DSpace at VNU: On a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz-Sobolev spaces

Applied Mathematics and Computation 219 (2013) 7820–7829

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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc

On a nonlinear and non-homogeneous problem without (A–R)
type condition in Orlicz–Sobolev spaces
N.T. Chung a,b,⇑, H.Q. Toan c
a

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
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
b

a r t i c l e

i n f o

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.

Keywords:
Nonlinear and non-homogeneous problems
Orlicz–Sobolev spaces

Existence
Multiplicity
Variational methods

1. Introduction and preliminaries
Let X be a bounded domain in RN ðN P 3Þ with smooth boundary @ X. Assume that a : ð0; 1Þ ! R is a function such that
the mapping u : R ! R, defined by

uðtÞ :¼

&

aðjtjÞt
0;

for t – 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

&

ÀdivðaðjrujÞruÞ ¼ f ðx; uÞ in X;
u¼0

on @ X;

ð1:1Þ

where f : X Â R ! R is a continuous function satisfying some suitable conditions.
It should be noticed that if aðjtjÞ ¼ jtjpÀ2 ; t 2 R; p > 1 then we obtain the well-known p-Laplace operator
Dp u ¼ divðjrujpÀ2 ruÞ and problem (1.1) becomes

&

ÀDp u ¼ f ðx; uÞ in X;
u¼0

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 nonlinearities is the following Ambrosetti–Rabinowitz type condition ((A–R) type condition for short): There exists l > p such that

⇑ Corresponding author at: Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam.
E-mail addresses: (N.T. Chung), (H.Q. Toan).
0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved.
/>

N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

0 < lFðx; tÞ :¼

Z

7821

t

f ðx; sÞds 6 f ðx; tÞt

ð1:3Þ

0

for all x 2 X and 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 ; d2 such that

Fðx; tÞ P d1 jtjl À d2 ;

8ðx; tÞ 2 X Â R:

This means that f is p-superlinear at infinity in the sense that

lim

jtj!þ1

Fðx; tÞ
¼ þ1:
jtjp

In recent years, there have been some authors considering p-superlinear problems of type (1.2) without the (A–R) type condition, 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-homogeneous problems of type (1.1) in Orlicz–Sobolev spaces. We assume that f is u0 -superlinear at infinity (see the condition ðf2 Þ

in Section 2) 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 ðC c Þ condition (see Definition 2.5). Our situation 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 is u0 -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].
For u : R ! R introduced at the start of the paper, we define

UðtÞ ¼

Z

t

uðsÞds; 8t 2 R:

0

We can see that U is a Young function, that is, Uð0Þ ¼ 0; U is convex, and limt!1 UðtÞ ¼ þ1. Furthermore, since UðtÞ ¼ 0 if
and only if t ¼ 0; limt!0 UðtÞ
¼ 0, and limt!1 UðtÞ
¼ þ1, the function U is then called an N-function. The function UÃ defined by
t
t
the formula

UÃ ðtÞ ¼

Z

t

uÀ1 ðsÞds for all t 2 R

0

is called the complementary function of U and it satisfies the condition

UÃ ðtÞ ¼ supfst À UðsÞ : s P 0g for all t P 0:
We observe that the function UÃ is also an N-function in the sense above and the following Young inequality holds

st 6 UðsÞ þ UÃ ðtÞ for all s; t P 0:
The Orlicz class defined by the N-function U is the set

K U ðXÞ :¼

&
'
Z
u : X ! R measurable :
UðjuðxÞjÞdx < 1
X

and the Orlicz space LU ðXÞ is then defined as the linear hull of the set K U ðXÞ. The space LU ðXÞ is a Banach space under the
following Luxemburg norm

&

'

Z 
uðxÞ
kukU :¼ inf k > 0 :
U
dx 6 1
k
X
or the equivalent Orlicz norm


&Z
'
Z


kukLU :¼ sup  uðxÞv ðxÞdx : v 2 K UÃ ðXÞ;
UÃ ðjv ðxÞjÞdx 6 1 :
X

X

For Orlicz spaces, the Hölder inequality reads as follows (see [21]):

