# On Ground States of Discretep(k)-Laplacian Systems in Generalized Orlicz Sequence Spaces

Authors: Juhong Kuang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Using the critical point theory, we establish sufficient conditions on the existence of ground states for discretep(k)-Laplacian systems. Our results considerably generalize some existing ones.


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## Body

## 1. Introduction and Main Results

The aim of this paper is to study the existence of ground state for discretep(k)-Laplacian system
(1)Δ[a(k)|Δu(k-1)|p(k)-2Δu(k-1)]-q(k)|u(k)|p(k)-2u(k)+f(k,u(k))=0,
where p(k)>1, for all k∈Z, a(k) and q(k) are real valued on Z. f:Z×R→R is continuous in the second variable. Moreover, Δ is the forward difference operator defined by Δu(k)=u(k+1)-u(k).We may think of (1) as being a discrete analogue of the following differential system:
(2)ddt(a(t)|u˙(t)|p(t)-2u˙(t))-q(t)|u(t)|p(t)-2u(t)+f(t,u(t))=0.For the casep(t)≡p, system (2) is a p-Laplacian system, which has been widely studied; to mention a few, see [1, 2]. Even for the special case p=2, system (2) can be regarded as the more general form of Emden-Fowler equation appearing in the study of astrophysics, fluid mechanics, gas dynamics, nuclear physics, relativistic mechanics, and chemically reacting system in terms of various special forms of f(t,u(t)) (see, e.g., [3]). The more general differential operator (2), namely, the so-called p(t)-Laplacian, has been studied by Fan et al. [4–7]. The p(t)-Laplacian operator can be used to describe the physical phenomena with “pointwise different properties.” The p(t)-Laplacian operator has more complicated properties than that of the p-Laplacian; for example, it is not homogeneous, and this makes some classic theories and methods, such as the theory of Sobolev spaces, not applicable.With the theory of nonlinear discrete dynamical systems being widely used to study discrete models appearing in many fields such as economics, ecology, computer science, neural networks, and cybernetics [8], the existence of solutions of discrete dynamical systems has become a hot topic; to mention a few, see [9–15]. For the case p(k)≡p, Iannizzotto and Tersian [16] obtained multiple homoclinic solutions for system (1) by using the critical point theorem, and for the special case p=2, Ma and Guo [13, 14] provided some sufficient conditions on the existence of homoclinic solutions for system (1). For the more general case—p(k)-Laplacian system (1)—Chen et al. [17] established some existence criteria to guarantee that the system has at least one or infinitely many homoclinic orbits. Motivated by Liu [2], which discussed the existence of ground state for p-Laplacian system, in this paper, we will consider the existence of ground state for the p(k)-Laplacian system (1).Now we are in a position to state our main results.Theorem 1.
Assume the following conditions hold:(a)
a(k)>0 for all k∈Z,(q)
q(k)>0 for all k∈Z and q(k)→+∞ as |k|→+∞,(p)
1<p-=inf⁡k∈Zp(k)≤sup⁡k∈Zp(k)=p+<+∞ for all k∈Z,(f)
f(k,u)=f1(k,u)-f2(k,u) is continuous in u for all k∈Z, and F(k,u)=∫0u‍f(k,s)ds for u∈R. Moreover,
(3)|f(k,u)|q(k)|u|p(k)-1⟶0asu⟶0uniformlyink∈Z;(f1)
there exists a constantβ>p+ such that
(4)uf1(k,u)≥βF1(k,u)>0∀k∈Z,u∈R∖{0},where F1(k,u)=∫0uf1(k,s)ds for u∈R;(f2)
there exists a constantτ∈(0,β) such that
(5)F2(k,u)≥0,uf2(k,u)≤τF2(k,u)hhhhhhhhhhhhhhhhh∀k∈Z,u∈R,where F2(k,u)=∫0uf2(k,s)ds for u∈R.
Then (1) has a ground state solution.Remark 2.
(q) implies that there existsq*>0 such that q(k)≥q* for all k∈Z.Remark 3.
We extend Theorem 3.1 in [14] to the more general case—p(k)-Laplacian system. Furthermore, we obtain the existence of the ground state.The rest of this paper is organized as follows. In Section2, we establish the variational structure associated with (1). Some preliminary results are also provided in this section. In Section 3, we give the proof of the main result.


