̂ .
Proof. We rewrite 𝐽 by
1
1
𝐽 (𝑢) = (𝐴𝑢+ , 𝑢+ )𝐸 + (𝐴𝑢− , 𝑢− )𝐸 − 𝜎 ∑ 𝜒𝑛 𝐹𝑛 (𝑢𝑛 ) .
2
2
𝑛∈Z
(29)
4
Abstract and Applied Analysis
Since 𝑢 ∈ M, we have
0 = (𝐽󸀠 (𝑢) ,
(b) For 𝑢 ∈ M, by (25), we have
2𝑠 + 𝑠2
𝑢 + (1 + 𝑠) V)
2
1 󵄩 󵄩2
𝑐 ≤ 󵄩󵄩󵄩𝑢+ 󵄩󵄩󵄩 −
2
2𝑠 + 𝑠2
2𝑠 + 𝑠2
=
(𝐴𝑢+ , 𝑢+ )𝐸 +
(𝐴𝑢− , 𝑢− )𝐸
2
2
(30)
−
+ (1 + 𝑠) (𝐴𝑢 , V)𝐸
− ∑ 𝜒𝑛 𝑓𝑛 (𝑢𝑛 ) (
𝑛∈Z
2𝑠 + 𝑠2
𝑢𝑛 + (1 + 𝑠) V𝑛 ) .
2
1
{(𝐴 (1 + 𝑠) 𝑢+ , (1 + 𝑠) 𝑢+ )𝐸 − (𝐴𝑢+ , 𝑢+ )𝐸 }
2
1
{(𝐴 ((1 + 𝑠) 𝑢− + V) , (1 + 𝑠) 𝑢− + V)𝐸 − (𝐴𝑢− , 𝑢− )𝐸 }
2
+ ∑ 𝜒𝑛 𝐹𝑛 (𝑢𝑛 ) − ∑ 𝜒𝑛 𝐹𝑛 (𝑢𝑛 + 𝑤𝑛 )
𝑛∈Z
=
𝑛∈Z
2𝑠 + 𝑠2
1
2𝑠 + 𝑠2
(𝐴𝑢+ , 𝑢+ )𝐸 +
(𝐴𝑢− , 𝑢− )𝐸 + (𝐴V, V)𝐸
2
2
2
Proof. Suppose by contradiction that there exist 𝑢(𝑘) ∈ W
and 𝑤(𝑘) ∈ 𝐸(𝑢(𝑘) ), 𝑘 ∈ N, such that 𝐽(𝑤(𝑘) ) > 0 for all 𝑘
and ‖𝑤(𝑘) ‖ → ∞ as 𝑘 → ∞. Without loss of generality,
we may assume that ‖𝑢(𝑘) ‖ = 1 for 𝑘 ∈ Z. Then, there exists
a subsequence, still denoted by the same notation, such that
𝑢(𝑘) → 𝑢 ∈ 𝐸+ . Set V(𝑘) = 𝑤(𝑘) /‖𝑤(𝑘) ‖ = 𝑠(𝑘) 𝑢(𝑘) + V(𝑘)− .
Then,
𝐽 (𝑤(𝑘) ) 1
2
󵄩2
󵄩
= ((𝑠(𝑘) ) − 󵄩󵄩󵄩󵄩V(𝑘)− 󵄩󵄩󵄩󵄩 )
0< 󵄩
2
󵄩
󵄩󵄩𝑤(𝑘) 󵄩󵄩
2
󵄩
󵄩
−
+ (1 + 𝑠) (𝐴𝑢− , V)𝐸 + ∑ 𝜒𝑛 𝐹𝑛 (𝑢𝑛 ) − ∑ 𝜒𝑛 𝐹𝑛 (𝑢𝑛 + 𝑤𝑛 )
𝑛∈Z
𝑛∈Z
1
𝑠
= (𝐴V, V)𝐸 + ∑ 𝜒𝑛 {𝑓𝑛 (𝑢𝑛 ) [𝑠 ( + 1) 𝑢𝑛 + (1 + 𝑠) V𝑛 ]
2
2
𝑛∈Z
+ 𝐹𝑛 (𝑢𝑛 ) − 𝐹𝑛 (𝑢𝑛 + 𝑤𝑛 ) } < 0.
(31)
The proof is complete.
Lemma 9. (a) There exists 𝛼 > 0 such that 𝑐 := inf M 𝐽(𝑢) ≥
inf 𝑆𝛼 𝐽(𝑢) > 0, where 𝑆𝛼 := {𝑢 ∈ 𝐸+ : ‖𝑢‖ = 𝛼}.
(b) ‖𝑢+ ‖ ≥ max{‖𝑢− ‖, √2𝑐} for every 𝑢 ∈ M.
Proof. (a) By (𝑓1 ) and (𝑓2 ), it is easy to show that for any 𝜀 >
0, there exists 𝑐𝜀 > 0 such that
󵄨
󵄨󵄨
𝑝−1
󵄨󵄨𝑓𝑛 (𝑢)󵄨󵄨󵄨 ≤ 𝜀 |𝑢| + 𝑐𝜀 |𝑢| ,
(34)
Lemma 10. Let W ⊂ 𝐸+ \ {0} be a compact subset. Then, there
exists 𝑅 > 0 such that 𝐽 ≤ 0 on 𝐸(𝑢) \ 𝐵𝑅 (0) for every 𝑢 ∈ W,
where 𝐵𝑅 (0) denotes the open ball with radius 𝑅 and center 0.
𝐽 (𝑢 + 𝑤) − 𝐽 (𝑢)
+
1 󵄩 󵄩2 󵄩 󵄩2
≤ (󵄩󵄩󵄩𝑢+ 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑢− 󵄩󵄩󵄩 ) .
2
Hence, ‖𝑢+ ‖ ≥ max{‖𝑢− ‖, √2𝑐}.
Together with Proposition 7, we know that
=
1 󵄩󵄩 − 󵄩󵄩2
󵄩𝑢 󵄩 − ∑ 𝜒 𝐹 (𝑢 )
2 󵄩 󵄩 𝑛∈Z 𝑛 𝑛 𝑛
󵄨
󵄨󵄨
2
𝑝
󵄨󵄨𝐹𝑛 (𝑢)󵄨󵄨󵄨 ≤ 𝜀|𝑢| + 𝑐𝜀 |𝑢| .
(32)
+
𝑞
‖⋅‖ is equivalent to the 𝐸 norm on 𝐸 and 𝐸 ⊂ 𝑙 for 2 ≤ 𝑞 ≤ ∞
with ‖𝑢‖𝑙𝑞 ≤ ‖𝑢‖𝐸 . Hence, for any 𝜀 ∈ (0, 1/2) and 𝑢 ∈ 𝐸+ , we
have
1
𝐽 (𝑢) ≥ ‖𝑢‖2 − 𝜀‖𝑢‖2 − 𝑐𝜀 𝜒‖𝑢‖𝑝 ,
2
(33)
which implies inf 𝑆𝛼 𝐽(𝑢) > 0 for some 𝛼 > 0 (small enough),
where 𝜒 = max{𝜒𝑛 }.
The first inequality is a consequence of Lemma 8 since for
̂ ∩ 𝑆𝛼 .
every 𝑢 ∈ M, there is 𝑠 > 0 such that 𝑠𝑢+ ∈ 𝐸(𝑢)
𝐹𝑛 (𝑤𝑛(𝑘) )
(35)
2
(V𝑛(𝑘) ) .
∑ 𝜒𝑛
(𝑘) 2
𝑛∈Z
(𝑤𝑛 )
By (25), we have
2
󵄩󵄩 (𝑘)− 󵄩󵄩2
󵄩2
󵄩
󵄩󵄩V 󵄩󵄩 ≤ (𝑠(𝑘) ) = 1 − 󵄩󵄩󵄩V(𝑘)− 󵄩󵄩󵄩 .
󵄩
󵄩
󵄩
󵄩
(36)
Consequently, we know that ‖V(𝑘)− ‖ ≤ 1/√2 and 1/√2 ≤
𝑠(𝑘) ≤ 1. Passing to a subsequence if necessary, we assume
that 𝑠(𝑘) → 𝑠 ∈ [1/√2, 1], V(𝑘) ⇀ V, V(𝑘)− ⇀ V∗− ∈ 𝐸− , and
V𝑛(𝑘) → V𝑛 for every 𝑛. Hence, V = 𝑠𝑢 + V∗− ≠ 0 and V∗− = V− . It
| = ‖𝑤(𝑘) ‖ ⋅ |V𝑛(𝑘)
| →
follows that for 𝑛0 ∈ Z with V𝑛0 ≠ 0, |𝑤𝑛(𝑘)
0
0
∞, as 𝑘 → ∞. Then, by (𝑓3 ), we have
∑ 𝜒𝑛
𝑛∈Z
𝐹𝑛 (𝑤𝑛(𝑘) )
2
(𝑤𝑛(𝑘) )
2
(V𝑛(𝑘) ) 󳨀→ ∞,
(37)
which contradicts with (35).
̂ consists
Lemma 11. For each 𝑢 ∈ 𝐸+ \ {0}, the set M ∩ 𝐸(𝑢)
of precisely one point which is the unique global maximum of
𝐽|𝐸(𝑢)
̂ .
̂ ≠ 0.