Z
X

uv dx 6 2kukLU ðXÞ kukLÃ ðXÞ
U

for all u 2 LU ðXÞ and

v 2 LU ðXÞ:
Ã

The Orlicz–Sobolev space W 1 LU ðXÞ building upon LU ðXÞ is the space defined by

W 1 LU ðXÞ :¼

&
'
@u
u 2 LU ðXÞ :
2 LU ðXÞ; i ¼ 1; 2; . . . ; N
@xi


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N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

and it is a Banach space with respect to the norm

kuk1;U :¼ kukU þ kjrujkU :
1
Now, we introduce the Orlicz–Sobolev space W 10 LU ðXÞ as the closure of C 1
0 ðXÞ in W LU ðXÞ. It turns out that the space
W 10 LU ðXÞ 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

t uðtÞ
UðtÞ

u0 :¼ inf
t>0

and u0 :¼ sup
t>0

t uðtÞ
:
UðtÞ

Throughout this paper, we assume that

1 < u0 6

t uðtÞ
6 u0 < 1;
UðtÞ

t P 0;

ð1:4Þ

which assures that U satisfies the D2 -condition, i.e.,

Uð2tÞ 6 K UðtÞ;

8t P 0;

ð1:5Þ

where K is a positive constant, see [19, Proposition 2.3].
In this paper, we also need the following condition

pffiffi
the function t # Uð t 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 the D2 -condition (1.5). Actually, condition (1.5) assures
that both LU ðXÞ and W 10 LU ðXÞ are separable, see [1]. Conditions (1.5) and (1.6) assure that LU ðXÞ is a uniformly convex space
and thus, a reflexive Banach space (see [19]); consequently, the Orlicz–Sobolev space W 10 LU ðXÞ is also a reflexive Banach
space. We also find that with the help of condition (1.4), the Orlicz–Sobolev space W 10 LU ðXÞ is continuously embedded in
1;u
the classical Sobolev space W 0 0 ðXÞ, as a result, W 10 LU ðXÞ is continuously and compactly embedded in the classical Lebesgue
q
Ã
space L ðXÞ for all 1 6 q < u0 , where

(

uÃ0 :¼

N u0
NÀu0

if u0 < N;

þ1

if u0 P N:

The following lemma plays an essential role in our arguments.
Proposition 1.1 (see [5,18,19]). Let u 2 W 10 LU ðXÞ. Then we have
R
0
(i) kuku 6 X UðjruðxÞjÞdx 6 kuku0 if kuk < 1,
R
0
u0
(ii) kuk 6 X UðjruðxÞjÞdx 6 kuku if kuk > 1.
2. Main results
In this section, we state and prove the main result of this paper. Let us introduce the following hypotheses:
ðf0 Þ f : X Â R ! R is a continuous function and satisfies the subcritical growth condition

jf ðx; tÞj 6 Cð1 þ jtjqÀ1 Þ;
0

8ðx; tÞ 2 X Â R;

where u < q < u and C is a positive constant;
0
ðf1 Þ f ðx; tÞ ¼ oðjtju À1 Þ; t ! 0, uniformly a.e. x 2 X;
Ã
0

¼ þ1 uniformly a.e. x 2 X, i.e., f is u0 -superlinear at infinity;
ðf2 Þ limjtj!þ1 Fðx;tÞ
u0
jtj

ðf3 Þ There exists a constant

l1 > 0 such that

Gðx; tÞ 6 Gðx; sÞ þ l1
for any x 2 X; 0 < t < s or s < t < 0, where Gðx; tÞ :¼ tf ðx; tÞ À u0 Fðx; tÞ and Fðx; tÞ :¼
ðf4 Þ f ðx; ÀtÞ ¼ Àf ðx; tÞ for all ðx; tÞ 2 X Â R.

Rt
0

f ðx; sÞds;

It should be noticed that the condition ðf3 Þ 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:
ðf30 Þ There exists t 0 > 0 such that

f ðx;tÞ
jtju

0 À2

t

is nondecreasing in t P t 0 and nonincreasing in t 6 Àt 0 for any x 2 X.