## 2. Variational Structure and Some Preliminary Results

In this section, we establish a variational structure which enables us to reduce the existence of solutions for (1) to the existence of critical points of the corresponding functional.LetS be the set of all two-sided sequences; that is,
(6)S={u={u(k)}:u(k)∈R,k∈Z}.
Then S is a vector space with au+bv={au(k)+bv(k)} for u,v∈S,a,b∈R.We definelp(k) as the set of all functions u∈S such that
(7)lp(k)={u∈S:∑k∈Z‍|u(k)|p(k)<+∞}
with the norm
(8)∥u∥p(k)=inf⁡{r>0:∑k∈Z‍|u(k)r|p(k)≤1}.
We also define
(9)E={u∈S:∑k∈Z‍[a(k)|Δu(k-1)|p(k)hhhhihhhhhhh+q(k)|u(k)|p(k)]<+∞∑k∈Z‍[a(k)|Δu(k-1)|p(k)}
with the norm
(10)∥u∥=inf⁡{r>0:∑k∈Z‍[a(k)|Δu(k-1)r|p(k)hhhhhhhhhhhh+q(k)|u(k)r|p(k)]≤1}.
We call the space E a sequence space; it is a special kind of generalized Orlicz sequence space. For the general theory of generalized Orlicz spaces, see [18, 19].Consider the functionalI on E defined by
(11)I(u)=∑k∈Z[a(k)p(k)|Δu(k-1)|p(k)+q(k)p(k)|u(k)|p(k)hhhhhh-F(k,u(k))a(k)p(k)].
Using the similar arguments as [17], we have the following lemmas.Lemma 4.
(
l
p
(
k
)
,
∥
·
∥
p
(
k
)
) is a reflexive Banach space. Let u∈lp(k) and
(12)ϕ(u)=∑k∈Z‍|u(k)|p(k);
one has (1)
if∥u∥p(k)>1, then ∥u∥p(k)p-≤ϕ(u)≤∥u∥p(k)p+;(2)
if∥u∥p(k)<1, then ∥u∥p(k)p+≤ϕ(u)≤∥u∥p(k)p-.Lemma 5.
(
E
,
∥
·
∥
) is a reflexive Banach space. Let u∈E and
(13)φ(u)=∑k∈Z‍[a(k)|Δu(k-1)|p(k)+q(k)|u(k)|p(k)];
one has(1)
if∥u∥>1, then ∥u∥p-≤φ(u)≤∥u∥p+;(2)
if∥u∥<1, then ∥u∥p+≤φ(u)≤∥u∥p-.Lemma 6.
I
∈
C
1
(
E
,
R
) and the Fréchet derivative is given by
(14)〈I′(u),v〉=∑k∈Z‍[a(k)|Δu(k-1)|p(k)-2hhhhhh×Δu(k-1)Δv(k-1)hhhhhh+q(k)|u(k)|p(k)-2u(k)v(k)]-∑k∈Z‍f(k,u(k))v(k),
for all u,v∈E. Moreover, the nonzero critical points of the functional I on E are the nontrivial solutions of (1).Lemma 7.
(f1) and (f2) imply that
(15)uf(k,u)≥βF(k,u)∀k∈Z,u∈R.
Moreover, for every k∈Z and u∈R, s-βF1(k,su) is nondecreasing on (0,+∞) and s-τF2(k,su) is nonincreasing on (0,+∞).Let(16)cmin⁡=inf⁡{I(u):I′(u)=0,u∈E∖{0}}.
Then u0≠0 with I(u0)=cmin⁡ is said to be a ground state solution of (1).As usual, we make use of the following basic notations. LetH be a Hilbert space and C1(H,R) denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on H.Definition 8.
LetI∈C1(H,R). A sequence {xj}⊂H is called a Palais-Smale sequence (P.S. sequence) for I if {I(xj)} is bounded and I′(xj)→0 as j→+∞. We say that I satisfies the Palais-Smale condition (P.S. condition) if any P.S. sequence for I possesses a convergent subsequence.LetBr be the open ball in H with radius r and center 0, and let ∂Br denote its boundary.Lemma 9 (mountain pass lemma).
LetH be a real Hilbert space and I∈C1(H,R) satisfies the P.S. condition. Assume that I(0)≤0 and the following two conditions hold.(J1)
There exist constantsa>0 and ρ>0 such that I∂Bρ≥a.(J2)
There exists ane∈H∖Bρ such that I(e)≤0.Then I possesses a critical value c≥a. Moreover, c can be characterized as
(17)c=inf⁡h∈Γmax⁡s∈[0,1]I(h(s)),
where
(18)Γ={h∈C([0,1],H):h(0)=0,h(1)=e}.