Proof. By Lemma 8, it suffices to show that M ∩ 𝐸(𝑢)
+
+
̂
̂
Since 𝐸(𝑢) = 𝐸(𝑢 /‖𝑢 ‖), we may assume that 𝑢 ∈ 𝑆+ . By
Lemma 10, there exists 𝑅 > 0 such that 𝐽 ≤ 0 on 𝐸(𝑢) \ 𝐵𝑅 (0)
provided that 𝑅 is large enough. By Lemma 9 (a), 𝐽(𝑡𝑢) > 0
̂
for small 𝑡 > 0. Moreover, 𝐽 ≤ 0 on 𝐸(𝑢)
\ 𝐵𝑅 (0). Hence,
0 < sup𝐸(𝑢)
̂ 𝐽 < ∞.
Abstract and Applied Analysis
5
̂
Let V(𝑘) ⇀ V in 𝐸(𝑢).
Then, V𝑛(𝑘) → V𝑛 as 𝑘 → ∞ for all 𝑛
after passing to a subsequence if necessary. Hence, 𝐹𝑛 (V𝑛(𝑘) ) →
𝐹𝑛 (V𝑛 ). Let 𝜑(V) = ∑𝑛∈Z 𝜒𝑛 𝐹𝑛 (V𝑛 ). Then,
𝜑 (V) = ∑ lim 𝜒𝑛 𝐹𝑛 (V𝑛(𝑘) )
𝑘→∞
𝑛∈Z
≤ lim inf ∑ 𝜒𝑛 𝐹𝑛 (V𝑛(𝑘) )
𝑘→∞
𝑛∈Z
(38)
= lim inf 𝜑 (V(𝑘) ) ;
𝑘→∞
that is, 𝜑 is a weakly lower semicontinuous. From the weak
lower semi-continuity of the norm, it is easy to see that 𝐽 is
̂
weakly upper semicontinuous on 𝐸(𝑢).
Therefore, 𝐽(𝑢0 ) =
̂
sup𝐸(𝑢)
̂ 𝐽 for some 𝑢0 ∈ 𝐸(𝑢) \ {0}. By the proof of Lemma 10,
󸀠
𝑢0 is a critical point of 𝐽|𝐸(𝑢)
̂ . It follows that (𝐽 (𝑢0 ), 𝑢0 ) =
(𝐽󸀠 (𝑢0 ), 𝑧) = 0 for all 𝑧 ∈ 𝐸 and hence 𝑢0 ∈ M. To summarize,
̂
𝑢0 ∈ M ∩ 𝐸(𝑢).
According to Lemma 11, for each 𝑢 ∈ 𝐸+ \ {0}, we may
̂
̂ : 𝐸+ \ {0} → M, 𝑢 󳨃→ 𝑚(𝑢),
where
define the mapping 𝑚
̂
̂ is the unique point of M ∩ 𝐸(𝑢).
𝑚(𝑢)
Lemma 12. 𝐽 is coercive on M; that is, 𝐽(𝑢) → ∞ as ‖𝑢‖ →
∞, 𝑢 ∈ M.
Proof. Suppose, by contradiction, that there exists a sequence
{𝑢(𝑘) } ⊂ M such that ‖𝑢(𝑘) ‖ → ∞ and 𝐽(𝑢(𝑘) ) ≤ 𝑑 for
some 𝑑 ∈ [𝑐, ∞). Let V(𝑘) = 𝑢(𝑘) /‖𝑢(𝑘) ‖. Then, there exists
a subsequence, still denoted by the same notation, such that
V(𝑘) ⇀ V and V𝑛(𝑘) → V𝑛 for every 𝑛 as 𝑘 → ∞.
First, we know that there exist 𝛿 > 0 and 𝑛𝑘 ∈ Z such that
󵄨󵄨 (𝑘)+ 󵄨󵄨
󵄨󵄨V𝑛 󵄨󵄨 ≥ 𝛿.
󵄨 𝑘 󵄨
(39)
Indeed, if not, then V(𝑘)+ → 0 in 𝑙∞ as 𝑘 → ∞. By
2
Lemma 9(b), 1/2 ≤ ‖V(𝑘)+ ‖ ≤ 1, which means that ‖V(𝑘)+ ‖𝑙2
is bounded. For 𝑞 > 2,
󵄩󵄩 (𝑘)+ 󵄩󵄩𝑞 󵄩󵄩 (𝑘)+ 󵄩󵄩𝑞−2 󵄩󵄩 (𝑘)+ 󵄩󵄩2
󵄩󵄩V 󵄩󵄩 𝑞 ≤ 󵄩󵄩V 󵄩󵄩 ∞ 󵄩󵄩V 󵄩󵄩 2 .
󵄩𝑙 󵄩
󵄩𝑙 󵄩
󵄩𝑙
󵄩
Then, V
(𝑘)+
(40)
𝐽 (𝑢(𝑘) ) 1 󵄩
󵄩2 󵄩
󵄩2
0 ≤ 󵄩 󵄩2 = (󵄩󵄩󵄩󵄩V(𝑘)+ 󵄩󵄩󵄩󵄩 − 󵄩󵄩󵄩󵄩V(𝑘)− 󵄩󵄩󵄩󵄩 )
󵄩󵄩𝑢(𝑘) 󵄩󵄩
2
󵄩 󵄩
− ∑ 𝜒𝑛
𝐹𝑛 (𝑢𝑛(𝑘) )
(43)
2
(𝑢𝑛(𝑘) )
𝑛∈Z
𝑉
× (V𝑛(𝑘) ) 󳨀→ −∞ as 𝑘 󳨀→ ∞,
a contradiction again. The proof is finished.
̂ : 𝐸+ \{0} → M is continuous.
Lemma 13. (a) The mapping 𝑚
̂ 𝑆+ : 𝑆+ → M is a
(b) The mapping 𝑚 = 𝑚|
+
homeomorphism between 𝑆 and M, and the inverse of 𝑚 is
given by 𝑚−1 (𝑢) = 𝑢+ /‖𝑢+ ‖, where 𝑆+ := {𝑢 ∈ 𝐸+ : ‖𝑢‖ = 1}.
(c) The mapping 𝑚−1 : M 󳨃→ 𝑆+ is the Lipschitz
continuous.
Proof. (a) Let (𝑢(𝑘) ) ⊂ 𝐸+ \ {0} be a sequence with 𝑢(𝑘) →
̂
̂ + /‖𝑤+ ‖), without loss of generality,
𝑢. Since 𝑚(𝑤)
= 𝑚(𝑤
̂ (𝑘) ) =
we may assume that ‖𝑢(𝑘) ‖ = 1 for all 𝑘. Then, 𝑚(𝑢
(𝑘) + (𝑘)
(𝑘) −
̂
̂
‖𝑚(𝑢
) ‖𝑢 + 𝑚(𝑢
) . By Lemma 10, there exists 𝑅 > 0
such that
̂ (𝑢(𝑘) )) = sup 𝐽 ≤ sup 𝐽
𝐽 (𝑚
𝐵𝑅 (0)
𝐸(𝑢(𝑘) )
󵄩 󵄩2
≤ sup 󵄩󵄩󵄩𝑢+ 󵄩󵄩󵄩 = 𝑅2
𝑢∈𝐵𝑅 (0)
(44)
for every 𝑘.
𝑞
→ 0 in all 𝑙 , 𝑞 > 2. By (32), for any 𝑠 ∈ R,
∑ 𝜒𝑛 𝐹𝑛 (𝑠V𝑛(𝑘)+ )
𝑛∈Z
Due to the periodicity of coefficients, both 𝐽 and M are
invariant under 𝑇-translation. Making such shifts, we can
assume that 1 ≤ 𝑛𝑘 ≤ 𝑇 − 1 in (39). Moreover, passing
to a subsequence if needed, we can assume that 𝑛𝑘 = 𝑛0 is
independent of 𝑘. Next, we may extract a subsequence, still
denoted by {V(𝑘) }, such that V𝑛(𝑘)+ → V𝑛+ for all 𝑛 ∈ Z. In
particular, for 𝑛 = 𝑛0 , inequality (39) shows that |V𝑛+0 | ≥ 𝛿 and
hence V+ ≠ 0.
Since |𝑢𝑛(𝑘) | → ∞ as 𝑘 → ∞, it follows again from (𝑓3 )
and Fatou’s lemma that
󵄩2
󵄩𝑞
󵄩
󵄩
≤ 𝜀𝑠 𝜒󵄩󵄩󵄩󵄩V(𝑘)+ 󵄩󵄩󵄩󵄩𝑙2 + 𝑐𝜀 𝑠𝑝 𝜒󵄩󵄩󵄩󵄩V(𝑘)+ 󵄩󵄩󵄩󵄩𝑙𝑝 ,
2
(41)
which implies that ∑𝑛∈Z 𝜒𝑛 𝐹𝑛 (𝑠V𝑛(𝑘)+ ) → 0 as 𝑘 → ∞.
̂ (𝑘) ) for 𝑠 ≥ 0, Lemma 8 implies that
Since 𝑠V(𝑘)+ ∈ 𝐸(𝑢
𝑑 ≥ 𝐽 (𝑢(𝑘) ) ≥ 𝐽 (𝑠V(𝑘)+ )
=
𝑠2 󵄩󵄩 (𝑘)+ 󵄩󵄩2
󵄩󵄩V 󵄩󵄩 − ∑ 𝜒𝑛 𝐹𝑛 (𝑠V𝑛(𝑘)+ )
󵄩
2󵄩
𝑛∈Z
≥
𝑠2
𝑠2
− ∑ 𝜒𝑛 𝐹𝑛 (𝑠V𝑛(𝑘)+ ) 󳨀→ ,
4 𝑛∈Z
4
as 𝑘 → ∞. This is a contradiction if 𝑠 > √4𝑑.
