On a nonlinear and non-homogeneous problem without A–Rtype condition in Orlicz–Sobolev spaces N.T.. It should be noticed that if aðjtjÞ ¼ jtjp2 ; t 2 R; p > 1 then we obtain the well-kno
Trang 1On a nonlinear and non-homogeneous problem without (A–R)
type condition in Orlicz–Sobolev spaces
N.T Chunga,b,⇑, H.Q Toanc
a
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
b
Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam
c
Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam
a r t i c l e i n f o
Keywords:
Nonlinear and non-homogeneous problems
Orlicz–Sobolev spaces
Existence
Multiplicity
Variational methods
a b s t r a c t
Using variational methods, we prove some existence and multiplicity results for a class of nonlinear and non-homogeneous problems without (A–R) type condition in Orlicz–Sobolev spaces
Ó 2013 Elsevier Inc All rights reserved
1 Introduction and preliminaries
LetXbe a bounded domain in RNðN P 3Þ with smooth boundary @X Assume that a : ð0; 1Þ ! R is a function such that the mappingu:R! R, defined by
uðtÞ :¼ aðjtjÞt for t – 0;
0; for t ¼ 0
is an odd, increasing homeomorphisms from R onto R
In this article, we are concerned with a class of nonlinear and non-homogeneous problems in Orlicz–Sobolev spaces of the form
div aðjð rujÞruÞ ¼ f ðx; uÞ inX;
u ¼ 0 on @X;
ð1:1Þ
where f :X R ! R is a continuous function satisfying some suitable conditions
It should be noticed that if aðjtjÞ ¼ jtjp2
; t 2 R; p > 1 then we obtain the well-known p-Laplace operator
Dpu ¼ divðjrujp2ruÞ and problem(1.1)becomes
Dpu ¼ f ðx; uÞ inX;
u ¼ 0 on @X:
ð1:2Þ
Since Ambrosetti and Rabinowitz proposed the mountain pass theorem in 1973 (see[2]), critical point theory has become one of the main tools for finding solutions to elliptic problems of variational type Especially, elliptic problems of type (1.2)have been intensively studied for many years One of the very important hypotheses usually imposed on the nonlin-earities is the following Ambrosetti–Rabinowitz type condition ((A–R) type condition for short): There existsl>p such that
0096-3003/$ - see front matter Ó 2013 Elsevier Inc All rights reserved.
http://dx.doi.org/10.1016/j.amc.2013.02.011
⇑ Corresponding author at: Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam E-mail addresses: ntchung82@yahoo.com (N.T Chung), hq_toan@yahoo.com (H.Q Toan).
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Applied Mathematics and Computation
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a m c
Trang 20 <lFðx; tÞ :¼
Z t
0
for all x 2Xand t 2 R n f0g This condition ensures that the energy functional associated to the problem satisfies the Palais– Smale condition ((PS) condition for short) Clearly, if the condition (A–R) is satisfied then there exist two positive constants
d1; d2such that
Fðx; tÞ P d1jtjl d2; 8ðx; tÞ 2X R:
This means that f is p-superlinear at infinity in the sense that
lim