N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

7823

The 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 ðC c Þ condition, we assume that the functions u and U satisfy the
following condition:
ðHÞ There exists a positive constant

l2 such that

HðtsÞ 6 HðtÞ þ l2
for all t P 0 and s 2 ½0; 1Š, where HðtÞ ¼ u0 UðtÞ À uðtÞt.
Before stating and proving the main results of this paper, we give some examples of functions u : 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) Let uðtÞ ¼ pjtjpÀ2 t; t 2 R; p > 1. A simple computation shows that u0 ¼ u0 ¼ p. In this case, the corresponding Orlicz
space LU ðXÞ is the classical Lebesgue space Lp ðXÞ while the Orlicz–Sobolev space W 10 LU ðXÞ is the classical Sobolev space
W 1;p
0 ðXÞ. We have HðtÞ ¼ 0 for all t 2 R and then the condition ðHÞ holds.
(2) Let uðtÞ ¼ logð1 þ t2 Þt; t 2 R. Then we can deduce that u0 ¼ 2 and u0 ¼ 4. Some simple computations show that the
function U is given by

1
2

UðtÞ ¼ ð1 þ t 2 Þ logð1 þ t 2 Þ À

t2
:
2 ln 10

Then

HðtÞ ¼ u0 UðtÞ À uðtÞt ¼ ð2 þ t 2 Þ logð1 þ t 2 Þ À

2t 2
P 0 for all t P 0:
ln 10

For each fixed t > 0, the function s 2 ½0; 1Š # HðtsÞ is continuous with respect to s. So, there exists s0 2 ½0; 1Š such that
Hðts0 Þ ¼ maxs2½0;1Š HðtsÞ. It is clear that s0 – 0. If s0 ¼ 1 then HðtsÞ 6 HðtÞ for all s 2 ½0; 1Š and t P 0. If s0 2 ð0; 1Þ, since
0Þ
limt!þ1 Hðts
¼ s20 < 1 there exists t > 3 large enough such that
HðtÞ

Hðts0 Þ
HðtÞ

< 1 for all t > t or Hðts0 Þ < HðtÞ for all t > t. Now, set

l2 :¼ 1 þ maxðt;sÞ2½0;tŠÂ½0;1Š HðtsÞ we have HðtsÞ < HðtÞ þ l2 for all t P 0 and s 2 ½0; 1Š and then the condition ðHÞ holds.

Definition 2.2. A function u 2 W 10 LU ðXÞ is said to be a weak solution of problem (1.1) if it holds that

Z

aðjrujÞru Á rv dx À

X

Z

f ðx; uÞv dx ¼ 0

X

for all v 2 W 10 LU ðXÞ.
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; ðf0 Þ—ðf3 Þ 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; ðf0 Þ; ðf2 Þ—ðf4 Þ are satisfied. Then problem (1.1) has infinitely
many weak solutions fuk g satisfying

Z

Uðjruk jÞdx À

X

Z

Fðx; uk Þdx ! þ1;

k ! 1:

X

Our Theorem 2.3 is 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 our Theorem 2.4 seems
to be new even in the special case uðtÞ ¼ pjtjpÀ2 t, i.e. the well-known problem with p-Laplace operator Dp u. We emphasize
that the extension from the p-Laplace operator to the differential operators involved in (1.1) is not trivial, since the new operators have a more complicated structure than the p-Laplace operator, for example they are non-homogeneous. By the presence of the hypothesis ðf2 Þ, 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 ða1 Þ
in [5, Theorem 3.1]):

(
max lim sup
jtj!0

supx2X Fðx; tÞ
jtju

0

; lim sup
jtj!þ1

supx2X Fðx; tÞ
0

jtju

)
6 0:

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N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

Moreover, 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 condition u0 > N. In order to prove the main theorems, we recall some useful concepts and results.
Definition 2.5. Let ðX; k Á kÞ be a real Banach space, J 2 C 1 ðX; RÞ. We say that J satisfies the ðC c Þ condition if any sequence
fum g & X such that Jðum Þ ! c and kJ 0 ðum Þkà ð1 þ kum kÞ ! 0 as m ! 1 has a convergent subsequence.
Proposition 2.6 (see [10]). Let ðX; k Á kÞ be a real Banach space, J 2 C 1 ðX; RÞ satisfies the ðC c Þ condition for any c > 0; Jð0Þ ¼ 0
and the following conditions hold:
(i) There exists a function / 2 X such that k/k > q and Jð/Þ < 0;
(ii) There exist two positive constants q and R such that JðuÞ P R for any u 2 X with kuk ¼ q.
Then the functional J has a critical value c P R, i.e. there exists u 2 X such that J 0 ðuÞ ¼ 0 and JðuÞ ¼ c.
In order to prove Theorem 2.4 we will use the following fountain theorem, see [24] for details. Let ðX; k Á kÞ be a real reflexive Banach space presenting by X ¼ Èj2N X j with dimðX j Þ < þ1 for any j 2 N. For each k 2 N, we set Y k ¼ Èkj¼0 X j and
Z k ¼ È1
j¼k X j .
Proposition 2.7 (see [24]). Let ðX; k Á kÞ be a real reflexive Banach space, J 2 C 1 ðX; RÞ satisfies the ðC c Þ condition for any c > 0 and
J is even. If for each sufficiently large k 2 N, there exist qk > rk > 0 such that the following conditions hold:
(i) ak :¼ inf fu2Zk :kuk¼rk g JðuÞ ! þ1 as k ! 1;
(ii) bk :¼ maxfu2Y k :kuk¼qk g JðuÞ 6 0.
Then the functional J has an unbounded sequence of critical values, i.e. there exists a sequence fuk g & X such that J 0 ðuk Þ ¼ 0
and Jðuk Þ ! þ1 as k ! þ1.
In the rest of this paper we will use the letter X to denote the Orlicz–Sobolev space W 10 LU ðXÞ. Let us define the energy
functional J : X ! R by the formula

