# Antiperiodic Solutions forp-Laplacian Systems via Variational Approach

Authors: Lifang Niu; Kaimin Teng
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We establish the existence of solutions forp-Laplacian systems with antiperiodic boundary conditions through using variational methods.


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## Body

## 1. Introduction

In this paper we investigate the existence of solutions of the following second orderp-Laplacian systems with antiperiodic boundary condition:
(1)-(|u′(t)|p-2u′(t))′=∇F(t,u(t))a.e.[0,T],u(0)=-u(T),u′(0)=-u′(T),
where T>0, F:[0,T]×RN→R (N≥1) satisfies the following fundamental assumption:(H)
F(t,x) is measurable in t for each x∈RN, continuous differentiable in x for almost every t∈[0,T], and there exists a∈C(R+,R+) and b∈L1(0,T;R+) such that
(2)|F(t,x)|≤a(|x|)b(t),|∇F(t,x)|≤a(|x|)b(t)for all x∈RN and a.e. t∈[0,T].In the last few decades, the following second order systems involving periodic boundary condition:(3)-u¨=∇F(t,u(t))a.e.[0,T],u(0)=u(T),u˙(0)=u˙(T),
have acted as one of the mainstream research problems in the field of differential equation. Under various assumptions of the potential F(t,x), there have been lots of existence and multiplicity of results in the literatures by using the tool of nonlinear analysis, such as degree theory, minimax methods, and Morse theory. Here we do not even try to review the huge bibliography, but we only list some references for our purpose; for example, we refer the readers to see [1–6] and the references therein.Comparing problem (1) with problem (3), we observe that the only difference is the boundary conditions. In order to use the variational methods, one of the main difficulties is the variational principle. For this matter, we try to modify some work space such that the variational principle can be established. Thanks to the work of Tian and Henderson [7], we borrow their ideas to give the variational principle for problem (1).Note that the study of antiperiodic solutions for nonlinear differential systems of the form(4)-(|u′(t)|p-2u′(t))′=∇F(t,u(t))a.e.t∈R
is closely related to the study of its periodic solutions. Indeed, if we assume that F(t+T,x)=F(t,x) and ∇F(t,-x)=-∇F(t,x), then let
(5)v(t)={u(t),t∈[0,T],-u(t-T),t∈[T,2T],
and we get that v is a solution of systems (4) with conditions v(0)=v(2T) and v′(0)=v′(2T). Since F(t,x) is 2T-periodic in t, v can be extended to be 2T-periodic over R, and hence v is a 2T-periodic solution of systems (4).In this paper, we will establish the existence of solutions for problem (1) by variational method. As far as we know, there are few papers studying the second order systems with antiperiodic boundary conditions by variational methods.This paper is organized as follows. In Section2, we introduce a variational principle for problem (1). In Section 3, we prove our main results.


## 2. Variational Principle

In the sequel, we denote by∥·∥p the Lp-norm (1≤p≤∞). Let C∞(0,T;RN) be the space of infinitely differentiable functions from [0,T] into RN. Let X={u∈C∞(0,T;RN),u(0)=-u(T)}; obviously, cos⁡((2k+1)π/T)t and sin⁡(((2k+1)π/T)t+π/2) (k=0,1,2,…) belong to X, and thus X≠∅.The following fundamental lemma proved in [7] is essential to establish the variational principle.Lemma 1.
Letu,v∈L1(0,T;RN). If, for every f∈X,
(6)∫0T‍(u(t),f′(t))dt=-∫0T‍(v(t),f(t))dt,
where (·,·) denotes the inner product on RN, then
(7)2T∫0T‍[u(t)+tv(t)]dt=∫0T‍v(t)dt
and there exists c∈RN such that
(8)u(t)=∫0t‍v(s)ds+ca.e.on⁡[0,T].Remark 2.