jtj!þ1
Fðx; tÞ
jtjp ¼ þ1:
In recent years, there have been some authors considering p-superlinear problems of type(1.2)without the (A–R) type con-dition, we refer to some interesting papers on this topic[6,7,10,14,16,17,20,23]and the references cited there
Our aim in this paper is to develop the ideas by Miyagaki et al.[20]and Li et al.[16]to a class of nonlinear and non-homo-geneous problems of type(1.1)in Orlicz–Sobolev spaces We assume that f isu0-superlinear at infinity (see the condition ðf2Þ
in Section2) but does not satisfy the (A–R) type condition(1.3)as in[8,12] To overcome the difficulties brought, we shall use the mountain pass theorem in[10]and the fountain theorem in[24]with the ðCcÞ condition (seeDefinition 2.5) Our situ-ation here is different from one’s introduced in the works[15,18], in which the authors consider problem(1.1)in the case when f isu0-sublinear at infinity
In order to study problem(1.1), let us introduce the functional spaces where it will be discussed We will give just a brief review of some basic concepts and facts of the theory of Orlicz and Orlicz–Sobolev spaces, useful for what follows, for more details we refer the readers to the books by Adams[1], Rao and Ren[21], the papers by Clément et al.[8,9], Donaldson[11], Gossez et al.[13], Miha˘ilescu et al.[18,19]and Cammaroto et al.[5]
Foru:R! R introduced at the start of the paper, we define
UðtÞ ¼
Z t
0
uðsÞds; 8t 2 R:
We can see thatUis a Young function, that is,Uð0Þ ¼ 0; Uis convex, and limt!1UðtÞ ¼ þ1 Furthermore, sinceUðtÞ ¼ 0 if and only if t ¼ 0; limt!0UðtÞt ¼ 0, and limt!1UðtÞt ¼ þ1, the functionUis then called an N-function The functionUdefined by the formula
UðtÞ ¼
Zt
0
u1ðsÞds for all t 2 R
is called the complementary function ofUand it satisfies the condition
UðtÞ ¼ supfst UðsÞ : s P 0g for all t P 0:
We observe that the functionUis also an N-function in the sense above and the following Young inequality holds
st 6UðsÞ þUðtÞ for all s; t P 0:
The Orlicz class defined by the N-functionUis the set
KUðXÞ :¼ u :X! R measurable :
Z
X
UðjuðxÞjÞdx < 1
and the Orlicz space LUðXÞ is then defined as the linear hull of the set KUðXÞ The space LUðXÞ is a Banach space under the following Luxemburg norm
kukU:¼ inf k > 0 :
Z
X
U uðxÞ k
dx 6 1
or the equivalent Orlicz norm
kukLU:¼ sup
Z
X
uðxÞvðxÞdx
:v2 KU ðXÞ;
Z
X
UðjvðxÞjÞdx 6 1
:
For Orlicz spaces, the Hölder inequality reads as follows (see[21]):
Z
X
uvdx 6 2kukLUðXÞkukL
U ðXÞ for all u 2 LUðXÞ and v2 LU ðXÞ:
The Orlicz–Sobolev space W1LUðXÞ building upon LUðXÞ is the space defined by
W1LUðXÞ :¼ u 2 LUðXÞ :@u
@xi
2 LUðXÞ; i ¼ 1; 2; ; N
Trang 3and it is a Banach space with respect to the norm
kuk1;U:¼ kukUþ kjrujkU:
Now, we introduce the Orlicz–Sobolev space W1LUðXÞ as the closure of C10ðXÞ in W1LUðXÞ It turns out that the space
W1LUðXÞ can be renormed by using as an equivalent norm
kuk :¼ kjrujkU:
For an easier manipulation of the spaces defined above, we define the numbers
u0:¼ inf
t>0
tuðtÞ
UðtÞ and u0:¼ sup
t>0
tuðtÞ
UðtÞ:
Throughout this paper, we assume that
1 <u06tuðtÞ
which assures thatUsatisfies theD2-condition, i.e.,
where K is a positive constant, see[19, Proposition 2.3]
In this paper, we also need the following condition
the function t #Uð ffiffi
t
p
Þ is convex for all t 2 ½0; 1Þ: ð1:6Þ
We notice that Orlicz–Sobolev spaces, unlike the Sobolev spaces they generalize, are in general neither separable nor reflexive A key tool to guarantee these properties is represented by theD2-condition(1.5) Actually, condition(1.5)assures that both LUðXÞ and W1LUðXÞ are separable, see[1] Conditions(1.5) and (1.6)assure that LUðXÞ is a uniformly convex space and thus, a reflexive Banach space (see[19]); consequently, the Orlicz–Sobolev space W1LUðXÞ is also a reflexive Banach space We also find that with the help of condition(1.4), the Orlicz–Sobolev space W1LUðXÞ is continuously embedded in the classical Sobolev space W1;u0
0 ðXÞ, as a result, W1LUðXÞ is continuously and compactly embedded in the classical Lebesgue space LqðXÞ for all 1 6 q <u
0, where
u
0:¼
N u0
N u 0 ifu0<N;
þ1 ifu0PN:
(
The following lemma plays an essential role in our arguments
Proposition 1.1 (see[5,18,19]) Let u 2 W1LUðXÞ Then we have
(i) kuku06R
XUðjruðxÞjÞdx 6 kuku0if kuk < 1,
(ii) kuku06R
XUðjruðxÞjÞdx 6 kuku0if kuk > 1
2 Main results
In this section, we state and prove the main result of this paper Let us introduce the following hypotheses:
ðf0Þ f :X R ! R is a continuous function and satisfies the subcritical growth condition
jf ðx; tÞj 6 Cð1 þ jtjq1Þ; 8ðx; tÞ 2X R;
whereu0<q <u
0and C is a positive constant;
ðf1Þ f ðx; tÞ ¼ oðjtju01Þ; t ! 0, uniformly a.e x 2X;
ðf2Þ limjtj!þ1Fðx;tÞ
jtju0 ¼ þ1 uniformly a.e x 2X, i.e., f isu0-superlinear at infinity;
ðf3Þ There exists a constantl1>0 such that
Gðx; tÞ 6 Gðx; sÞ þl1
for any x 2X; 0 < t < s or s < t < 0, where Gðx; tÞ :¼ tf ðx; tÞ u0Fðx; tÞ and Fðx; tÞ :¼Rt
0f ðx; sÞds;
ðf4Þ f ðx; tÞ ¼ f ðx; tÞ for all ðx; tÞ 2X R
It should be noticed that the condition ðf3Þ is a consequence of the following condition, which was firstly introduced by Miyagaki et al.[20]for problem(1.2)in the case p ¼ 2 and developed by Li et al.[16]in the case when p > 1 is arbitrary:
ðf0
3Þ There exists t0>0 such that f ðx;tÞ
u 0 2 is nondecreasing in t P t0and nonincreasing in t 6 t0for any x 2X
Trang 4The readers may consult the proof and comments on this assertion in the papers by Li et al.[16]or Miyagaki et al.[20]and the references cited there
In order to prove the energy functional verifying the ðCcÞ condition, we assume that the functionsuandUsatisfy the following condition:
ðHÞ There exists a positive constantl2such that
HðtsÞ 6 HðtÞ þl2
for all t P 0 and s 2 ½0; 1, where HðtÞ ¼u0UðtÞ uðtÞt