(42)
̂ (𝑘) ) is bounded. Passing to
It follows from Lemma 12 that 𝑚(𝑢
a subsequence if needed, we may assume that
󵄩󵄩
+󵄩
̂ (𝑘) ) 󵄩󵄩󵄩󵄩 󳨀→ 𝑡,
𝑡(𝑘) := 󵄩󵄩󵄩𝑚(𝑢
󵄩
󵄩
−
̂ (𝑘) ) ⇀ 𝑢∗−
𝑚(𝑢
(45)
in 𝐸 as 𝑘 → ∞,
where 𝑡 ≥ √2𝑐 > 0 by Lemma 9(b). Moreover, by Lemma 11,
̂ (𝑢(𝑘) )) ≥ 𝐽 (𝑡(𝑘) 𝑢(𝑘) + 𝑚(𝑢)
̂ − ) 󳨀→ 𝐽 (𝑡𝑢 + 𝑚(𝑢)
̂ −)
𝐽 (𝑚
̂ (𝑢)) .
= 𝐽 (𝑚
(46)
6
Abstract and Applied Analysis
Therefore, using the weak lower semicontinuity of the norm
and 𝜑 (defined in Lemma 11), we get
̂ (𝑤) = 𝐽 (𝑢𝑡 ) − 𝐽 (𝑢)
̂ (𝑤𝑡 ) − Ψ
Ψ
̂ (𝑢(𝑘) ))
̂ (𝑢)) ≤ lim 𝐽 (𝑚
𝐽 (𝑚
𝑘󳨀→∞
2
1
= lim ( (𝑡(𝑘) ) −
𝑘→∞ 2
= 𝐽 (𝑠𝑡 𝑤𝑡 + 𝑢𝑡− ) − 𝐽 (𝑠0 𝑤 + 𝑢− )
− 󵄩2
1 󵄩󵄩󵄩
̂ (𝑘) ) 󵄩󵄩󵄩󵄩
󵄩󵄩𝑚(𝑢
󵄩
󵄩
2
̂ (𝑢𝑛(𝑘) )))
− ∑ 𝜒𝑛 𝐹𝑛 (𝑚
≤ 𝐽 (𝑠𝑡 𝑤𝑡 + 𝑢𝑡− ) − 𝐽 (𝑠𝑡 𝑤 + 𝑢𝑡− )
with some 𝜂𝑡 ∈ (0, 1). Similarly,
1 󵄩󵄩 − 󵄩󵄩2
−
󵄩𝑢 󵄩 − ∑ 𝜒 𝐹 (𝑡𝑢𝑛 + 𝑢∗,𝑛 )
2 󵄩 ∗ 󵄩 𝑛∈Z 𝑛 𝑛
̂ (𝑤𝑡 ) − Ψ
̂ (𝑤) = 𝐽 (𝑠𝑡 𝑤𝑡 + 𝑢− ) − 𝐽 (𝑠0 𝑤 + 𝑢− )
Ψ
𝑡
≥ 𝐽 (𝑠0 𝑤𝑡 + 𝑢− ) − 𝐽 (𝑠0 𝑤 + 𝑢− )
̂ (𝑢)) ,
= 𝐽 (𝑡𝑢 + 𝑢∗− ) ≤ 𝐽 (𝑚
which implies that all inequalities above must be equalities
̂ − and hence
̂ (𝑘) )− → 𝑢∗− . By Lemma 11, 𝑢∗− = 𝑚(𝑢)
and 𝑚(𝑢
(𝑘)
̂
̂
) → 𝑚(𝑢).
𝑚(𝑢
(b) This is an immediate consequence of (a).
(c) For 𝑢, V ∈ M, by (b), we have
V+ 󵄩󵄩
󵄩 󵄩󵄩 𝑢+
󵄩󵄩 −1
󵄩󵄩𝑚 (𝑢) − 𝑚−1 (V)󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩 + − + 󵄩󵄩󵄩󵄩
󵄩 󵄩󵄩 ‖𝑢 ‖ ‖V ‖ 󵄩󵄩
󵄩
󵄩 +󵄩 󵄩 +󵄩 + 󵄩
󵄩󵄩 +
󵄩 𝑢 − V+ (󵄩󵄩󵄩V 󵄩󵄩󵄩 − 󵄩󵄩󵄩𝑢 󵄩󵄩󵄩) V 󵄩󵄩󵄩
󵄩󵄩
= 󵄩󵄩󵄩󵄩
+
‖𝑢+ ‖ ‖V+ ‖ 󵄩󵄩󵄩
󵄩󵄩 ‖𝑢+ ‖
≤
(48)
2 󵄩󵄩
2
+󵄩
󵄩󵄩(𝑢 − V) 󵄩󵄩󵄩 ≤ √ ‖𝑢 − V‖ .
+
𝑐
‖𝑢 ‖
̂ : 𝐸+ \ {0} → R and
We will consider the functional Ψ
+
Ψ : 𝑆 → R defined by
̂ := 𝐽 (𝑚
̂ (𝑤)) ,
Ψ
̂ 𝑆+ .
Ψ := Ψ|
with some 𝜏𝑡 ∈ (0, 1). Combining these inequalities and the
continuity of function 𝑡 󳨃→ 𝑠𝑡 , we have
󵄩󵄩󵄩𝑚(𝑤)
̂ (𝑤𝑡 ) − Ψ
̂ (𝑤)
̂ + 󵄩󵄩󵄩󵄩 󸀠
Ψ
̂ (𝑤)) 𝑧.
𝐽 (𝑚
= 𝑠0 𝐽󸀠 (𝑢) 𝑧 = 󵄩
𝑡󳨀→0
𝑡
‖𝑤‖
(54)
lim
̂ is bounded linear in 𝑧 and
Hence, the Gâteaux derivative of Ψ
̂ is of class 𝐶1 (see [15]).
continuous in 𝑤. It follows that Ψ
̂
(b) It follows from (a) by noting that 𝑚(𝑤) = 𝑚(𝑤)
since
𝑤 ∈ 𝑆+ .
(c) Let {𝑤𝑛 } be a Palais-Smale sequence for Ψ, and let 𝑢𝑛 =
𝑚(𝑤𝑛 ) ∈ M. Since for every 𝑛 ∈ Z, we have an orthogonal
splitting 𝐸 = 𝑇𝑤𝑛 𝑆+ ⊕ 𝐸(𝑤𝑛 ); using (b), we have
󵄩󵄩 󸀠
󵄩
󵄩󵄩Ψ (𝑤𝑛 )󵄩󵄩󵄩 = sup Ψ󸀠 (𝑤𝑛 ) 𝑧
󵄩
󵄩
+
𝑧∈𝑇𝑤𝑛 𝑆
‖𝑧‖=1
󵄩
+󵄩
= 󵄩󵄩󵄩󵄩𝑚(𝑤𝑛 ) 󵄩󵄩󵄩󵄩 sup 𝐽󸀠 (𝑚 (𝑤𝑛 )) 𝑧
(49)
𝑧∈𝑇𝑤𝑛 𝑆+
‖𝑧‖=1
󵄩 󵄩
= 󵄩󵄩󵄩𝑢𝑛+ 󵄩󵄩󵄩
󵄩󵄩 ̂ + 󵄩󵄩
𝑚(𝑤) 󵄩󵄩 󸀠
̂󸀠 (𝑤) 𝑧 = 󵄩󵄩
̂ (𝑤)) 𝑧 ∀𝑤, 𝑧 ∈ 𝐸+ , 𝑤 ≠ 0. (50)
𝐽 (𝑚
Ψ
‖𝑤‖
(b) Ψ ∈ 𝐶1 (𝑆+ , R) and
= {V ∈ 𝐸+ : (𝑤, V) = 0} .
(53)
= 𝐽󸀠 (𝑠0 [𝑤 + 𝜏𝑡 (𝑤𝑡 − 𝑤)] + 𝑢− ) 𝑠0 𝑡𝑧,
̂ ∈ 𝐶1 (𝐸+ \ {0}, R) and
Lemma 14. (a) Ψ
󵄩 ̂ + 󵄩󵄩 󸀠
+
Ψ󸀠 (𝑤) 𝑧 = 󵄩󵄩󵄩𝑚(𝑤)
󵄩󵄩 𝐽 (𝑚 (𝑤)) 𝑧 ∀𝑧 ∈ 𝑇𝑤 𝑆
(52)
= 𝐽󸀠 (𝑠𝑡 [𝑤 + 𝜂𝑡 (𝑤𝑡 − 𝑤)] + 𝑢𝑡− ) 𝑠𝑡 𝑡𝑧
(47)
𝑛∈Z
1
≤ 𝑡2 −
2
the function 𝑡 󳨃→ 𝑠𝑡 is continuous. Then, 𝑠0 = ‖𝑢+ ‖/‖𝑤‖. By
Lemma 11 and the mean value theorem, we have
(55)
sup 𝐽󸀠 (𝑢𝑛 ) 𝑧,
𝑧∈𝑇𝑤𝑛 𝑆+
‖𝑧‖=1
because 𝐽󸀠 (𝑢𝑛 )V = 0 for all V ∈ 𝐸(𝑤𝑛 ) and 𝐸(𝑤𝑛 ) is orthogonal
to 𝑇𝑤𝑛 𝑆+ . Using (b) again, we have
(51)
󵄩
󵄩󵄩 󸀠
󵄩
󵄩
󵄩󵄩Ψ (𝑤𝑛 )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑢𝑛+ 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝐽󸀠 (𝑢𝑛 )󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩 󵄩
= 󵄩󵄩󵄩𝑢𝑛+ 󵄩󵄩󵄩
(c) {𝑤𝑛 } is a Palais-Smale sequence for Ψ if and only if
{𝑚(𝑤𝑛 )} is a Palais-Smale sequence for 𝐽.