From Lemma1, we have the following facts.(i)
A functionv satisfying (6) is called a weak derivative of u. By a Fourier series argument, the weak derivative, if it exists, is unique. We denote by u′ the weak derivative of u.(ii)
We will identify the equivalence classu and its continuous representation
(9)u^(t)=∫0t‍u′(s)ds+c.(iii)
Equations (7) and (8) imply that u(0)=-u(T)=c. For this matter, we only show that u(T)+c=0. Indeed, using integration by parts, we have
(10)u(T)=∫0T‍v(s)ds+c=2T∫0T‍[u(t)+tv(t)]dt+c=2T∫0T‍u(t)dt+2∫0T‍v(t)dt-2T∫0T‍∫0t‍v(s)dsdt+c=2T∫0T‍u(t)dt+2∫0T‍v(t)dt-2T∫0T‍u(t)dt+3c=2u(T)+c.By (9), we have
(11)u(t)=u(τ)+∫τt‍u′(s)ds,fort,τ∈[0,T].(iv)
Ifu′ is continuous on [0,T], then by (9) u′ is the classical derivative of u=u^.(v)
It follows from (9) and Rademacher theorem that u′ is the classical derivative of u a.e. on [0,T].The Sobolev spaceW^T1,p is the space of functions u∈Lp(0,T;RN) having a weak derivative u′∈Lp(0,T;RN). Obviously, if u∈W^T1,p, u(t)=∫0tu′(s)ds+c and u(0)=-u(T)=c. The norm over W^T1,p is defined by
(12)∥u∥W^T1,p=(∫0T‍|u(t)|pdt+∫0T‍|u′(t)|pdt)1/p.
It is easy to see that W^T1,p is a reflexive Banach space and X⊂W^T1,p.With the proof of Lemma 3.3 in [7] and Theorem 8.8 in [8], we have the following embedding theorem.Lemma 3.
Let1/p+1/q=1(1<p<+∞). Then(i)
the embeddingW^T1,p↪Lq(0,T;RN) is compact;(ii)
there exits a constantc>0 such that
(13)∥u∥∞≤c∥u∥W^T1,p.Moreover, the embedding W^T1,p↪C(0,T;RN) is compact.Proof.
From Theorem 8.2 in [8], for u∈W^T1,p, u(t)=u(0)+∫0tu′(s)ds and u(t)=u(T)-∫tTu′(s)ds. By Hölder’s inequality, we have
(14)u(t)=12[u(0)+u(T)+∫0t‍u′(s)ds-∫tT‍u′(s)ds]≤12T1/q(∫0T‍|u′(t)|pdt)1/p.
On the other hand, letting B be a unit ball in W^T1,p, for u∈B, we have
(15)|u(t)-u(s)|=|∫st‍u′(τ)dτ|≤∥u′∥p|t-s|1/q≤|t-s|1/q,∀t,s∈(0,T).
It follows from the Ascoli-Arzelà theorem that B has a compact closure in C(0,T;RN).By Lemma1, the norm ∥·∥W^T1,p in W^T1,p is equivalent to the norm defined as
(16)∥u∥=(∫0T‍|u′(t)|pdt)1/p.
Indeed, by (14), one has
(17)∫0T‍|u(t)|pdt≤T∥u∥∞p≤T1+p/q2p∫0T‍|u′(t)|pdt.
Therefore, we have
(18)2T+2∥u∥W^T1,p≤∥u∥≤∥u∥W^T1,p,∀u∈W^T1,p
and the equivalence of the norm is proved.Remark 4.
There is some differences between antiperiodic boundary value problem and periodic boundary value problem. One is the norm on the work space, and the other is the decomposition of spaceW^T1,p not like WT1,p=RN⊕W~T1,p, where W~T1,p={u∈WT1,p:∫0Tu(t)dt=0}. And it is well known that the norms ∥u′∥p and ∥u∥WT1,p are equivalent to W~T1,p.The energy functionalφ:W^T1,p→R corresponding to problem (1) is defined by
(19)φ(u)=1p∫0T‍|u′(t)|pdt-∫0T‍F(t,u(t))dt.
Under the assumption of (H), we have the following.Lemma 5.
Let(H) hold. Then φ∈C1(W^T1,p,R) and
(20)〈φ′(u),v〉=∫0T‍|u′(t)|p-2(u′(t),v′(t))dt-∫0T‍(∇F(t,u(t)),v(t))dt
for every v∈W^T1,p. Moreover, if φ′(u)=0, then u is a solution of problem (1); that is, u∈W^T1,p satisfies the equation and antiperiodic condition in (1).Proof.
Similarly as the proof of Theorem 1.4 in [1], we obtain that φ∈C1(W^T1,p,R) and (28) holds. If
(21)0=〈φ′(u),v〉=∫0T‍|u′(t)|p-2(u′(t),v′(t))dt-∫0T‍(∇F(t,u(t)),v(t))dt=0,
for all v∈W^T1,p and hence for all v∈X, by Lemma 1, there exists a constant c∈RN such that
(22)-|u′(t)|p-2u′(t)=∫0t‍∇F(s,u(s))ds+c,a.e.on⁡[0,T].