## 3. Proof of Main Result

In order to prove Theorem1, we first prove the following lemmas.Lemma 10.
The embeddingE↪lp(k) is compact.Proof.
Let{uj} be a bounded sequence in E; that is, there exists M>0 such that ∥uj∥<M for all j∈Z+. By reflexivity, passing to a subsequence we have uj⇀u in E for some u∈E. We may assume u=0, in particular uj(k)→0 as j→+∞ for all k∈Z. For all ϵ>0, we can find h∈Z+ such that
(19)q(k)>1+Mϵ∀|k|>h.
By continuity of the finite sum, there exists ν0∈Z+ such that
(20)∑|k|≤h‍|uj(k)|p(k)≤ϵ1+M∀j>ν0.
So for all j≥ν0 we have
(21)∑k∈Z‍|uj(k)|p(k)≤ϵ1+M+ϵ1+M∑|k|>h‍q(k)|uj(k)|p(k)≤ϵ1+M+ϵ1+M∑k∈Z‍q(k)|uj(k)|p(k).
Since
(22)∑k∈Z‍q(k)|uj(k)|p(k)≤φ(uj)≤{∥uj∥p-,if∥uj∥<1,1,if∥uj∥=1,∥uj∥p+,if∥uj∥>1,
letting M0=max⁡{Mp-,1,Mp+}, we have
(23)∑k∈Z‍|uj(k)|p(k)≤1+M01+Mϵ∀j≥ν0.
Thus, uj(k)→0 in lp(k), and the proof is completed.Lemma 11.
Assume thatu∈lp(k) and v∈lr(k). Moreover, p(k) satisfies condition (p) and 1/r(k)+1/p(k)=1 for all k∈Z. Then
(24)∑k∈Z‍u(k)v(k)≤(1p-+1r-)∥u∥p(k)∥v∥r(k),
where r-=inf⁡{r(k):k∈Z} and r-=p+/(p+-1).Proof.
Let(25)r1=∥u∥p(k),r2=∥v∥r(k);
then
(26)∑k∈Z‍u(k)v(k)r1r2≤∑k∈Z‍[1p(k)(u(k)r1)p(k)+1r(k)(v(k)r2)r(k)]≤1p-+1r-.
The proof is completed.Lemma 12.
Assume that all the conditions of Theorem1 hold. Then the functional I satisfies the P.S. condition.Proof.
Assume that{uj}j∈N⊂E is a sequence such that {I(uj)} is a bounded and I′(uj)→0 as j→+∞. Then there exists a positive constant M′ such that |I(uj)|≤M′ for all j∈Z+.
First, we show that∥uj∥ is bounded. Now we may assume that ∥uj∥>1; otherwise, ∥uj∥ is bounded obviously. When j is large enough, we have
(27)M′+1β∥uj∥≥I(uj)-1β〈I′(uj),uj〉=∑k∈Z‍(1p(k)-1β)[a(k)|Δuj(k-1)|p(k)hhhihhhhhhhhhhh+q(k)|uj(k)|p(k)]+∑k∈Z‍[1βf(k,uj)uj(k)-F(k,uj(k))]≥(1p+-1β)∥uj∥p-.
It follows from β>p+ and p->1 that there exists a constant M*>0 such that
(28)∥uj∥≤M*,∀j∈Z+.
By Lemma 10, we can choose a subsequence, still denoted by {uj}, such that
(29)uj⇀u*inE,(30)uj⟶u*inlp(k),
for some u*∈E.
Next, we prove that(31)lim⁡j→+∞∑k∈Z(f(k,uj(k))-f(k,u*(k)))×(uj(k)-u*(k))=0.
By (f), for any0<ϵ<min⁡{1/2,1/2p+}, there exists a positive constant ρ<1 with q*(1/p-)ρ(p+/p-)<1 such that
(32)f(k,u)≤ϵq(k)|u|p(k)-1∀k∈Z,|u|≤ρ,
Since u*∈E, there exists a positive integer T such that
(33)|u*(k)|≤ρ2∀k>T;
we have
(34)|f(k,u*(k))|≤ϵq(k)|u*(k)|p(x)-1∀k>T.
By (30), there exists ν1∈Z+ such that
(35)|uj(k)-u*(k)|≤ρ2∀j>ν1,k∈Z.