JðuÞ ¼

Z

UðjrujÞ À

X

Z

Fðx; uÞdx:

ð2:1Þ

X

By Proposition 1.1 and the continuous embeddings obtained from the hypothesis ðf0 Þ, some standard arguments assure that
the functional J is well-defined on X and J 2 C 1 ðXÞ with the derivative given by

J 0 ðuÞðv Þ ¼

Z

aðjrujÞru Á rv dx À

X

Z

f ðx; uÞv dx
X

for all u; v 2 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 ðf0 Þ–ðf2 Þ are satisfied. Then we have the following assertions:
(i) There exists / 2 X; / > 0 such that Jðt/Þ ! À1 as t ! þ1;
(ii) There exist q > 0 and R > 0 such that JðuÞ P R for any u 2 X with kuk ¼ q.

Proof. (i) From ðf2 Þ, it follows that for any M > 0 there exists a constant C M ¼ CðMÞ > 0 depending on M, such that
0

Fðx; tÞ P Mjtju À C M ;

8x 2 X; 8t 2 R:

ð2:2Þ

Take / 2 X with / > 0, from (2.2) and Proposition 1.1 we get

Jðt/Þ ¼

Z

Uðjrt/jÞdx À

Z

X

X

0

Fðx; t/Þdx 6 kt/ku À M

Z

0

jt/ju dx þ C M jXj 6 t u
X

0



Z
0
0
k/ku À M j/ju dx þ C M jXj;

ð2:3Þ

X

where t > 1 is large enough to ensure that kt/k > 1, and jXj denotes the Lebesgue measure of X. From (2.3), if M is large
enough such that
0

k/ku À M

Z
X

then we have

0

j/ju dx < 0;


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N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

lim Jðt/Þ ¼ À1;

t!þ1

which ends the proof of (i).
0
(ii) Since the embeddings X,!Lu ðXÞ and X,!Lq ðXÞ are continuous, there exist constants C 1 ; C 2 > 0 such that

kukLu0 ðXÞ 6 C 1 kuk;
Let 0 <  <

1
u0

2C 1

kukLq ðXÞ 6 C 2 kuk:

ð2:4Þ

, where C 1 is given by (2.4). From ðf0 Þ and ðf1 Þ, we have
0

Fðx; tÞ 6 jtju þ CðÞjtjq ;

8ðx; tÞ 2 X Â R:

ð2:5Þ

From (2.5), for all u 2 X with kuk < 1, we have

Z

Z

0

Fðx; uÞdx P kuku À 
X
X


0
0
1
À CðÞC q2 kukqÀu kuku ;
P
2

J k ðuÞ ¼

UðjrujÞdx À

Z

0

juju dx À CðÞ
X

Z
X

0

0

0

u
jujq dx P kuku À C u
À CðÞC q2 kukq
1 kuk

ð2:6Þ

where C 2 > 0 is given by (2.4). From (2.6) and the fact that q > u0 , we can choose R > 0 and q > 0 such that JðuÞ P R > 0 for
all u 2 X with kuk ¼ q. The proof of Lemma 2.8 is complete. h
Lemma 2.9. Assume that the conditions (1.4) and (1.6), ðHÞ; ðf0 Þ, ðf2 Þ–ðf3 Þ are satisfied. Then the functional J satisfies the ðC c Þ

condition for any c > 0.
Proof. Let fum g & X be a ðC c Þ sequence of the functional J, that is,

kJ 0 ðum Þkà ð1 þ kum kÞ ! 0 as m ! 1;