From Remark 2, we know that |u′(t)|p-2u′(t) has a weak derivative
(23)-(|u′(t)|p-2u′(t))′=∇F(t,u(t))a.e.on⁡[0,T].
By Remark 2, we get
(24)|u′(0)|p-2u′(0)=-|u′(T)|p-2u′(T).
By (24), if u′(0)=u′(T)=0, clearly, we have u′(0)=-u′(T). If not, we see that u′(0)·u′(T)<0. Hence, by calculation, we get
(25)|u′(0)|=|u′(T)|.
This implies that u′(0)=-u′(T) and hence the conclusion is proved.Lemma 6.
The functionalφ:W^T1,p→R is weakly lower semicontinuous.Proof.
Assumingun⇀u in W^T1,p, then by (ii) of Lemma 3, we have un→u in C([0,T];RN). By hypothesis (H), we have
(26)|∫0T‍[F(t,un(t))-F(t,u(t))]dt|=|∫0T‍∫01‍(∇F(t,u(t)+s(un(t)-u(t))),un(t)-u(t)∇F(t,u(t)+s(un(t)-u(t))))dsdt∫0T‍∫01‍(∇F(t,u(t)+s(un(t)-u(t))),|≤∫0T‍∫01‍b(t)a(|u+s(un-u)|)|un-u|dsdt≤max⁡s∈[0,M]a(s)∥b∥L1∥un-u∥L∞⟶0,
where M>0 is a constant such that |un+s(un-u)|≤M for every n∈N, all s∈[0,1], and t∈[0,T]. It follows from the weak lower semicontinuity of the norm function in Banach space that
(27)liminf⁡n→∞∫0T‍|un′(t)|pdt≥∫0T‍|u′(t)|pdt.
Accordingly, the conclusion is completed.Next, we study the eigenvalue problem(28)-(|u′(t)|p-2u′(t))′=λ|u(t)|p-2u(t),t∈[0,T],u(0)=-u(T),u′(0)=-u′(T).Definition 7.
One saysλ∈R is an eigenvalue of problem (28) if there exists u∈W^T1,p, u≢0, such that
(29)∫0T‍(|u′(t)|p-2u′(t),v′(t))dt=λ∫0T‍(|u(t)|p-2u(t),v(t))dt,foreveryv∈W^T1,p.Lemma 8.
The eigenvalue problem (28) possesses the following properties:(a)
all the eigenvalues are positive real numbers;(b)
for eachu∈W^T1,p, one has λ1∫0T|u(t)|pdt≤∫0T|u′(t)|pdt, where λ1=inf⁡u∈W^T1,p,∥u∥Lp=1⁡∫0T|u′(t)|pdt.Proof.
(a) Lettingeλ(t) be the eigenfunction corresponding to the eigenvalue λ, we have
(30)∫0T‍|eλ′(t)|pdt=λ∫0T‍|eλ(t)|pdt.
Hence, λ≥0 and λ=0 if and only if eλ′(t)=0, implying that eλ(t)=C, a.e. on [0,T], where C is a constant. Since eλ∈W^T1,p (u(0)=-u(T)), we see that eλ(t)≡0, for all t∈[0,T]; this contradicts with the definition of eigenfunction.
(b) By standard minimization arguments, we can prove the conclusion. We omit it.Remark 9.
Note that the eigenvalue of one dimensional vectorp-Laplacian operator under antiperiodic boundary condition possesses some similar properties as the p-Laplacian operator -Δp under Dirichlet boundary condition on bounded domain.


## 3. Main Results and Proof

In this section, we will give some existence and multiplicity of results for problem (1).Theorem 10.
Assume thatF(t,u) satisfies hypothesis (H) and the following conditions.(F0)
There existf∈L1([0,T];R+) and g∈L1([0,T];R+) such that
(31)F(t,x)≤f(t)|x|α+g(t),∀x∈RN,a.e.t∈[0,T]with 0≤α<p. Then problem (1) has at least one solution in W^T1,p.Proof.