Before stating and proving the main results of this paper, we give some examples of functionsu:R! R which are odd, increasing homeomorphism from R onto R and satisfy conditions(1.4) and (1.6)and ðHÞ, the readers can find them in[5,18] Example 2.1
(1) LetuðtÞ ¼ pjtjp2t; t 2 R; p > 1 A simple computation shows thatu0¼u0¼ p In this case, the corresponding Orlicz space LUðXÞ is the classical Lebesgue space LpðXÞ while the Orlicz–Sobolev space W1LUðXÞ is the classical Sobolev space
W1;p0 ðXÞ We have HðtÞ ¼ 0 for all t 2 R and then the condition ðHÞ holds
(2) LetuðtÞ ¼ logð1 þ t2Þt; t 2 R Then we can deduce thatu0¼ 2 andu0¼ 4 Some simple computations show that the functionUis given by
UðtÞ ¼1
2ð1 þ t
2
Þ logð1 þ t2Þ t
2
2 ln 10:
Then
HðtÞ ¼u0UðtÞ uðtÞt ¼ ð2 þ t2Þ logð1 þ t2Þ 2t
2
ln 10P0 for all t P 0:
For each fixed t > 0, the function s 2 ½0; 1 # HðtsÞ is continuous with respect to s So, there exists s02 ½0; 1 such that Hðts0Þ ¼ maxs2½0;1HðtsÞ It is clear that s0–0 If s0¼ 1 then HðtsÞ 6 HðtÞ for all s 2 ½0; 1 and t P 0 If s02 ð0; 1Þ, since limt!þ1HðtsHðtÞ0Þ¼ s2<1 there exists t > 3 large enough such thatHðts0 Þ
HðtÞ <1 for all t > t or Hðts0Þ < HðtÞ for all t > t Now, set
l2:¼ 1 þ maxðt;sÞ2½0;t½0;1HðtsÞ we have HðtsÞ < HðtÞ þl2for all t P 0 and s 2 ½0; 1 and then the condition ðHÞ holds
Definition 2.2 A function u 2 W1LUðXÞ is said to be a weak solution of problem(1.1)if it holds that
Z
X
aðjrujÞru r vdx
Z
X
f ðx; uÞvdx ¼ 0
for allv2 W1LUðXÞ
Our main results in this paper are given by the following two theorems
Theorem 2.3 Assume that the conditions(1.4) and (1.6)and ðHÞ; ðf0Þ—ðf3Þ are satisfied Then problem(1.1)has a non-trivial weak solution
Theorem 2.4 Assume that the conditions(1.4) and (1.6)and ðHÞ; ðf0Þ; ðf2Þ—ðf4Þ are satisfied Then problem(1.1)has infinitely many weak solutions fukg satisfying
Z
X
UðjrukjÞdx
Z
X
Fðx; ukÞdx ! þ1; k ! 1:
OurTheorem 2.3is exactly an extension from the results Miyagaki et al.[20]and Li et al.[16]to problem(1.1)considered
in Orlicz–Sobolev spaces (note that in this paper, we do not use the parameter k as in[16,20]), while ourTheorem 2.4seems
to be new even in the special caseuðtÞ ¼ pjtjp2t, i.e the well-known problem with p-Laplace operatorDpu We emphasize that the extension from the p-Laplace operator to the differential operators involved in(1.1)is not trivial, since the new oper-ators have a more complicated structure than the p-Laplace operator, for example they are non-homogeneous By the pres-ence of the hypothesis ðf2Þ, it is clear that our results in this paper are also different from the earlier ones in the paper by Cammaroto et al.[5]since the authors required in[5]that f ðx; tÞ satisfies the following condition (see the condition ða1Þ
in[5, Theorem 3.1]):
max lim sup
jtj!0
supx2XFðx; tÞ jtju0 ; lim supjtj!þ1
supx2XFðx; tÞ jtju0
60:
Trang 5Moreover, the method for study of problem(1.1)in the paper[5]is essentially based on the three critical points theorem by