(d) 𝑤 ∈ 𝑆+ is a critical point of Ψ if and only if 𝑚(𝑤) ∈ M
is a nontrivial critical point of 𝐽. Moreover, the corresponding
values of Ψ and 𝐽 coincide and inf 𝑆+ Ψ = inf M 𝐽 = 𝑐.
̂
Proof. (a) We put 𝑢 = 𝑚(𝑤)
∈ M, so we have 𝑢 =
(‖𝑢+ ‖/‖𝑤‖)𝑤 + 𝑢− . Let 𝑧 ∈ 𝐸+ . Choose 𝛿 > 0 such that 𝑤𝑡 :=
̂ 𝑡 ) ∈ M. We may
𝑤 + 𝑡𝑧 ∈ 𝐸+ \ {0} for |𝑡| < 𝛿 and put 𝑢𝑡 = 𝑚(𝑤
write 𝑢𝑡 = 𝑠𝑡 𝑤𝑡 + 𝑢𝑡− with 𝑠𝑡 > 0. From the proof of Lemma 13,
𝐽󸀠 (𝑢𝑛 ) (𝑧 + V)
‖𝑧 + V‖
𝑧∈𝑇𝑤𝑛 𝑆+ ,V∈𝐸(𝑤𝑛 )
sup
𝑧+V ≠ 0
󵄩 󵄩
≤ 󵄩󵄩󵄩𝑢𝑛+ 󵄩󵄩󵄩
(56)
𝐽󸀠 (𝑢𝑛 ) (𝑧) 󵄩󵄩 󸀠
󵄩
= 󵄩󵄩󵄩Ψ (𝑤𝑛 )󵄩󵄩󵄩󵄩 .
+
‖𝑧‖
𝑧∈𝑇𝑤𝑛 𝑆 \{0}
sup
Therefore,
󵄩󵄩 󸀠
󵄩
󵄩
󵄩
󵄩󵄩Ψ (𝑤𝑛 )󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝑢𝑛+ 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝐽󸀠 (𝑢𝑛 )󵄩󵄩󵄩 .
󵄩
󵄩
󵄩
󵄩
(57)
Abstract and Applied Analysis
7
According to Lemma 9(b) and Lemma 12, √2𝑐 ≤ ‖𝑢𝑛+ ‖ ≤
sup𝑛 ‖𝑢𝑛+ ‖ < ∞. Hence, {𝑤𝑛 } is a Palais-Smale sequence for
Ψ if and only if {𝑢𝑛 } is a Palais-Smale sequence for 𝐽.
(d) By (57), Ψ󸀠 (𝑤) = 0 if and only if 𝐽󸀠 (𝑚(𝑤)) = 0. The
other part is clear.
Proof of Theorem 1. (1) We know that 𝑐 > 0 by Lemma 9(a). If
𝑢0 ∈ M satisfies 𝐽(𝑢0 ) = 𝑐, then 𝑚−1 (𝑢0 ) ∈ 𝑆+ is a minimizer
of Ψ and therefore a critical point of Ψ and also a critical point
of 𝐽 by Lemma 14. We shall show that there exists a minimizer
𝑢 ∈ M of 𝐽|M . Let {𝑤(𝑘) } ⊂ 𝑆+ be a minimizing sequence
for Ψ. By Ekeland’s variational principle, we may assume that
Ψ(𝑤(𝑘) ) → 𝑐 and Ψ󸀠 (𝑤(𝑘) ) → 0 as 𝑘 → ∞. Then, 𝐽(𝑢(𝑘) ) →
𝑐 and 𝐽󸀠 (𝑢(𝑘) ) → 0 as 𝑘 → ∞ by Lemma 14(c), where 𝑢(𝑘) :=
𝑚(𝑤(𝑘) ) ∈ M. By Lemma 12, {𝑢(𝑘) } is bounded, and hence
{𝑢(𝑘) } has a weakly convergent subsequence.
First, we show that there exist 𝛿 > 0 and 𝑛𝑘 ∈ Z such that
󵄨󵄨 (𝑘) 󵄨󵄨
󵄨󵄨𝑢𝑛 󵄨󵄨 ≥ 𝛿.
󵄨 𝑘󵄨
(58)
Indeed, if not, then 𝑢(𝑘) → 0 in 𝑙∞ as 𝑘 → ∞. From the
simple fact that for 𝑞 > 2,
󵄩󵄩 (𝑘) 󵄩󵄩𝑞 󵄩󵄩 (𝑘) 󵄩󵄩𝑞−2 󵄩󵄩 (𝑘) 󵄩󵄩2
󵄩󵄩𝑢 󵄩󵄩 𝑞 ≤ 󵄩󵄩𝑢 󵄩󵄩 ∞ 󵄩󵄩𝑢 󵄩󵄩 2 ,
󵄩 󵄩𝑙 󵄩 󵄩𝑙 󵄩 󵄩𝑙
(59)
we have 𝑢(𝑘) → 0 in all 𝑙𝑞 , 𝑞 > 2. By (32), we know that
𝑛∈Z
(60)
(62)
that is, 𝑢 is a nontrivial critical point of 𝐽.
Finally, we show that 𝐽(𝑢) = 𝑐. By Lemma 6 and Fatou’s
lemma, we have
1
𝑐 = lim (𝐽 (𝑢(𝑘) ) − 𝐽󸀠 (𝑢(𝑘) ) 𝑢(𝑘) )
𝑘→∞
2
1
= lim ∑ 𝜒𝑛 ( 𝑓𝑛 (𝑢𝑛(𝑘) ) 𝑢𝑛(𝑘) − 𝐹𝑛 (𝑢𝑛(𝑘) ))
𝑘→∞
2
𝑛∈Z
1
≥ ∑ 𝜒𝑛 ( 𝑓𝑛 (𝑢𝑛 ) 𝑢𝑛 − 𝐹𝑛 (𝑢𝑛 ))
2
𝑛∈Z
(63)
1
= 𝐽 (𝑢) − 𝐽󸀠 (𝑢) 𝑢 = 𝐽 (𝑢) ≥ 𝑐.
2
Hence, 𝐽(𝑢) = 𝑐. That is, 𝑢 is a nontrivial ground state solution
of (7).
(2) If 𝛽 = ∞, by way of contradiction, we assume that (7)
has a nontrivial solution 𝑢 ∈ 𝐸. Then, 𝑢 is a nonzero critical
point of 𝐽 in 𝐸. Thus, 𝐽󸀠 (𝑢) = 0. But by Lemma 6,
(64)
Now, we are ready to prove Theorem 2. From now on, we
always assume that 𝑓𝑛 is odd in 𝑢. We need some notations.
For 𝑎 ≥ 𝑏 ≥ 𝑐, denote
𝐽𝑎 = {𝑢 ∈ M : 𝐽 (𝑢) ≤ 𝑎} ,
𝐽𝑏 := {𝑢 ∈ M : 𝐽 (𝑢) ≥ 𝑏} ,
𝐽𝑏𝑎 = 𝐽𝑎 ∩ 𝐽𝑏 ,
which implies that ∑𝑛∈Z 𝜒𝑛 𝑓𝑛 (𝑢𝑛(𝑘) )𝑢𝑛(𝑘)+ = 𝑜(‖ 𝑢(𝑘)+ ‖) as 𝑘 →
∞. Therefore,
𝑛∈Z
∀V ∈ 𝐸;
This is a contradiction, so the conclusion holds.
This completes the proof of Theorem 1.