First we show thatφ is coercive on W^T1,p. In fact, from (F0) and Lemma 3, we have
(32)φ(u)=1p∫0T‍|u′(t)|pdt-∫0T‍F(t,u(t))dt≥1p∫0T‍|u′(t)|pdt-∥u∥L∞α∫0T‍f(t)dt-∫0T‍g(t)dt≥1p∫0T‍|u′(t)|pdt-C(∫0T‍|u′(t)|pdt)α/p-C.
Therefore, φ(u)→+∞ as ∥u∥→∞. As a result, we get a bounded minimizing sequence in W^T1,p. Combining Lemma 6, by Theorem 1.1 and Corollary 1.1 in [1], problem (1) has at least one solution which minimizes φ on W^T1,p.Remark 11.
By hypothesis(H), we see that F(t,x) is summable over t∈[0,T] in the neighborhood of zero, and thus the condition (F0) can be weaken to hold for |x| large.Ifα=p, we may assume the function f(t) in (F0) satisfies that
(33)∫0T‍f(t)dt<2ppTp-1,
as the same proof of Theorem 10, and we can get the following theorem.Theorem 12.
Under the above assumptions ofF(t,x), then problem (1) has at least one solution in W^T1,p.Theorem 13.
Assume thatF(t,u) satisfies hypothesis (H) and the following conditions:(F1)
lim⁡|x|→0⁡(F(t,x)/|x|p)=0, and lim⁡|x|→+∞⁡(F(t,x)/|x|p)=+∞ uniformly for a.e. t∈[0,T];(F2)
there exist constantsr>p and μ>r-p such that limsup⁡|x|→+∞⁡(F(t,x)/|x|r)<+∞ uniformly for a.e. t∈[0,T], and liminf⁡|x|→+∞⁡(((∇F(t,x),x)-pF(t,x))/|x|μ)>0 uniformly for a.e. t∈[0,T].
Then problem (1) possesses at least one nontrivial solution in W^T1,p.In order to prove Theorem13, we need the following results.Lemma 14.
Suppose(H), (F1), and (F2) hold. Then functional φ satisfies the (C)-condition; that is, for every sequence {un}⊂W^T1,p, {un} has a convergent subsequence if φ(un)is bounded and (1+∥un∥)∥φ′(un)∥→0 as n→∞.Proof.
Suppose{un}⊂W^T1,p, φ(un) is bounded, and (1+∥un∥)∥φ′(un)∥→0 as n→∞, and then there exists a constant M>0 such that
(34)|φ(un)|≤M,(1+∥un∥)∥φ′(un)∥≤M
for every n∈N. On one hand, by (F2), there exist constants C>0 and δ>0 such that
(35)F(t,x)≤C|x|r,∀|x|≥δ,a.e.t∈[0,T].
It follows from (H) and (35) that
(36)F(t,x)≤max⁡s∈[0,δ]a(s)b(t)+C|x|r,∀x∈RN,a.e.t∈[0,T].
Hence, by (34), (36), and Hölder inequality, we have
(37)1p∥un∥p=φ(un)+∫0T‍F(t,un(t))dt≤C∫0T‍|un(t)|rdt+C.
On the other hand, by (F2), there are constants C>0 and δ1>0 such that
(38)(∇F(t,x),x)-pF(t,x)≥C|x|μ,∀|x|≥δ1,a.e.t∈[0,T].
By (H), one has
(39)|(∇F(t,x),x)-pF(t,x)|≤Cb(t),∀|x|≤δ1,a.e.t∈[0,T],
where C=(p+δ1)  max⁡s∈[0,δ1]⁡a(s). Hence, from (38) and (39), we get
(40)(∇F(t,x),x)-pF(t,x)≥C(|x|μ-δ1μ)-Cb(t),
for all x∈RN and a.e. t∈[0,T]. Hence, by (34) and (40), one has
(41)(p+1)M≥pφ(un)-〈φ′(un),un〉=∫0T‍[(∇F(t,un(t)),un(t))-pF(t,un(t))]dt≥C∫0T‍|un(t)|μdt-CTδ1μ-C∫0T‍b(t)dt.
So {un} is bounded in Lμ(0,T;RN). If μ≥r, by (37) and Hölder inequality, it is easy to obtain that {un} is bounded in W^T1,p. If μ<r, by Lemma 3, we have
(42)∫0T‍|un(t)|rdt≤∥un∥L∞r-μ∫0T‍|un(t)|μdt≤C∥un∥r-μ∫0T‍|un(t)|μdt.