This, combined with (33) and (32), gives us
(36)|uj(k)|≤|uj(k)-u*(k)|+|u*(k)|≤ρ∀j>ν1,k>T,|f(k,uj(k))|≤ϵq(k)|uj(k)|p(k)-1∀j>ν1,k>T.
Then for j>ν1,
(37)∑|k|>T‍|f(k,uj(k))-f(k,u*(k))||uj(k)-u*(k)|≤ϵ∑|k|>T‍(q(k)|uj(k)|p(k)-1+q(k)|u*(k)|p(k)-1)×(|uj(k)|+|u*(k)|)≤ϵ∑k∈Z‍q(k)1/r(k)|uj(k)|p(k)-1q(k)1/p(k)|uj(k)|+ϵ∑k∈Z‍q(k)1/r(k)|uj(k)|p(k)-1q(k)1/p(k)|u*(k)|+ϵ∑k∈Z‍q(k)1/r(k)|u*(k)|p(k)-1q(k)1/p(k)|uj(k)|+ϵ∑k∈Z‍q(k)1/r(k)|u*(k)|p(k)-1q(k)1/p(k)|u*(k)|,
where 1/r(k)+1/p(k)=1 for all k∈Z.
Letv1={v1(k)} and v1(k)=q(k)1/r(k)|uj(k)|p(k)-1, v2={v2(k)} and v2(k)=q(k)1/p(k)|uj(k)|, h1={h1(k)} and h1(k)=q(k)1/r(k)|u*(k)|p(k)-1, and h2={h2(k)} and h2(k)=q(k)1/p(k)|u*(k)|.
It is easy to check thatv1,h1∈lr(k) and v2,h2∈lp(k). Then using Lemma 11, for j>ν1, we have
(38)∑|k|>T‍|f(k,uj(k))-f(k,u*(k))||uj(k)-u*(k)|≤(ϵp-+ϵr-)×(∥v1∥r(k)∥v2∥p(k)+∥v1∥r(k)∥h2∥p(k)hhhhhh+∥h1∥r(k)∥v2∥p(k)+∥h1∥r(k)∥h2∥p(k)).
Now we show that ∥v1∥r(k) is bounded. We may assume that ∥v1∥r(k)>1; otherwise, ∥v1∥r(k) is bounded obviously. By Lemmas 4 and 5, we have
(39)∥v1∥r(k)≤[∑k∈Z‍q(k)|uj(k)|p(k)]1/r-≤∥uj∥p+/r-≤M*p+/r-.
Let M1={1,M*p+/r-}; then ∥v1∥r(k)≤M1; that is, ∥v1∥r(k) is bounded. Using the similar arguments as above, we obtain that ∥v2∥p(k), ∥h1∥r(k), and ∥h2∥p(k) are bounded; that is, there exist three positive constants M2, M3, and M4 such that
(40)∥v2∥p(k)≤M2,∥h1∥r(k)≤M3,∥h2∥p(k)≤M4.
This, combined with (38), gives us
(41)∑|k|>T|f(k,uj(k))-f(k,u*(k))||uj(k)-u*(k)|≤ϵ(1p-+1r-)×(M1M2+M1M4+M3M2+M3M4)∀j>ν1.
By continuity of the finite sum and (30), there exists ν2∈Z+ such that
(42)∑|k|≤T|f(k,uj(k))-f(k,u*(k))|‍×|uj(k)-u*(k)|≤ϵ∀j≥ν2.
Let ν=max⁡{ν1,ν2}. Combining (41) and (42) together, we have
(43)∑k∈Z‍|f(k,uj(k))-f(k,u*(k))||uj(k)-u*(k)|≤ϵ(1p-+1r-)×(M1M2+M1M4+M3M2+M3M4)+ϵ∀j>ν.
Thus, (31) holds.
Finally, we show that{uj} possesses a convergent subsequence. Since I′(uj)→0 and uj⇀u*, it follows at once that
(44)lim⁡j→+∞〈I′(uj)-I′(u*),uj-u*〉=0.
This, combined with (31), gives us
(45)lim⁡j→+∞{∑k∈Z‍a(k)[|Δuj(k-1)|p(k)-2Δuj(k-1)hhhhhhhhhhhhh-|Δu*(k-1)|p(k)-2Δu*(k-1)|Δuj(k-1)|p(k)-2]×(Δuj(k-1)-Δu*(k-1))+∑k∈Z‍q(k)[|uj(k)|p(k)-2uj(k)hhhhhhhhhhhhhh-|u*(k)|p(k)-2u*(k)|Δuj(k-1)|p(k)-2]hhhhhhhhhh×(uj(k)-u*(k))∑k∈Z‍}=0.