B Ricceri[22] Regarding the problem(1.1)with Neumann boundary conditions, we refer the readers to[3,4], in which the authors studied the multiplicity of weak solutions under the conditionu0>N In order to prove the main theorems, we re-call some useful concepts and results
Definition 2.5 Let ðX; k kÞ be a real Banach space, J 2 C1
ðX; RÞ We say that J satisfies the ðCcÞ condition if any sequence
fumg X such that JðumÞ ! c and kJ0ðumÞkð1 þ kumkÞ ! 0 as m ! 1 has a convergent subsequence
Proposition 2.6 (see [10]) Let ðX; k kÞ be a real Banach space, J 2 C1ðX; RÞ satisfies the ðCcÞ condition for any c > 0; Jð0Þ ¼ 0 and the following conditions hold:
(i) There exists a function / 2 X such that k/k >qand Jð/Þ < 0;
(ii) There exist two positive constantsqand R such that JðuÞ P R for any u 2 X with kuk ¼q
Then the functional J has a critical value c P R, i.e there exists u 2 X such that J0ðuÞ ¼ 0 and JðuÞ ¼ c
In order to proveTheorem 2.4we will use the following fountain theorem, see[24]for details Let ðX; k kÞ be a real reflex-ive Banach space presenting by X ¼ j2NXj with dimðXjÞ < þ1 for any j 2 N For each k 2 N, we set Yk¼ k
j¼0Xj and
Zk¼ 1
j¼kXj
Proposition 2.7 (see [24]) Let ðX; k kÞ be a real reflexive Banach space, J 2 C1ðX; RÞ satisfies the ðCcÞ condition for any c > 0 and
J is even If for each sufficiently large k 2 N, there existqk>rk>0 such that the following conditions hold:
(i) ak:¼ inffu2Z k :kuk¼r k gJðuÞ ! þ1 as k ! 1;
(ii) bk:¼ maxfu2Y k :kuk¼ q k gJðuÞ 6 0
Then the functional J has an unbounded sequence of critical values, i.e there exists a sequence fukg X such that J0ðukÞ ¼ 0 and JðukÞ ! þ1 as k ! þ1
In the rest of this paper we will use the letter X to denote the Orlicz–Sobolev space W1LUðXÞ Let us define the energy functional J : X ! R by the formula
JðuÞ ¼
Z
X
UðjrujÞ
Z
X
ByProposition 1.1and the continuous embeddings obtained from the hypothesis ðf0Þ, some standard arguments assure that the functional J is well-defined on X and J 2 C1
ðXÞ with the derivative given by
J0ðuÞðvÞ ¼
Z
X
aðjrujÞru r vdx
Z
X
f ðx; uÞvdx
for all u;v2 X, see for example [Lemma 4.2][19] Thus, weak solutions of problem(1.1)are exactly the critical points of the functional J
Lemma 2.8 Assume that the conditions ðf0Þ–ðf2Þ are satisfied Then we have the following assertions:
(i) There exists / 2 X; / > 0 such that Jðt/Þ ! 1 as t ! þ1;
(ii) There existq>0 and R > 0 such that JðuÞ P R for any u 2 X with kuk ¼q
Proof (i) From ðf2Þ, it follows that for any M > 0 there exists a constant CM¼ CðMÞ > 0 depending on M, such that
Fðx; tÞ P Mjtju0 CM; 8x 2X; 8t 2 R: ð2:2Þ
Take / 2 X with / > 0, from(2.2)andProposition 1.1we get
Jðt/Þ ¼
Z
X
Uðjrt/jÞdx
Z
X
Fðx; t/Þdx 6 kt/ku0 M
Z
X
jt/ju0dx þ CMjXj 6 tu0 k/ku0 M
Z
X
j/ju0dx
þ CMjXj; ð2:3Þ
where t > 1 is large enough to ensure that kt/k > 1, and jXj denotes the Lebesgue measure ofX From(2.3), if M is large enough such that