Hence, by (37) and ν>r-p, we obtain {un} is bounded in W^T1,p too. Thus, {un} is bounded in W^T1,p. Since W^T1,p is a reflexive Banach space, by Lemma 3, there exists a subsequence, still denoted by {un}, such that
(43)un⇀uinW^T1,p,un⟶uinC([0,T];RN),asn⟶∞.
Next, we will show that un→u in W^T1,p. Indeed, from (43) and hypothesis (H), it is easy to obtain that
(44)〈φ′(un)-φ′(u),un-u〉⟶0,asn⟶∞,∫0T‍(∇F(t,un(t))-∇F(t,u(t)),un(t)-u(t))dt⟶0,asn⟶∞.
Hence, by (44), we get
(45)∫0T‍(|un′(t)|p-2un′(t)-|u′(t)|p-2u′(t),un′(t)-u′(t))dt⟶0,asn⟶∞.
Therefore, it follows from (45) and Lemmas 3.2 and 3.3 in [9] that un→u in W^T1,p. The proof of Lemma 14 is completed.Proof of Theorem13.
By(F1), for every ɛ>0, there exists δ2>0 such that
(46)F(t,x)≤ɛ|x|p,∀|x|≤δ2,a.e.t∈[0,T].Combined with (35) and (46), we have
(47)F(t,x)≤ɛ|x|p+C|x|r,∀x∈RN,a.e.t∈[0,T].
Thus, by Lemma 3, for u∈W^T1,p, one has
(48)φ(u)=1p∫0T‍|u′(t)|pdt-∫0T‍F(t,u(t))dt≥(1p-ɛC)∥u∥p-C∥u∥r.
Since r>p, then there exist ρ>0 and α>0 such that
(49)φ(u)≥α,∀u∈W^T1,pwith∥u∥=ρ.
On the other hand, by (F1), for any M1>0, there exists δ3>0 such that
(50)F(t,x)≥M1|x|p,∀|x|≥δ3,a.e.t∈[0,T].
It follows from (H) that
(51)F(t,x)≤max⁡s∈[0,δ3]a(s)b(t),∀|x|≤δ3,a.e.t∈[0,T].
Therefore, we obtain
(52)F(t,x)≥M1(|x|p-δ3p)-max⁡s∈[0,δ3]a(s)b(t)
for all x∈RN and a.e. t∈[0,T].
Now, we choosee0∈W^T1,p being the eigenfunction corresponding to λ1 which is defined in Lemma 8. For ζ>0, from (52), we have
(53)φ(ζe0)=ζpp∫0T‍|e0′(t)|pdt-∫0T‍F(t,ζe0(t))dt≤ζpp∫0T‍|e0′(t)|pdt-M1ζp×∫0t‍|e0(t)|pdt+M1δ3pT+C=ζp(1p-M1λ1)∥e0∥p+M1δ3pT+C.
We choose M1=2(λ1/p), and the above inequality implies that
(54)φ(ζe0)⟶-∞,asζ⟶+∞.
In view of Lemma 14, (49), and (54), noting that φ(0)=0 and applying the mountain pass theorem under the (C)-condition, there exists a critical point u∈W^T1,p of φ, such that φ(u)≥α. Hence u is a nontrivial solution of problem (1), and this completes the proof.We can weaken the condition(F1) in the following condition (F1′):
(55)limsup⁡|x|→0pF(t,x)|x|p<λ1,liminf⁡|x|→+∞pF(t,x)|x|p>λ1
uniformly for a.e. t∈[0,T].Theorem 15.
Suppose thatF(t,x) satisfies hypotheses (H), (F1′), and (F2). Then problem (1) has at least one nontrivial solution in W^T1,p.Proof.
Checking the proof of Theorem13, we only need to verify that (49) and (54) hold. In fact, by (F1′), there exist two constants γ<λ1 and δ4>0 such that
(56)F(t,x)≤γp|x|p,∀|x|≤δ4,a.e.t∈[0,T].
From (35) and (56), we have
(57)F(t,x)≤γp|x|p+C|x|r,∀x∈RN,a.e.t∈[0,T].
Thus, for u∈W^T1,p, one has
(58)φ(u)=1p∫0T‍|u′(t)|pdt-∫0T‍F(t,u(t))dt≥1p(1-γλ1)∥u∥p-C∥u∥r.
Since r>p and 1-γ/λ1>0, then (49) holds.