󵄨 󵄨𝑝−1 󵄨
󵄨
+ 𝑐𝜀 𝜒 ∑ 󵄨󵄨󵄨󵄨𝑢𝑛(𝑘) 󵄨󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨󵄨𝑢𝑛(𝑘)+ 󵄨󵄨󵄨󵄨
󵄩
󵄩
𝑜 (󵄩󵄩󵄩󵄩𝑢(𝑘)+ 󵄩󵄩󵄩󵄩) = (𝐽󸀠 (𝑢(𝑘) ) , 𝑢(𝑘)+ )
󵄩2
󵄩
= 󵄩󵄩󵄩󵄩𝑢(𝑘)+ 󵄩󵄩󵄩󵄩 − ∑ 𝜒𝑛 𝑓𝑛 (𝑢𝑛(𝑘) ) 𝑢𝑛(𝑘)+
𝑘→∞
𝑛∈Z
𝑛∈Z
󵄩
󵄩 󵄩 󵄩
≤ 𝜀𝜒󵄩󵄩󵄩󵄩𝑢(𝑘) 󵄩󵄩󵄩󵄩𝑙2 ⋅ 󵄩󵄩󵄩󵄩𝑢(𝑘)+ 󵄩󵄩󵄩󵄩𝑙2
󵄩
󵄩 󵄩𝑝−1 󵄩
+ 𝑐𝜀 𝜒󵄩󵄩󵄩󵄩𝑢(𝑘) 󵄩󵄩󵄩󵄩𝑙𝑝 ⋅ 󵄩󵄩󵄩󵄩𝑢(𝑘)+ 󵄩󵄩󵄩󵄩𝑙𝑝
󵄩
󵄩 󵄩 󵄩
≤ 𝜀𝜒󵄩󵄩󵄩󵄩𝑢(𝑘) 󵄩󵄩󵄩󵄩𝑙2 ⋅ 󵄩󵄩󵄩󵄩𝑢(𝑘)+ 󵄩󵄩󵄩󵄩
󵄩
󵄩 󵄩𝑝−1 󵄩
+ 𝑐𝜀 𝜒󵄩󵄩󵄩󵄩𝑢(𝑘) 󵄩󵄩󵄩󵄩𝑙𝑝 ⋅ 󵄩󵄩󵄩󵄩𝑢(𝑘)+ 󵄩󵄩󵄩󵄩 ,
(𝐽󸀠 (𝑢) , V) = lim (𝐽󸀠 (𝑢(𝑘) ) , V) = 0,
(𝐽󸀠 (𝑢) , 𝑢) = ((𝐿 − 𝜔) 𝑢, 𝑢) − ∑ 𝜒𝑛 𝑓𝑛 (𝑢𝑛 ) 𝑢𝑛 < 0.
󵄨 󵄨 󵄨
󵄨
∑ 𝜒𝑛 𝑓𝑛 (𝑢𝑛(𝑘) ) 𝑢𝑛(𝑘)+ ≤ 𝜀𝜒 ∑ 󵄨󵄨󵄨󵄨𝑢𝑛(𝑘) 󵄨󵄨󵄨󵄨 ⋅ 󵄨󵄨󵄨󵄨𝑢𝑛(𝑘)+ 󵄨󵄨󵄨󵄨
𝑛∈Z
Next, we may extract a subsequence, still denoted by
{𝑢(𝑘) }, such that 𝑢(𝑘) ⇀ 𝑢 and 𝑢𝑛(𝑘) → 𝑢𝑛 for all 𝑛 ∈ Z.
Particularly, for 𝑛 = 𝑛0 , inequality (58) shows that |𝑢𝑛0 | ≥ 𝛿,
so 𝑢 ≠ 0. Moreover, we have
Ψ𝑎 = {𝑤 ∈ 𝑆+ : Ψ (𝑤) ≤ 𝑎} ,
Ψ𝑏 := {𝑤 ∈ 𝑆+ : Ψ (𝑤) ≥ 𝑏} ,
(65)
Ψ𝑏𝑎 = Ψ𝑎 ∩ Ψ𝑏 ,
(61)
󵄩2
󵄩
󵄩
󵄩
= 󵄩󵄩󵄩󵄩𝑢(𝑘)+ 󵄩󵄩󵄩󵄩 − 𝑜 (󵄩󵄩󵄩󵄩𝑢(𝑘)+ 󵄩󵄩󵄩󵄩) .
Then, ‖ 𝑢(𝑘)+ ‖2 → 0 as 𝑘 → ∞, contrary to Lemma 9(b).
From the periodicity of the coefficients, we know that 𝐽
and 𝐽󸀠 are both invariant under 𝑇-translation. Making such
shifts, we can assume that 1 ≤ 𝑛𝑘 ≤ 𝑇 − 1 in (58). Moreover,
passing to a subsequence, we can assume that 𝑛𝑘 = 𝑛0 is
independent of 𝑘.
𝐾 = {𝑤 ∈ 𝑆+ : Ψ󸀠 (𝑤) = 0} ,
𝐾𝑎 = {𝑤 ∈ 𝐾 : Ψ (𝑤) = 𝑎} ,
] (𝑎) = sup {‖𝑢‖ : 𝑢 ∈ 𝐽𝑎 } .
It is easy to see that ](𝑎) < ∞ for every 𝑎 by Lemma 12.
Proof of Theorem 2. It is easy to see that mappings 𝑚, 𝑚−1 are
equivariant with respect to the Z-action by Lemma 13; hence,
the orbits O(𝑢) ⊂ M consisting of critical points of 𝐽 are in
1-1 correspondence with the orbits O(𝑤) ⊂ 𝑆+ consisting of
8
Abstract and Applied Analysis
critical points of Ψ by Lemma 14(d). Next, we may choose a
subset F ⊂ 𝐾 such that F = −F and F consists of a unique
representative of Z-orbits. So, we only need to prove that the
set F is infinite. By contradiction, we assume that
F is a finite set.
(66)
Let
Γ𝑗 = {𝐴 ⊂ 𝑆+ : 𝐴 = −𝐴, 𝐴 is closed and 𝛾 (𝐴) ≥ 𝑗} , (67)
where 𝛾 denotes genus and 𝑗 ∈ N. We consider the sequence
of the Lusternik-Schnirelmann values of Ψ defined by
𝑑
𝑐𝑘 = inf {𝑑 ∈ R : 𝛾 (Ψ ) ≥ 𝑘, 𝑘 ∈ N} .
(68)
𝑐𝑘 < 𝑐𝑘+1 .
(69)
Firstly, we show that
𝜅 = inf {‖V − 𝑤‖ : V, 𝑤 ∈ 𝐾, V ≠ 𝑤} > 0.
(70)
In fact, there exist V(𝑘) , 𝑤(𝑘) ∈ F, and 𝑔𝑘 , 𝑙𝑘 ∈ Z such that
V(𝑘) ∗ 𝑔𝑘 ≠ 𝑤(𝑘) ∗ 𝑙𝑘 for all 𝑘 and
󵄩󵄩 (𝑘)
󵄩
󵄩󵄩V ∗ 𝑔𝑘 − 𝑤(𝑘) ∗ 𝑙𝑘 󵄩󵄩󵄩 󳨀→ 𝜅 as 𝑘 󳨀→ ∞.
󵄩
󵄩
(71)
Let 𝑚𝑘 = 𝑔𝑘 − 𝑙𝑘 . Passing to a subsequence, V(𝑘) = V ∈ F,
𝑤(𝑘) = 𝑤 ∈ F, and either 𝑚𝑘 = 𝑚 ∈ Z for all 𝑘 or |𝑚𝑘 | → ∞.
In the first case, 0 < ‖V(𝑘) ∗ 𝑔𝑘 − 𝑤(𝑘) ∗ 𝑙𝑘 ‖ = ‖V − 𝑤 ∗ 𝑚‖ = 𝜅
for all 𝑘. In the second case, 𝑤 ∗ 𝑚𝑘 ⇀ 0 and therefore 𝜅 =
lim𝑘 → ∞ ‖V − 𝑤 ∗ 𝑚𝑘 ‖ ≥ ‖V‖ = 1. By (70), 𝛾(𝐾𝑐𝑘 ) = 0 or 1.
Next, we consider a pseudogradient vector field of Ψ [18];
that is, there exists a Lipschitz continuous map 𝑉: 𝑆+ \ 𝐾 →
𝑇𝑤 𝑆+ and for all 𝑤 ∈ 𝑆+ \ 𝐾,
󵄩
󵄩
‖𝑉 (𝑤)‖ < 2 󵄩󵄩󵄩󵄩Ψ󸀠 (𝑤)󵄩󵄩󵄩󵄩 ,
1󵄩
󵄩2
⟨𝑉 (𝑤) , Ψ󸀠 (𝑤)⟩ > 󵄩󵄩󵄩󵄩Ψ󸀠 (𝑤)󵄩󵄩󵄩󵄩 .
2
(72)
Let 𝜂 : D → 𝑆+ \ 𝐾 be the corresponding Ψ-decreasing flow
defined by
𝑑
𝜂 (𝑡, 𝑤) = −𝑉 (𝜂 (𝑡, 𝑤)) ,
𝑑𝑡
(73)
𝜂 (0, 𝑤) = 𝑤,
where D = {(𝑡, 𝑤) : 𝑤 ∈ 𝑆+ \ 𝐾, 𝑇− (𝑤) < 𝑡 < 𝑇+ (𝑤)} ⊂ R ×
(𝑆+ \𝐾), and 𝑇− (𝑤) < 0, 𝑇+ (𝑤) > 0 are the maximal existence
times of the trajectory 𝑡 → 𝜂(𝑡, 𝑤) in negative and positive
direction. By the continuity property of the genus, there exists
𝛿 > 0 such that 𝛾(𝑈) = 𝛾(𝐾𝑐𝑘 ), where 𝑈 = 𝑁𝛿 (𝐾𝑐𝑘 ) := {𝑤 ∈
𝑆+ : dist(𝑤, 𝐾𝑐𝑘 ) < 𝛿} and 𝛿 < 𝜅/2. Following the deformation
argument (Lemma A.3), we choose 𝜀 = 𝜀(𝛿) > 0 such that
lim Ψ (𝜂 (𝑡, 𝑤)) < 𝑐𝑘 − 𝜀
𝑡 → 𝑇+ (𝑤)
for 𝑤 ∈ Ψ𝑐𝑘 +𝜀 \ 𝑈.