The following two inequalities are taken from [20] and will play an important role in the proof of our main result:
(46)[(|ξ|p-2ξ-|η|p-2η)(ξ-η)]×(|ξ|+|η|)2-p≥|ξ-η|2,1<p<2,2p[(|ξ|p-2ξ-|η|p-2η)(ξ-η)]≥|ξ-η|p,p≥2,
for every ξ and η in R. We define
(47)Rj(k)=a(k)[|Δuj(k-1)|p(k)-2Δuj(k-1)hhhhhh-|Δu*(k-1)|p(k)-2Δu*(k-1)|Δuj(k-1)|p(k)-2]×(Δuj(k-1)-Δu*(k-1))∀k∈Z.Qj(k)=q(k)[|uj(k)|p(k)-2uj(k)-|u*(k)|p(k)-2u*(k)]×(uj(k)-u*(k))∀k∈Z.
This, combined with (45), produces at once
(48)lim⁡j→+∞∑k∈Z‍Rj(k)=0,lim⁡j→+∞∑k∈Z‍Qj(k)=0.
Now we show that φ(uj-u*)→0 as j→+∞. That is,
(49)lim⁡j→+∞∑k∈Z‍a(k)|Δuj(k-1)-Δu*(k-1)|p(k)=0,(50)lim⁡j→+∞∑k∈Z‍q(k)|uj(k)-u*(k)|p(k)=0.
Let us first prove (50). Since {uj} and u* are bounded in E, there exists a constant M*>1 such that φ(uj)≤M* and φ(u*)≤M* for all j∈Z+. We denote
(51)W1={k∈Z:1<p(k)<2},W2={k∈Z:p(k)≥2}.
By (46), we have
(52)∑k∈W2‍q(k)|uj(k)-u*(k)|p(k)≤∑k∈W2‍2p+q(k)(|uj(k)|p(k)-2uj(k)hhhhhhhhhhhhhhh-|u*(k)|p(k)-2u*(k)|uj(k)|p(k)-2)hhhhhhh×(uj(k)-u*(k))≤2p+∑k∈Z‍Qj(k),(53)∑k∈W1‍q(k)|uj(k)-u*(k)|p(k)=∑k∈W1‍q(k)[|uj(k)-u*(k)|2]p(k)/2≤∑k∈W1(Qj(k))p(k)/2‍×{q(k)[|uj(k)|+|u*(k)|]p(k)}(2-p(k))/2≤∑k∈W1‍(Qj(k))p(k)/2×[2p+q(k)(|uj(k)|p(k)+|u*(k)|p(k))](2-p(k))/2≤2p+(2-p-)/2×∑k∈Z(Qj(k))p(k)/2‍×[q(k)|uj(k)|p(k)+q(k)|u*(k)|p(k)](2-p(k))/2.
Let(54)p1(k)=2p(k),r1(k)=22-p(k),vj*(k)=(Qj(k))p(k)/2,wj*(k)=[q(k)|uj(k)|p(k)+q(k)|u*(k)|p(k)](2-p(k))/2.
Then it is easy to check that vj*={vj*(k)}∈lp1(k) and wj*={wj*(k)}∈lr1(k). By Lemma 11 and (53), we have
(55)∑k∈W1‍q(k)|uj(k)-u*(k)|p(k)≤2p+(2-p-)/2(1+p+2-p-2)∥vj*∥p1(k)∥wj*∥r1(k).
Since
(56)lim⁡j→+∞∑k∈Z‍Qj(k)=0,∑k∈Z‍[q(k)|uj(k)|p(k)+q(k)|u*(k)|p(k)]≤φ(uj)+φ(u*)≤2M*.
It is easy to see that ∥vj*∥p1(k)→0 as j→+∞ and ∥wj*∥r1(k) is bounded for all j∈Z+. This, combined with (52) and (55), gives us
(57)lim⁡j→+∞∑k∈Z‍q(k)|uj(k)-u*(k)|p(k)=lim⁡j→+∞∑k∈W1‍q(k)|uj(k)-u*(k)|p(k)+lim⁡j→+∞∑k∈W2‍q(k)|uj(k)-u*(k)|p(k)=0.