By(F1′), there exist two constants γ1>λ1 and δ5>0 such that
(59)F(t,x)≥γ1p|x|p,∀|x|≥δ5,a.e.t∈[0,T].
Hence, by (H) and (59), we get(60)F(t,x)≥γ1p(|x|p-δ5p)-max⁡s∈[0,δ5]a(s)b(t)
for all x∈RN and a.e. t∈[0,T]. Therefore, (60) and γ1>λ1 imply that
(61)φ(ζe0)⟶-∞,asζ⟶+∞.
Hence, we complete the proof.Next, we consider the asymptotically quadratic case. For this purpose, we supposeF satisfies the following conditions:(F3)
limsup⁡|x|→+∞⁡(F(t,x)/|x|p)=+∞ uniformly for a.e. t∈[0,T];(F4)
there existsl(t)∈L1(0,T;R+) such that (∇F(t,x),x)-pF(t,x)≥l(t) for all x∈RN and a.e. t∈[0,T] and
(62)lim⁡|x|→+∞[(∇F(t,x),x)-pF(t,x)]=+∞uniformly for a.e. t∈[0,T].Theorem 16.
Suppose(H), (F1), (F3), and (F4) hold. Then problem (1) possesses at least one nontrivial solution in W^T1,p.Proof.
Paralleling to the proof of Theorem13, we only need to verify that φ satisfies the (C)-condition. Suppose {un}⊂W^T1,p, φ(un) is bounded, and (1+∥un∥)∥φ′(un)∥→0 as n→∞. Hence, it is easy to get that
(63)∫0T‍[(∇F(t,un(t)),un(t))-pF(t,un(t))]dt=pφ(un)-〈φ′(un),un〉≤C.
We next show that{un} is bounded in W^T1,p. If not, without loss of generality, we can assume that ∥un∥→+∞ as n→∞. Letting vn=un/∥un∥, then ∥vn∥=1, and so going to a sequence if necessary, we assume that vn⇀v in W^T1,p and vn→v in C(0,T;RN).
By(F3), there exist two constants a1>0 and R1>0 such that
(64)F(t,x)≤a1|x|p,∀|x|≥R1,a.e.t∈[0,T].
From (H) and (64), we get(65)F(t,x)≤a1|x|p+b(t)max⁡s∈[0,R1]a(s)
for all x∈RN and a.e. t∈[0,T]. Hence, by (65), one has
(66)φ(un)∥un∥p=1p-1∥un∥p∫0T‍F(t,un(t))dt≥1p-a1∫0T‍|vn(t)|pdt-C∥un∥p,
which implies that
(67)∥v∥Lp≥C>0
and thus v≢0. Therefore, there exists a subset E⊂[0,T] with meas⁡(E)>0 such that v(t)≠0 on E.
By(F4), from Lemma 2 in [5], then for ɛ=(1/2)meas⁡(E)>0, there exists a subset Eɛ⊂[0,T] with meas⁡([0,T]∖Eɛ)<ɛ such that
(68)lim⁡|x|→+∞[(∇F(t,x),x)-pF(t,x)]=+∞
uniformly for t∈Eɛ. Obviously, meas⁡(E∩Eɛ)>0. If not, we assume meas⁡(E∩Eɛ)=0. Since E=(E∩Eɛ)∪(E∖Eɛ), thus we have
(69)0<meas⁡(E)≤meas⁡(E∩Eɛ)+meas⁡([0,T]∖Eɛ)<ɛ=12meas⁡(E),
which leads to a contradiction. Hence, we have proved that
(70)un(t)⟶+∞,asn⟶∞,fora.e.t∈E∩Eɛ.
From(F4), we have
(71)∫0T‍[(∇F(t,un(t)),un(t))-pF(t,un(t))]dt=∫E∩Eɛ‍[(∇F(t,un(t)),un(t))-pF(t,un(t))]dt+∫[0,T]∖E∩Eɛ‍[(∇F(t,un(t)),un(t))-pF(t,un(t))]dt≥∫E∩Eɛ‍[(∇F(t,un(t)),un(t))-pF(t,un(t))]dt+∫[0,T]∖E∩Eɛ‍l(t)dt.
By (68) and (70), we get
(72)∫0T‍[(∇F(t,un(t)),un(t))-pF(t,un(t))]dt⟶+∞
as n→∞. This contradicts with (63). The proof is completed.



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*Source: 324518-2014-08-24.xml*