k/ku0 M
Z
X
j/ju0dx < 0;
then we have
Trang 6t!þ1Jðt/Þ ¼ 1;
which ends the proof of (i)
(ii) Since the embeddings X,!Lu0ðXÞ and X,!LqðXÞ are continuous, there exist constants C1;C2>0 such that
kukLu 0
ðXÞ6C1kuk; kukLq ðXÞ6C2kuk: ð2:4Þ
Let 0 << 1
2Cu0, where C1is given by(2.4) From ðf0Þ and ðf1Þ, we have
Fðx; tÞ 6jtju0þ CðÞjtjq; 8ðx; tÞ 2X R: ð2:5Þ
From(2.5), for all u 2 X with kuk < 1, we have
JkðuÞ ¼
Z
X
UðjrujÞdx
Z
X
Fðx; uÞdx P kuku0
Z
X
juju0dx CðÞ
Z
X
jujqdx P kuku0Cu10kuku0 CðÞCq2kukq
P 1
2 CðÞCq2kukqu0
where C2>0 is given by(2.4) From(2.6)and the fact that q >u0, we can choose R > 0 andq>0 such that JðuÞ P R > 0 for all u 2 X with kuk ¼q The proof ofLemma 2.8is complete h
Lemma 2.9 Assume that the conditions(1.4) and (1.6), ðHÞ; ðf0Þ, ðf2Þ–ðf3Þ are satisfied Then the functional J satisfies the ðCcÞ condition for any c > 0
Proof Let fumg X be a ðCcÞ sequence of the functional J, that is,
JðumÞ ! c; kJ0ðumÞkð1 þ kumkÞ ! 0 as m ! 1;
which shows that
c ¼ JðumÞ þ oð1Þ; J0
where oð1Þ ! 0 as m ! 1
We shall prove that the sequence fumg is bounded in X Indeed, if fumg is unbounded in X, we may assume that kumk ! 1
as m ! 1 We define the sequence fwmg by wm¼ um
ku m k; m ¼ 1; 2; It is clear that fwmg X and kwmk ¼ 1 for any m Therefore, up to a subsequence, still denoted by fwmg, we have fwmg converges weakly to w 2 X and
wm! w strongly in LqðXÞ; m ! 1; ð2:9Þ
wm! w strongly in Lu0ðXÞ; m ! 1: ð2:10Þ
LetX–:¼ fx 2X:wðxÞ – 0g If x 2X–then it follows from(2.8)that jumðxÞj ¼ jwmðxÞjkumk ! þ1 as m ! 1 Moreover, from
ðf2Þ, we have
lim
m!1
Fðx; umðxÞÞ
jumðxÞju0 jwmðxÞj
u 0
Using the condition ðf2Þ, there exists t0>0 such that
Fðx; tÞ
for all x 2Xand jtj > t0>0 Since Fðx; tÞ is continuous onX ½t0;t0, there exists a positive constant C3such that
for all ðx; tÞ 2X ½t0;t0 From(2.12) and (2.13)there exists C42 R such that
for all ðx; tÞ 2X R From(2.14), for all x 2Xand m, we have
Fðx; umðxÞÞ C4
kumku0 P0
or
Trang 7Fđx; umđxỡỡ
jumđxỡju0 jwmđxỡj
u 0
C4
kumku0P0; 8x 2X; 8m: đ2:15ỡ
By(2.7),Proposition 1.1we have
c Ử Jđumỡ ợ ođ1ỡ Ử
Z
X
Uđjrumjỡdx
Z
X
Fđx; umỡdx ợ ođ1ỡ P kumku0
Z
X
Fđx; umỡdx ợ ođ1ỡ
or
Z
X
Fđx; umỡdx P kumku0 c ợ ođ1ỡ ! ợ1 as m ! 1: đ2:16ỡ
We also have
c Ử Jđumỡ ợ ođ1ỡ Ử
Z
X
Uđjrumjỡdx
Z
X
Fđx; umỡdx ợ ođ1ỡ 6 kumku0
Z
X
Fđx; umỡdx ợ ođ1ỡ
or
kumku0P
Z
X
Fđx; umỡdx ợ c ođ1ỡ > 0 for m large enough: đ2:17ỡ
We claim that jXỜj Ử 0 In fact, if jXỜj Ờ 0, then by(2.11), (2.15) and (2.17)and the Fatou lemma, we have
ợ Ử đợ1ỡjXỜj
Ử
Z
X Ờ
lim inf
m!1
Fđx; umđxỡỡ
jumđxỡju0 jwmđxỡj
u 0
dx Z
X Ờ
lim sup
m!1
C4
kumku0dx
Ử
Z
X Ờ
lim inf
m!1
Fđx; umđxỡỡ
jumđxỡju0 jwmđxỡj
u 0
C4
kumku0
! dx
6lim inf
m!1
Z
X Ờ
Fđx; umđxỡỡ
jumđxỡju0 jwmđxỡj
u 0
C4
kumku0
! dx
6lim inf
m!1
Z
X
Fđx; umđxỡỡ
jumđxỡju0 jwmđxỡj
u 0
C4
kumku0
! dx
Ử lim inf
m!1
Z
X
Fđx; umđxỡỡ
kumku0 dx lim supm!1
Z
X
C4
kumku0dx
Ử lim inf
m!1
Z
X
Fđx; umđxỡỡ
kumku0 dx
6lim inf
m!1
R