Jðum Þ ! c;
which shows that

c ¼ Jðum Þ þ oð1Þ;

J 0 ðum Þðum Þ ¼ oð1Þ;

ð2:7Þ

where oð1Þ ! 0 as m ! 1.
We shall prove that the sequence fum g is bounded in X. Indeed, if fum g is unbounded in X, we may assume that kum k ! 1
as m ! 1. We define the sequence fwm g by wm ¼ kuumm k ; m ¼ 1; 2; . . . It is clear that fwm g & X and kwm k ¼ 1 for any m.
Therefore, up to a subsequence, still denoted by fwm g, we have fwm g converges weakly to w 2 X and

wm ðxÞ ! wðxÞ;

a:e: in X;

m ! 1;

wm ! w strongly in Lq ðXÞ;

m ! 1;

0

wm ! w strongly in Lu ðXÞ;

m ! 1:

ð2:8Þ
ð2:9Þ
ð2:10Þ

Let X– :¼ fx 2 X : wðxÞ – 0g. If x 2 X– then it follows from (2.8) that jum ðxÞj ¼ jwm ðxÞjkum k ! þ1 as m ! 1. Moreover, from
ðf2 Þ, we have

lim

Fðx; um ðxÞÞ

m!1

jum ðxÞj

u0

0

jwm ðxÞju ¼ þ1;

x 2 X– :

ð2:11Þ

Using the condition ðf2 Þ, there exists t 0 > 0 such that

Fðx; tÞ
jtju

0

>1

ð2:12Þ

for all x 2 X and jtj > t 0 > 0. Since Fðx; tÞ is continuous on X  ½Àt0 ; t0 Š, there exists a positive constant C 3 such that

jFðx; tÞj 6 C 3

ð2:13Þ

for all ðx; tÞ 2 X  ½Àt 0 ; t 0 Š. From (2.12) and (2.13) there exists C 4 2 R such that

Fðx; tÞ P C 4

ð2:14Þ

for all ðx; tÞ 2 X Â R. From (2.14), for all x 2 X and m, we have

Fðx; um ðxÞÞ À C 4
kum ku
or

0

P0


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N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

Fðx; um ðxÞÞ
jum ðxÞj

u0

C4

0

jwm ðxÞju À

0

kum ku

8x 2 X; 8m:

P 0;

ð2:15Þ

By (2.7), Proposition 1.1 we have

c ¼ Jðum Þ þ oð1Þ ¼

Z

Uðjrum jÞdx À

X

Z

Fðx; um Þdx þ oð1Þ P kum ku0 À

Z

X

Fðx; um Þdx þ oð1Þ

X

or

Z

Fðx; um Þdx P kum ku0 À c þ oð1Þ ! þ1 as m ! 1:

ð2:16Þ

X

We also have

c ¼ Jðum Þ þ oð1Þ ¼

Z

Uðjrum jÞdx À

X

Z

0

Fðx; um Þdx þ oð1Þ 6 kum ku À

X

Z

Fðx; um Þdx þ oð1Þ

X

or
0

kum ku P

Z

Fðx; um Þdx þ c À oð1Þ > 0 for m large enough:

ð2:17Þ

X

We claim that jX– j ¼ 0. In fact, if jX– j – 0, then by (2.11), (2.15) and (2.17) and the Fatou lemma, we have

þ ¼ ðþ1ÞjX– j
Z
Z
Fðx; um ðxÞÞ
C4
u0
¼
lim inf
jw
ðxÞj
dx
À
lim sup
dx
m
0
0
u
m!1
m!1

X–
X–
kum ku
jum ðxÞj
!
Z
0
Fðx; um ðxÞÞ
C4
dx
¼
lim inf
jwm ðxÞju À
0
u0
m!1
X–
kum ku
jum ðxÞj
!
Z
0
Fðx; um ðxÞÞ
C4
6 lim inf
jwm ðxÞju À
dx
0
u0
m!1

X–
kum ku
jum ðxÞj
!
Z
0
Fðx; um ðxÞÞ
C4
dx
jwm ðxÞju À
6 lim inf
0
u0
m!1
X
kum ku
jum ðxÞj
Z
Z
Fðx; um ðxÞÞ
C4
¼ lim inf
dx À lim sup
dx
0
u
u0
m!1
m!1
X