𝑟 : 𝑤 ∈ Ψ𝑐𝑘 +𝜀 \ 𝑈 󳨀→ [0, ∞) ,
(75)
𝑟 (𝑤) = inf {𝑡 ∈ [0, 𝑇+ (𝑤)) : Ψ (𝜂 (𝑡, 𝑤)) ≤ 𝑐𝑘 − 𝜀} ,
which satisfies 𝑟(𝑤) < 𝑇+ (𝑤) for every 𝑤 ∈ Ψ𝑐𝑘 +𝜀 \ 𝑈. Since
𝑐𝑘 − 𝜀 is not a critical value of Ψ by (74), it is easy to see that 𝑟
is a continuous and even map. It follows that the map
𝑔 : Ψ𝑐𝑘 +𝜀 \ 𝑈 󳨀→ Ψ𝑐𝑘 −𝜀 ,
𝑔 (𝑤) = 𝜂 (𝑟 (𝑤) , 𝑤)
(76)
is odd and continuous. Then, 𝛾(Ψ𝑐𝑘 +𝜀 \ 𝑈) ≤ 𝛾(Ψ𝑐𝑘 −𝜀 ) ≤ 𝑘 − 1,
and consequently,
Now, we claim that
𝐾𝑐𝑘 ≠ 0,
Then, for every 𝑤 ∈ Ψ𝑐𝑘 +𝜀 \ 𝑈, there exists 𝑡 ∈ [0, 𝑇+ (𝑤)) such
that Ψ(𝜂(𝑡, 𝑤)) < 𝑐𝑘 − 𝜀. Hence, we may define the entrance
time map
(74)
𝛾 (Ψ𝑐𝑘 +𝜀 ) ≤ 𝛾 (𝑈) + 𝑘 − 1 = 𝛾 (𝐾𝑐𝑘 ) + 𝑘 − 1.
(77)
So, 𝛾(𝐾𝑐𝑘 ) ≥ 1. Therefore, 𝐾𝑐𝑘 ≠ 0. Moreover, the definition
of 𝑐𝑘 and of 𝑐𝑘+1 implies that 𝛾(𝐾𝑐𝑘 ) ≥ 1 if 𝑐𝑘 < 𝑐𝑘+1 and
𝛾(𝐾𝑐𝑘 ) > 1 if 𝑐𝑘 = 𝑐𝑘+1 . Since 𝛾(F) = 𝛾(𝐾𝑐𝑘 ) ≤ 1, 𝑐𝑘 <
𝑐𝑘+1 . Therefore, there is an infinite sequence {±𝑤𝑘 } of pairs
of geometrically distinct critical points of Ψ with Ψ(𝑤𝑘 ) = 𝑐𝑘 ,
which contradicts with (66). Therefore, the set F is infinite.
This completes the proof of Theorem 2.
Appendix
Here, we give a proof of (74). We state the discrete property
of the Palais-Smale sequences. It yields nice properties of the
corresponding pseudogradient flow.
Lemma A.1. Let 𝑑 ≥ 𝑐. If {𝑤1(𝑘) }, {𝑤2(𝑘) } ⊂ Ψ𝑑 are two PalaisSmale sequences for Ψ, then either ‖𝑤1(𝑘) − 𝑤2(𝑘) ‖ → 0 as
𝑘 → ∞ or lim sup𝑘 → ∞ ‖𝑤1(𝑘) − 𝑤2(𝑘) ‖ ≥ 󰜚(𝑑) > 0, where 󰜚(𝑑)
depends on 𝑑 but not on the particular choice of the PalaisSmale sequences.
Proof. Set 𝑢1(𝑘) = 𝑚(𝑤1(𝑘) ) and 𝑢2(𝑘) = 𝑚(𝑤2(𝑘) ). Then, {𝑢1(𝑘) },
{𝑢2(𝑘) } ⊂ 𝐽𝑑 are the bounded Palais-Smale sequences for 𝐽. We
fix 𝑝 in (𝑓2 ) and consider the following two cases.
(i) ‖ 𝑢1(𝑘) − 𝑢2(𝑘) ‖𝑙𝑝 → 0 as 𝑘 → ∞.
By a straightforward calculation and (32), for any 𝜀 > 0,
there exist 𝐶1 , 𝐶2 > 0, and 𝑘0 such that for all 𝑘 ≥ 𝑘0 ,
󵄩2
󵄩󵄩 (𝑘)
󵄩󵄩(𝑢 − 𝑢(𝑘) )+ 󵄩󵄩󵄩
2
󵄩󵄩
󵄩󵄩 1
+
= 𝐽󸀠 (𝑢1(𝑘) ) (𝑢1(𝑘) − 𝑢2(𝑘) ) − 𝐽󸀠 (𝑢2(𝑘) ) (𝑢2(𝑘) − 𝑢2(𝑘) )
+
(𝑘)
(𝑘)
+ ∑ 𝜒𝑛 [𝑓𝑛 (𝑢1𝑛
) − 𝑓𝑛 (𝑢2𝑛
)] (𝑢1(𝑘) − 𝑢2(𝑘) )
𝑛∈Z
󵄩󵄩
+󵄩
󵄩2
≤ 𝜀󵄩󵄩󵄩(𝑢1(𝑘) − 𝑢2(𝑘) ) 󵄩󵄩󵄩
󵄩
󵄩
+
Abstract and Applied Analysis
9
󵄨 (𝑘) 󵄨󵄨 󵄨󵄨 (𝑘) 󵄨󵄨
󵄨󵄨 + 󵄨󵄨𝑢2𝑛 󵄨󵄨)
+ 𝜒 ∑ [𝜀 (󵄨󵄨󵄨󵄨𝑢1𝑛
󵄨 󵄨 󵄨
If 𝑢1 = 0, then 𝑢2 ≠ 0 and
𝑛∈Z
󵄩󵄩
󵄩󵄩 (𝑢(𝑘) )+
󵄩
󵄩󵄩 (𝑘)
1
(𝑘) 󵄩
󵄩
−
lim inf 󵄩󵄩󵄩𝑤1 − 𝑤2 󵄩󵄩󵄩 = lim inf 󵄩󵄩󵄩󵄩 󵄩
󵄩󵄩 (𝑘) + 󵄩󵄩󵄩
𝑘→∞
𝑘→∞ 󵄩
󵄩󵄩 󵄩󵄩(𝑢1 ) 󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩 + 󵄩󵄩 √
󵄩𝑢 󵄩
2𝑐
≥ 󵄩 2󵄩 ≥
.
𝛼2
] (𝑑)
󵄨 (𝑘) 󵄨󵄨𝑝−1 󵄨󵄨 (𝑘) 󵄨󵄨𝑝−1
󵄨󵄨 + 󵄨󵄨𝑢2𝑛 󵄨󵄨 )]
+ 𝑐𝜀 (󵄨󵄨󵄨󵄨𝑢1𝑛
󵄨
󵄨 󵄨
󵄨󵄨
+ 󵄨󵄨
× 󵄨󵄨󵄨(𝑢1(𝑘) − 𝑢2(𝑘) ) 󵄨󵄨󵄨
󵄨
󵄨
󵄩󵄩 (𝑘)
󵄩
+
󵄩
≤ 𝜀 󵄩󵄩󵄩(𝑢1 − 𝑢2(𝑘) ) 󵄩󵄩󵄩
󵄩
󵄩
+󵄩
󵄩󵄩 (𝑘) 󵄩󵄩 󵄩󵄩 (𝑘) 󵄩󵄩 󵄩󵄩󵄩 (𝑘)
󵄩
+ 𝜒𝜀 (󵄩󵄩󵄩𝑢1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑢2 󵄩󵄩󵄩) 󵄩󵄩(𝑢1 − 𝑢2(𝑘) ) 󵄩󵄩󵄩
󵄩
󵄩
󵄩
𝑝−1
𝑝−1
+󵄩
󵄩 󵄩
󵄩 󵄩
󵄩
󵄩
+ 𝜒𝑐𝜀 (󵄩󵄩󵄩󵄩𝑢1(𝑘) 󵄩󵄩󵄩󵄩𝑙𝑝 + 󵄩󵄩󵄩󵄩𝑢2(𝑘) 󵄩󵄩󵄩󵄩𝑙𝑝 ) 󵄩󵄩󵄩(𝑢1(𝑘) − 𝑢2(𝑘) ) 󵄩󵄩󵄩 𝑝
󵄩𝑙
󵄩
󵄩
󵄩
󵄩
󵄩󵄩 (𝑘)
+
+
󵄩
󵄩
󵄩
≤ 𝜀 󵄩󵄩󵄩(𝑢1 − 𝑢2(𝑘) ) 󵄩󵄩󵄩 + 𝜒𝜀𝐶1 󵄩󵄩󵄩(𝑢1(𝑘) − 𝑢2(𝑘) ) 󵄩󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩󵄩 (𝑘)
󵄩
+ 𝜒𝑐𝜀 𝐶2 󵄩󵄩󵄩𝑢1 − 𝑢2(𝑘) 󵄩󵄩󵄩󵄩𝑙𝑝 .