XFđx; umđxỡỡdx R
XFđx; umỡdx ợ c ođ1ỡ:
đ2:18ỡ
From(2.16) and (2.18), we obtain
ợ1 6 1;
which is a contradiction This shows that jXỜj Ử 0 and thus wđxỡ Ử 0 a.e inX
Since Jđtumỡ is continuous in t 2 ơ0; 1, for each m there exists tm2 ơ0; 1, m Ử 1; 2; , such that
Jđtmumỡ :Ử max
It is clear that tm>0 and Jđtmumỡ P c > 0 Ử Jđ0ỡ Ử Jđ0:umỡ If tm<1 then d
dtJđtumỡjtỬtmỬ 0 which gives J0đtmumỡđtmumỡ Ử 0 If
tmỬ 1, then J0đumỡđumỡ Ử ođ1ỡ So we always have
J0
Let fRkg be a positive sequence of real numbers such that Rk>1 for any k and limk!1RkỬ ợ1 Then kRkwmk Ử Rk>1 for any
k and m Fix k, since wm! 0 strongly in the spaces LqđXỡ and Lu0đXỡ as m ! 1, using(2.5), we deduce that there exists a constant C5>0 such that
Z
X
Fđx; Rkwmỡdx 6 C5
Z
X
Rkjwmju0dx ợ C5
Z
X
Rqkjwmjqdx ! 0 as m ! 1;
which yield
lim
m!1
Z
X
Trang 8Since kumk ! 1 as m ! 1, we also have kumk > Rkor 0 < R k
ku m k<1 for m large enough Hence, using(2.21),Proposition 1.1, it follows that
JðtmumÞ P J Rk
kumkum
¼ JðRkwmÞ ¼
Z
X
UðjrRkwmjÞdx
Z
X
Fðx; RkwmÞdx P kRkwmku0
Z
X
Fðx; RkwmÞdx
¼ Ru0
Z
X
Fðx; RkwmÞdx P1
2
u 0
for any m large enough From(2.22), letting m; k ! 1 we have
lim
On the other hand, using the conditions ðf3Þ; ðHÞ and relations(1.4) and (2.7), for all m large enough, we have
JðtmumÞ ¼ JðtmumÞ 1
u0J0ðtmumÞðtmumÞ þ oð1Þ
¼
Z
X
UðjrtmumjÞdx
Z
X
Fðx; tmumÞdx 1
u0
Z
X
aðjrtmumjÞjrtmumj2dx þ 1
u0
Z
X
f ðx; tmumÞtmumdx þ oð1Þ
6 1
u0
Z
X
HðtmjrumjÞdx þ 1
u0
Z
X
Gðx; tmumÞdx þ oð1Þ
6 1
u0
Z
X
HðjrumjÞ þl2
dx þ 1
u0
Z
X
Gðx; umÞ þl1
dx þ oð1Þ
¼
Z
X
UðjrumjÞdx
Z
X
Fðx; umÞdx 1
u0
Z
X
aðjrumjÞjrumj2dx
Z
X
f ðx; umÞumdx
þl1þl2
u0 jXj þ oð1Þ
¼ JðumÞ 1
u0J0ðumÞðumÞ þl1þl2
u0 jXj þ oð1Þ ! c þl1þl2
u0 jXj as m ! 1: ð2:24Þ
From(2.23) and (2.24)we obtain a contradiction This shows that the sequence fumg is bounded in X
Now, by conditions(1.4) and (1.6), the Banach space X is reflexive Thus, there exists u 2 X such that passing to a subsequence, still denoted by fumg, it converges weakly to u in X and converges strongly to u in the space LqðXÞ Using the condition ðf0Þ and the Hölder inequality, we have
Z
X
f ðx; umÞðum uÞdx
Z
X
jf ðx; umÞjjum ujdx 6 C
Z
X
ð1 þ jumjq1Þjum ujdx 6 C 1 þ kumkq1Lq
ðXÞ
kum ukL q
ðXÞ
! 0 as m ! 1;
which yield
lim
m!1
Z
X
From(2.7) and (2.25)we get
lim
m!1
Z
X
aðjrumjÞrumrðum uÞdx ¼ 0: ð2:26Þ
By(2.26)and the fact that fumg converges weakly to u in X, we can apply the result of Miha˘ilescu[18, Lemma 5]in order to deduce that the sequence fumg converges strongly to u in X Therefore, the functional J satisfies the ðCcÞ condition for any
c > 0 The proof ofLemma 2.9is complete
Proof ofTheorem 2.3 ByLemmas 2.8 and 2.9, the functional J satisfies all the assumptions of the mountain pass theorem Therefore, the functional J has a critical value c P R > 0 Hence, problem(1.1)has at least one non-trivial weak solution in X Next, because X is a reflexive and separable Banach space, there exist fejg X and fe