X kum k
kum k
Z
Fðx; um ðxÞÞ
dx
¼ lim inf
0
m!1
X
ku ku
R m
X Fðx; um ðxÞÞdx
:
6 lim inf R
m!1
X Fðx; um Þdx þ c À oð1Þ

ð2:18Þ

From (2.16) and (2.18), we obtain

þ1 6 1;
which is a contradiction. This shows that jX– j ¼ 0 and thus wðxÞ ¼ 0 a.e. in X.
Since Jðtum Þ is continuous in t 2 ½0; 1Š, for each m there exists t m 2 ½0; 1Š, m ¼ 1; 2; . . ., such that

Jðt m um Þ :¼ maxJðtum Þ:

ð2:19Þ

t2½0;1Š

It is clear that t m > 0 and Jðtm um Þ P c > 0 ¼ Jð0Þ ¼ Jð0:um Þ. If t m < 1 then
tm ¼ 1, then J 0 ðum Þðum Þ ¼ oð1Þ. So we always have

d
Jðtum Þjt¼tm
dt

J 0 ðtm um Þðtm um Þ ¼ oð1Þ:

¼ 0 which gives J 0 ðt m um Þðtm um Þ ¼ 0. If

ð2:20Þ

Let fRk g be a positive sequence of real numbers such that Rk > 1 for any k and limk!1 Rk ¼ þ1. Then kRk wm k ¼ Rk > 1 for any
0
k and m. Fix k, since wm ! 0 strongly in the spaces Lq ðXÞ and Lu ðXÞ as m ! 1, using (2.5), we deduce that there exists a
constant C 5 > 0 such that

Z

Fðx; Rk wm Þdx 6 C 5

X

Z

0

Rk jwm ju dx þ C 5

X

Z

X

Rqk jwm jq dx ! 0 as m ! 1;

which yield

lim

m!1

Z
X

Fðx; Rk wm Þdx ¼ 0:

ð2:21Þ


7827

N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

Since kum k ! 1 as m ! 1, we also have kum k > Rk or 0 < kuRmk k < 1 for m large enough. Hence, using (2.21), Proposition 1.1, it
follows that




Z
Z
Z
Rk
UðjrRk wm jÞdx À Fðx; Rk wm Þdx P kRk wm ku0 À Fðx; Rk wm Þdx
um ¼ JðRk wm Þ ¼
kum k
X
X
X
Z
1 u0
u0
¼ Rk À Fðx; Rk wm Þdx P Rk
2
X

Jðt m um Þ P J

ð2:22Þ

for any m large enough. From (2.22), letting m; k ! 1 we have

lim Jðt m um Þ ¼ þ1:

ð2:23Þ

m!1

On the other hand, using the conditions ðf3 Þ; ðHÞ and relations (1.4) and (2.7), for all m large enough, we have

1
Jðt m um Þ ¼ Jðt m um Þ À 0 J 0 ðt m um Þðt m um Þ þ oð1Þ
u
Z
Z
Z
Z
1
1
¼
Uðjrt m um jÞdx À Fðx; t m um Þdx À 0
aðjrtm um jÞjrt m um j2 dx þ 0
f ðx; tm um Þt m um dx þ oð1Þ
u X
u X
X
X
Z
Z
1
1
6 0
Hðt jru jÞdx þ 0
Gðx; tm um Þdx þ oð1Þ
u X m m
u X

Z
Z
À
Á
À
Á
1
1
Hðjrum jÞ þ l2 dx þ 0
Gðx; um Þ þ l1 dx þ oð1Þ
6 0
u X
u X
Z

Z
Z
Z
1
l þl
¼
Uðjrum jÞdx À Fðx; um Þdx À 0
aðjrum jÞjrum j2 dx À f ðx; um Þum dx þ 1 0 2 jXj þ oð1Þ

u
X
X
l
þ
l

l
þ
l
¼ Jðum Þ À 0 J 0 ðum Þðum Þ þ 1 0 2 jXj þ oð1Þ ! c þ 1 0 2 jXj as m ! 1:
u
u
u
X

X

u

1

ð2:24Þ

From (2.23) and (2.24) we obtain a contradiction. This shows that the sequence fum g 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 fum g, it converges weakly to u in X and converges strongly to u in the space Lq ðXÞ. Using the
condition ðf0 Þ and the Hölder inequality, we have