Using the similar arguments, we have (49). So φ(uj-u*)→0 as j→+∞. By Lemma 5, it follows that ∥uj-u*∥→0 as j→+∞, and the proof is completed.Proof of Theorem1.
The proof consists of two steps.
Step  1. We use Lemma 9 to show that (1) has a nontrivial solution in E.
First we prove thatI satisfies (J1) of Lemma 9. It follows from (32) that
(58)|F(k,u(k))|≤12p+q(k)|u(k)|p(k),
for |u(k)|≤ρ and k∈Z. Then, let δ=q*(1/p-)ρ(p+/p-)<1, for all u∈∂Bδ∩E, we have |u(k)|≤ρ and
(59)I(u)=∑k∈Z‍[a(k)p(k)|Δu(k-1)|p(k)+q(k)p(k)|u(k)|p(k)]-∑k∈Z‍F(k,u(k))≥∑k∈Z‍1p+[a(k)|Δu(k-1)|p(k)+q(k)|u(k)|p(k)]-∑k∈Z‍12p+q(k)|u(k)|p(k)≥∑k∈Z‍12p+[a(k)|Δu(k-1)|p(k)+q(k)|u(k)|p(k)]≥12p+∥u∥p+=12p+δp+>0,
and hence I satisfies (J1) of Lemma 9.
Next, we prove thatI satisfies (J2) of Lemma 9. Let e={e(k)}∈E and
(60)e(k)={0,ifk≠0,1,ifk=0.
Then
(61)F(k,e(k))={0,ifk≠0,F(0,1),ifk=0.
By Lemma 7, for s>1, we have
(62)∑k∈Z‍F1(k,se(k))=F1(0,s)≥sβF1(0,1),F1(0,1)>0,∑k∈Z‍F2(k,se(k))=F2(0,s)≤sτF2(0,1),F2(0,1)≥0.
Then
(63)I(se)=∑k∈Z‍[a(k)p(k)|sΔe(k-1)|p(k)+q(k)p(k)|se(k)|p(k)]-∑k∈Z‍[F1(k,se(k))-F2(k,se(k))]≤a(0)p(0)sp(0)+a(1)p(1)sp(1)+q(0)p(0)sp(0)-sβF1(0,1)+sτF2(0,1).
Since F1(0,1)>0 and p(0),p(1), τ are smaller than β, we can choose s* large enough such that I(s*e)<0. So we have verified all assumptions of Lemma 9; we know that I possesses a critical value α≥(1/2p+)δp+>0, where
(64)α=inf⁡h∈Γmax⁡s∈[0,1]I(h(s)),Γ={h∈C([0,1],E):h(0)=0,h(1)=s*e}.
A critical point u* of I corresponding to α is nonzero as α>0.
Step  2. We prove that (1) has a ground state in E.
Let(65)K={u∈E:I′(u)=0,u≠0}
be the critical set of I. Obviously, K is a nonempty set. Denote
(66)c~=inf⁡{I(u):u∈K}.
Since u∈K, we have
(67)I(u)=I(u)-1β〈I′(u),u〉≥(1p+-1β)φ(u)≥0.
Then 0≤c~≤I(u).
Suppose that{uj}⊂K such that I(uj)→c~. Obviously, {uj} is a P.S. sequence. By Lemma 12, we can choose a subsequence, still denoted by {uj}, such that
(68)lim⁡j→∞∥uj-u0∥=0,u0∈E.
Then I(uj)→I(u0) and I(u0)=c~. Now we prove that u0 is nonzero. If u0=0, then there exists a positive integer W such that for all j>W, we have
(69)∥uj∥≤q*(1/p-)ρ(p+/p-)<1.
By (32), it follows that
(70)〈I′(uj),uj〉=∑k∈Z‍[a(k)|Δuj(k-1)|p(k)+q(k)|uj(k)|p(k)-f(k,uj(k))uj(k)a(k)|Δuj(k-1)|p(k)]≥∑k∈Z‍[|uj(k)|p(k)a(k)|Δuj(k-1)|p(k)+q(k)|uj(k)|p(k)]-12∑k∈Z‍q(k)|uj(k)|p(k)≥12φ(uj)≥12∥uj∥p+>0,
which is in contradiction to I′(uj)=0. Thus, u0 is the ground state solution of (1). The proof is completed.



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*Source: 808102-2014-04-13.xml*