jg Xsuch that
X ¼ spanfej:j ¼ 1; 2; ; g; X¼ spanfe
j :j ¼ 1; 2; ; g
and
ei;e
j
D E
¼ 1; if i ¼ j;
0; if i – j:
For convenience, we write Xj¼ spanfejg and define for each k 2 N the subspaces Yk¼ k
j¼1Xjand Zk¼ 1
j¼kXj The following result is useful for our arguments
Trang 9Lemma 2.10 Ifu0<q <u
0then we have
ak:¼ sup kukLq ðXÞ:kuk ¼ 1; u 2 Zk
! 0 as k ! 1:
Proof Obviously, for any k 2 N; 0 <akþ16ak, soak!aP0 as k ! 1 Let uk2 Zk; k ¼ 1; 2; satisfy
kukk ¼ 1 and 0 6ak kukkL q ðXÞ<1
Then there exists a subsequence of fukg, still denoted by fukg such that fukg converges weakly to u in X and
e
j;u
D E
¼ lim
k!1 e
j;uk
D E
; j ¼ 1; 2;
Since Zkis a closed subspace of X, by Mazur’s theorem, we have u 2 Zkfor any k Consequently, we get u 2 \1
k¼1Zk¼ f0g, and
so fukg converges weakly to 0 in X as k ! 1 Sinceu0<q <u
0, the embedding X,!LqðXÞ is compact, then fukg converges strongly to 0 in Lq
ðXÞ Hence, by(2.27), we have limk!1ak¼ 0 h
Lemma 2.11 Assume that the conditions ðf0Þ and ðf2Þ are satisfied Then there existqk>rk>0 such that
(i) ak:¼ inffu2Z k :kuk¼r k gJðuÞ ! þ1 as k ! 1;
(ii) bk:¼ maxfu2Y k :kuk¼ q k gJðuÞ 6 0
Proof (i) By ðf0Þ, there exists C6>0 such that
for all ðx; tÞ 2X R; u0<q <u
0 From(2.28), for any u 2 Zkwith kuk > 1 we have
JðuÞ ¼
Z
X
UðjrujÞdx
Z
X
Fðx; uÞdx P kuku0 C6
Z
X
jujqdx C6
Z
X
jujdx P kuku0 C7aq
kkukq C8kuk; ð2:29Þ
where
ak:¼ sup kukL q
ðXÞ:kuk ¼ 1; u 2 Zk
:
Now, for any u 2 Zkwith kuk ¼ rk¼ ð2C7aq
kÞu 0 q1 we have
JðuÞ P kuku0 C7aq
kkukq C8kuk ¼ 2C7aq
k
u0
u0 q C7aq
k 2C7aq k
q
u0 q C8 2C6aq
k
1
u0 q
¼1
2 2C7aq
k
u0
u0 q C8 2C7aq
k
1
u0 q¼1
2r
u0
ByLemma 2.10,ak! 0 as k ! 1 andu
0>q >u0>u0>1 we have that rk! þ1 as k ! 1 Therefore, by(2.30)it follows that ak! þ1 as k ! 1
ðiiÞ By(2.2), for any w 2 Ykwith kwk ¼ 1 and t > 1, we have
JðtwÞ ¼
Z
X
UðjrtwjÞdx
Z
X
Fðx; twÞdx 6 ktwku0 M
Z
X
jtwju0dx þ CMjXj
¼ tu0 kwku0 M
Z
X
jwju0dx
It is clear that we can choose M > 0 large enough such that
kwku0 M
Z
X
jwju0dx < 0:
For this choice, it follows from(2.31)that
lim
t!þ1JðtwÞ ¼ 1:
Hence, there exists t > rk>1 large enough such that JðtwÞ 6 0 and thus, if we setqk¼ t we conclude that
bk:¼ max
fu2Y k :kuk¼ q k gJðuÞ 6 0:
The proof ofLemma 2.11is complete h
Trang 10Proof Proof ofTheorem 2.4 ByLemma 2.11, the functional J satisfies all the assumptions of the fountain theorem By Lemma 2.9, the functional J satisfies the ðCcÞ condition for every c > 0 Moreover, from the condition ðf4Þ; J is even Hence,
we can apply the fountain theorem in order to obtain a sequence of critical points fukg X of J such that JðukÞ ! þ1 as
k ! 1 The proof ofTheorem 2.4is complete h
Acknowledgments
The authors thank the referees for their suggestions and helpful comments which improved the presentation of the ori-ginal manuscript This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED)
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