+ 2
𝑢1(𝑘) ⇀ 𝑢1 ∈ 𝐸,
we may pass to subse-
𝑢2(𝑘) ⇀ 𝑢2 ∈ 𝐸,
where 𝑢1 ≠ 𝑢2 by (A.2) and 𝐽󸀠 (𝑢1 ) = 𝐽󸀠 (𝑢2 ) = 0, and
󵄩󵄩 (𝑘) + 󵄩󵄩
󵄩󵄩󵄩 (𝑘) + 󵄩󵄩󵄩
󵄩󵄩(𝑢 ) 󵄩󵄩 󳨀→ 𝛼2 ,
󵄩󵄩(𝑢1 ) 󵄩󵄩 󳨀→ 𝛼1 ,
󵄩󵄩 1 󵄩󵄩
󵄩
󵄩
(A.7)
Lemma A.2. For every 𝑤 ∈ 𝑆+ , the limit lim𝑡 → 𝑇+ (𝑤) 𝜂(𝑡, 𝑤)
exists and is a critical point of Ψ.
This implies lim sup𝑘 → ∞ ‖(𝑢1(𝑘) − 𝑢2(𝑘) ) ‖ ≤ lim sup𝑘 → ∞ (1 +
+
+
𝜒𝐶1 )𝜀‖(𝑢1(𝑘) − 𝑢2(𝑘) ) ‖. Hence, ‖(𝑢1(𝑘) − 𝑢2(𝑘) ) ‖ → 0. Similarly,
−
‖(𝑢1(𝑘) − 𝑢2(𝑘) ) ‖ → 0. Therefore, ‖𝑢1(𝑘) − 𝑢2(𝑘) ‖ → 0 as 𝑘 →
∞. By Lemma 13(c), we have ‖𝑤1(𝑘) − 𝑤2(𝑘) ‖ = ‖𝑚−1 (𝑢1(𝑘) ) −
𝑚−1 (𝑢2(𝑘) )‖ → 0 as 𝑘 → ∞.
(ii) ‖𝑢1(𝑘) − 𝑢2(𝑘) ‖𝑙𝑝 󴀀󴀂󴀠 0 as 𝑘 → ∞.
There exist 𝛿 > 0 and 𝑛𝑘 ∈ Z such that
󵄨󵄨󵄨𝑢(𝑘) − 𝑢(𝑘) 󵄨󵄨󵄨 ≥ 𝛿.
(A.2)
󵄨󵄨 1𝑛𝑘
2𝑛𝑘 󵄨󵄨
For bounded sequences
quences so that
+
Similarly, if 𝑢2 = 0, then 𝑢1 ≠ 0 and lim inf 𝑘 → ∞ ‖𝑤1(𝑘) −𝑤2(𝑘) ‖ ≥
√2𝑐/](𝑑).
The proof is complete.
(A.1)
{𝑢1(𝑘) }, {𝑢2(𝑘) },
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩󵄩 (𝑘) + 󵄩󵄩 󵄩󵄩󵄩
󵄩󵄩(𝑢 ) 󵄩󵄩 󵄩󵄩
󵄩󵄩 2 󵄩󵄩 󵄩󵄩
(𝑢2(𝑘) )
(A.3)
Proof. Fix 𝑤 ∈ 𝑆+ and set 𝑑 = Ψ(𝑤). We distinguish two cases
to finish the proof.
Case 1 (𝑇+ (𝑤) < ∞). For 0 ≤ 𝑠 < 𝑡 < 𝑇+ (𝑤), by (72) and (73),
we have
󵄩󵄩󵄩𝜂 (𝑡, 𝑤) − 𝜂 (𝑠, 𝑤)󵄩󵄩󵄩
󵄩
󵄩
𝑡
󵄩
󵄩
≤ ∫ 󵄩󵄩󵄩𝑉 (𝜂 (𝜏, 𝑤))󵄩󵄩󵄩 𝑑𝜏
𝑠
𝑡
≤ 2√2 ∫ √⟨Ψ󸀠 ( 𝜂 (𝜏, 𝑤)) , 𝑉 (𝜂 (𝜏, 𝑤))⟩𝑑𝜏
𝑠
𝑡
≤ 2√2 (𝑡 − 𝑠)(∫ ⟨Ψ󸀠 (𝜂 (𝜏, 𝑤)) , 𝑉 (𝜂 (𝜏, 𝑤))⟩ 𝑑𝜏)
1/2
𝑠
= 2√2 (𝑡 − 𝑠)[Ψ (𝜂 (𝑠, 𝑤)) − Ψ (𝜂 (𝑡, 𝑤))]
1/2
≤ 2√2 (𝑡 − 𝑠)[Ψ (𝑤) − 𝑐]1/2 .
(A.8)
(A.4)
where √2𝑐 ≤ 𝛼𝑖 ≤ ](𝑑), 𝑖 = 1, 2 by Lemma 9(b).
If 𝑢1 ≠ 0 and 𝑢2 ≠ 0. Then, 𝑢1 , 𝑢2 ∈ M and 𝑤1 = 𝑚−1 (𝑢1 ) ∈
𝐾, 𝑤2 = 𝑚−1 (𝑢2 ) ∈ 𝐾, 𝑤1 ≠ 𝑤2 . Therefore,
󵄩󵄩
+ 󵄩
󵄩
󵄩󵄩 (𝑢(𝑘) )+
(𝑢2(𝑘) ) 󵄩󵄩󵄩
󵄩
󵄩󵄩 (𝑘)
1
(𝑘) 󵄩
󵄩󵄩
󵄩
󵄩
󵄩
󵄩
󵄩
−
lim inf 󵄩󵄩𝑤1 − 𝑤2 󵄩󵄩 = lim inf 󵄩󵄩 󵄩
+󵄩
+󵄩
󵄩
󵄩 󵄩󵄩
󵄩 󵄩󵄩
𝑘→∞
𝑘→∞ 󵄩
󵄩󵄩 󵄩󵄩󵄩(𝑢1(𝑘) ) 󵄩󵄩󵄩 󵄩󵄩󵄩(𝑢2(𝑘) ) 󵄩󵄩󵄩 󵄩󵄩󵄩
󵄩󵄩
󵄩 󵄩
󵄩󵄩
󵄩󵄩 𝑢+ 𝑢+ 󵄩󵄩
󵄩
󵄩 󵄩
󵄩
≥ 󵄩󵄩󵄩󵄩 1 − 2 󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩𝛽1 𝑤1 − 𝛽2 𝑤2 󵄩󵄩󵄩 ,
󵄩󵄩 𝛼1 𝛼2 󵄩󵄩
(A.5)
where 𝛽1 = ‖𝑢1+ ‖/𝛼1 ≥ √2𝑐/](𝑑) and 𝛽2 = ‖𝑢2+ ‖/𝛼2 ≥
√2𝑐/](𝑑). Since ‖𝑤1 ‖ = ‖𝑤2 ‖ = 1, we have
󵄩
󵄩 󵄩
󵄩
lim inf 󵄩󵄩󵄩󵄩𝑤1(𝑘) − 𝑤2(𝑘) 󵄩󵄩󵄩󵄩 ≥ 󵄩󵄩󵄩𝛽1 𝑤1 − 𝛽2 𝑤2 󵄩󵄩󵄩
𝑘→∞
󵄩 √2𝑐𝜅
󵄩
≥ min {𝛽1 , 𝛽2 } 󵄩󵄩󵄩𝑤1 − 𝑤2 󵄩󵄩󵄩 ≥
.
] (𝑑)
(A.6)
Since 𝑇+ (𝑤) < ∞, this implies that lim𝑡 → 𝑇+ (𝑤) 𝜂(𝑡, 𝑤) exists
and is a critical point of Ψ, otherwise the trajectory 𝑡 →
𝜂(𝑡, 𝑤) could be continued beyond 𝑇+ (𝑤).
Case 2 (𝑇+ (𝑤) = ∞). To prove that lim𝑡 → 𝑇+ (𝑤) 𝜂(𝑡, 𝑤) exists,
we claim that for every 𝜀 > 0, there exists 𝑡𝜀 > 0 such that
‖𝜂(𝑡𝜀 , 𝑤) − 𝜂(𝑡, 𝑤)‖ < 𝜀 for 𝑡 ≥ 𝑡𝜀 . If not, then there exist
0 < 𝜀0 < (1/2)󰜚(𝑑) (󰜚(𝑑) is the same number in Lemma A.1)
and a sequence {𝑡𝑛 } ⊂ [0, ∞) with 𝑡𝑛 → ∞ such that
‖𝜂(𝑡𝑛 , 𝑤) − 𝜂(𝑡𝑛+1 , 𝑤)‖ = 𝜀0 for every 𝑛. Choose the smallest
𝑡𝑛1 ∈ (𝑡𝑛 , 𝑡𝑛+1 ) such that ‖𝜂(𝑡𝑛 , 𝑤) − 𝜂(𝑡𝑛1 , 𝑤)‖ = 𝜀0 /3. Let
𝜄𝑛 = min𝑠∈[𝑡𝑛 ,𝑡𝑛1 ] ‖Ψ󸀠 (𝜂(𝑠, 𝑤))‖. By (72) and (73), we have
𝜀0 󵄩󵄩 1
󵄩
= 󵄩󵄩𝜂 (𝑡 , 𝑤) − 𝜂 (𝑡𝑛 , 𝑤)󵄩󵄩󵄩󵄩
3 󵄩 𝑛
𝑡1
𝑛
󵄩
󵄩
≤ ∫ 󵄩󵄩󵄩𝑉 (𝜂 (𝜏, 𝑤))󵄩󵄩󵄩 𝑑𝜏
𝑡
𝑛
1
𝑡𝑛
󵄩
󵄩
≤ 2 ∫ 󵄩󵄩󵄩󵄩Ψ󸀠 (𝜂 (𝜏, 𝑤))󵄩󵄩󵄩󵄩 𝑑𝜏
𝑡
𝑛
10
Abstract and Applied Analysis
1
≤
2 𝑡𝑛 󵄩󵄩 󸀠
󵄩2
∫ 󵄩󵄩Ψ (𝜂 (𝜏, 𝑤))󵄩󵄩󵄩󵄩 𝑑𝜏
𝜄𝑛 𝑡𝑛 󵄩
≤
4 𝑡𝑛
∫ ⟨Ψ󸀠 (𝜂 (𝜏, 𝑤)) , 𝑉 (𝜂 (𝜏, 𝑤))⟩ 𝑑𝜏
𝜄𝑛 𝑡𝑛
Choose 𝜀 < 𝛿𝜏2 /8𝑀 such that (a) holds. By Lemma A.1 and
̃ ∈ 𝐾𝑑 as
(a), the only way that (b) can fail is that 𝜂(𝑡, 𝑤) → 𝑤
𝑡 → 𝑇+ (𝑤) for some 𝑤 ∈ Ψ𝑑+𝜀 \ 𝑁𝛿 (𝐾𝑑 ). In this case, we let
1
𝑡1 = sup {𝑡 ∈ [0, 𝑇+ (𝑤)) : 𝜂 (𝑡, 𝑤) ∉ 𝑁𝛿 (̃
𝑤)} ,
𝑤)} .
𝑡2 = inf {𝑡 ∈ (𝑡1 , 𝑇+ (𝑤)) : 𝜂 (𝑡, 𝑤) ∈ 𝑁𝛿/2 (̃
4
= (Ψ (𝜂 (𝑡𝑛 , 𝑤)) − Ψ (𝜂 (𝑡𝑛1 , 𝑤))) .
𝜄𝑛
(A.9)
Since Ψ(𝜂(𝑡𝑛 , 𝑤)) − Ψ(𝜂(𝑡𝑛1 , 𝑤)) → 0 as 𝑛 → ∞, 𝜄𝑛 → 0
and there exist ̃𝑡𝑛1 ∈ [𝑡𝑛 , 𝑡𝑛1 ] such that Ψ󸀠 (𝑤𝑛1 ) → 0, where
𝑤𝑛1 = 𝜂(̃𝑡𝑛1 , 𝑤). Similarly, we choose the largest 𝑡𝑛2 ∈ (𝑡𝑛1 , 𝑡𝑛+1 )
such that ‖𝜂(𝑡𝑛+1 , 𝑤) − 𝜂(𝑡𝑛2 , 𝑤)‖ = 𝜀0 /3. Then, there exist ̃𝑡𝑛2 ∈
[𝑡𝑛2 , 𝑡𝑛+1 ] such that Ψ󸀠 (𝑤𝑛2 ) → 0, where 𝑤𝑛2 = 𝜂(̃𝑡𝑛2 , 𝑤). Since
‖𝑤𝑛1 −𝜂(𝑡𝑛 , 𝑤)‖ ≤ 𝜀0 /3 and ‖𝑤𝑛2 −𝜂(𝑡𝑛+1 , 𝑤)‖ ≤ 𝜀0 /3, {𝑤𝑛1 }, {𝑤𝑛2 }
are two the Palais-Smale sequences such that
𝜀0 󵄩󵄩 1
󵄩
≤ 󵄩󵄩󵄩𝑤𝑛 − 𝑤𝑛2 󵄩󵄩󵄩󵄩
3
󵄩
󵄩
≤ 󵄩󵄩󵄩󵄩𝑤𝑛1 − 𝜂 (𝑡𝑛 , 𝑤)󵄩󵄩󵄩󵄩
(A.10)
󵄩󵄩
󵄩󵄩
󵄩󵄩 󵄩󵄩󵄩 2
+ 󵄩󵄩𝜂 (𝑡𝑛 , 𝑤) − 𝜂 (𝑡𝑛+1 , 𝑤)󵄩󵄩 + 󵄩󵄩𝑤𝑛 − 𝜂 (𝑡𝑛+1 , 𝑤)󵄩󵄩󵄩
≤ 2𝜀0 < 󰜚 (𝑑) ,
which contradicts with Lemma A.1. This proves the claim.
Therefore, lim𝑡 → 𝑇+ (𝑤) 𝜂(𝑡, 𝑤) exists, and, obviously, it must be
a critical point of Ψ. This completes the proof.
Lemma A.3. Let 𝑑 ≥ 𝑐. Then, for every 𝛿 > 0, there exists
𝜀 = 𝜀(𝛿) > 0 such that
𝑑+𝜀
(a) Ψ𝑑−𝜀
∩ 𝐾 = 𝐾𝑑 ,
Proof. (a) According to (66), for 𝜀 > 0 small enough, it is easy
to see that (a) is satisfied.
(b) Without loss of generality, we may assume that
𝑁𝛿 (𝐾𝑑 ) ⊂ Ψ𝑑+1 and 𝛿 < 󰜚(𝑑 + 1). Set
(A.11)
We claim that 𝜏 > 0. Indeed, if not, then there exists a
sequence {𝑤1(𝑘) } ⊂ 𝑁𝛿 (𝐾𝑑 ) \ 𝑁𝛿/2 (𝐾𝑑 ) such that Ψ󸀠 (𝑤1(𝑘) ) →
0. By the Z-invariance of Ψ and assumption (66), we may
assume 𝑤1(𝑘) ∈ 𝑁𝛿 (𝑤0 ) \ 𝑁𝛿/2 (𝑤0 ) for some 𝑤0 ∈ 𝐾𝑑 after
passing to a subsequence. Let 𝑤2(𝑘) → 𝑤0 . Then, Ψ󸀠 (𝑤2(𝑘) ) →
0 and
𝛿
󵄩
󵄩
≤ lim sup 󵄩󵄩󵄩󵄩𝑤1(𝑘) − 𝑤2(𝑘) 󵄩󵄩󵄩󵄩 ≤ 𝛿 < 󰜚 (𝑑 + 1) ,
2
𝑛→∞
(A.12)
which contradicts with Lemma A.1. This proves the claim.
Let
󵄩
󵄩
𝑀 = sup {󵄩󵄩󵄩󵄩Ψ󸀠 (𝑤)󵄩󵄩󵄩󵄩 : 𝑤 ∈ 𝑁𝛿 (𝐾𝑑 ) \ 𝑁𝛿/2 (𝐾𝑑 )} .
Then,
𝛿 󵄩󵄩
󵄩
= 󵄩𝜂 (𝑡 , 𝑤) − 𝜂 (𝑡2 , 𝑤)󵄩󵄩󵄩
2 󵄩 1
𝑡
2
󵄩
󵄩
≤ ∫ 󵄩󵄩󵄩𝑉 (𝜂 (𝜏, 𝑤))󵄩󵄩󵄩 𝑑𝜏
𝑡
1
≤ 2∫
𝑡2
𝑡1
󵄩󵄩 󸀠
󵄩
󵄩󵄩Ψ (𝜂 (𝜏, 𝑤))󵄩󵄩󵄩 𝑑𝜏
󵄩
󵄩
≤ 2𝑀 (𝑡2 − 𝑡1 ) ,
Ψ (𝜂 (𝑡2 , 𝑤)) − Ψ (𝜂 (𝑡1 , 𝑤))
(A.15)
𝑡2
= − ∫ ⟨Ψ󸀠 (𝜂 (𝜏, 𝑤)) , 𝑉 (𝜂 (𝜏, 𝑤))⟩ 𝑑𝑠
𝑡1
1 𝑡2 󵄩
󵄩2
≤ − ∫ 󵄩󵄩󵄩󵄩Ψ󸀠 (𝜂 (𝑠, 𝑤))󵄩󵄩󵄩󵄩 𝑑𝑠
2 𝑡1
1
𝛿𝜏2
≤ − 𝜏2 (𝑡2 − 𝑡1 ) ≤ −
.
2
8𝑀
It follows that Ψ(𝜂(𝑡2 , 𝑤)) ≤ 𝑑 + 𝜀 − (𝛿𝜏2 /8𝑀) < 𝑑 and
̃, a contradiction again. This completes
therefore 𝜂(𝑡2 , 𝑤) 󴀀󴀂󴀠 𝑤
the proof.
Acknowledgments
(b) lim𝑡 → 𝑇+ (𝑤) Ψ(𝜂(𝑡, 𝑤)) < 𝑑 − 𝜀 for 𝑤 ∈ Ψ𝑑+𝜀 \ 𝑁𝛿 (𝐾𝑑 ).
󵄩
󵄩
𝜏 = inf {󵄩󵄩󵄩󵄩Ψ󸀠 (𝑤)󵄩󵄩󵄩󵄩 : 𝑤 ∈ 𝑁𝛿 (𝐾𝑑 ) \ 𝑁𝛿/2 (𝐾𝑑 )} .
(A.14)
(A.13)
The authors would like to thank the anonymous referees for
their constructive comments and suggestions, which considerably improved the presentation of the paper. This work
is supported by the Program for Changjiang Scholars and
Innovative Research Team in University (no. IRT1226), the
National Natural Science Foundation of China (no. 11171078),
and the Specialized Fund for the Doctoral Program of Higher
Education of China (no. 20114410110002).
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