Z
 Z
Z




 f ðx; um Þðum À uÞdx 6

kum À ukLq ðXÞ
jf ðx; um Þjjum À ujdx 6 C ð1 þ jum jqÀ1 Þjum À ujdx 6 C 1 þ kum kqÀ1


Lq ðXÞ
X

X

X

! 0 as m ! 1;
which yield

lim

m!1

Z

f ðx; um Þðum À uÞdx ¼ 0:

ð2:25Þ

X

From (2.7) and (2.25) we get

lim

m!1

Z

aðjrum jÞrum Á rðum À uÞdx ¼ 0:

ð2:26Þ

X

By (2.26) and the fact that fum g converges weakly to u in X, we can apply the result of Miha˘ilescu [18, Lemma 5] in order to
deduce that the sequence fum g converges strongly to u in X. Therefore, the functional J satisfies the ðC c Þ condition for any
c > 0. The proof of Lemma 2.9 is complete.
Proof of Theorem 2.3. By Lemmas 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 fej g & X and feÃj g & X Ã such that

X ¼ spanfej : j ¼ 1; 2; . . . ; g;

X Ã ¼ spanfeÃj : j ¼ 1; 2; . . . ; g

and

D

E & 1; if i ¼ j;
ei ; eÃj ¼
0; if i – j:

For convenience, we write X j ¼ spanfej g and define for each k 2 N the subspaces Y k ¼ Èkj¼1 X j and Z k ¼ È1

X . The following
j¼k j
result is useful for our arguments.


7828

N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

Lemma 2.10. If u0 < q < uÃ0 then we have

n

o

ak :¼ sup kukLq ðXÞ : kuk ¼ 1; u 2 Z k ! 0 as k ! 1:
Proof. Obviously, for any k 2 N; 0 < akþ1 6 ak , so ak ! a P 0 as k ! 1. Let uk 2 Z k ; k ¼ 1; 2; . . . satisfy

kuk k ¼ 1 and 0 6 ak À kuk kLq ðXÞ <

1
:
k

ð2:27Þ

Then there exists a subsequence of fuk g, still denoted by fuk g such that fuk g converges weakly to u in X and

D
E

D
E
eÃj ; u ¼ lim eÃj ; uk ;

j ¼ 1; 2; . . .

k!1

Since Z k is a closed subspace of X, by Mazur’s theorem, we have u 2 Z k for any k. Consequently, we get u 2 \1
k¼1 Z k ¼ f0g, and
so fuk g converges weakly to 0 in X as k ! 1. Since u0 < q < uÃ0 , the embedding X,!Lq ðXÞ is compact, then fuk g converges
strongly to 0 in Lq ðXÞ. Hence, by (2.27), we have limk!1 ak ¼ 0. h
Lemma 2.11. Assume that the conditions ðf0 Þ and ðf2 Þ are satisfied. Then there exist qk > r k > 0 such that
(i) ak :¼ inf fu2Zk :kuk¼rk g JðuÞ ! þ1 as k ! 1;
(ii) bk :¼ maxfu2Y k :kuk¼qk g JðuÞ 6 0.

Proof. (i) By ðf0 Þ, there exists C 6 > 0 such that

jFðx; tÞj 6 C 6 ðjtj þ jtjq Þ

ð2:28Þ

for all ðx; tÞ 2 X Â R; u0 < q < uÃ0 .
From (2.28), for any u 2 Z k with kuk > 1 we have

JðuÞ ¼

Z

UðjrujÞdx À

X

Z

Fðx; uÞdx P kuku0 À C 6

X

Z

jujq dx À C 6

X

Z
X

jujdx P kuku0 À C 7 aqk kukq À C 8 kuk;

ð2:29Þ

where

n

o

ak :¼ sup kukLq ðXÞ : kuk ¼ 1; u 2 Z k :
1

Now, for any u 2 Z k with kuk ¼ rk ¼ ð2C 7 aqk Þu0 Àq we have

À
Á u0
À
Á q
À
Á 1
JðuÞ P kuku0 À C 7 aqk kukq À C 8 kuk ¼ 2C 7 aqk u0 Àq À C 7 aqk 2C 7 aqk u0 Àq À C 8 2C 6 aqk u0 Àq
Á u0
À
Á 1
1À
1 u
¼ 2C 7 aqk u0 Àq À C 8 2C 7 aqk u0 Àq ¼ r k 0 À C 7 r k :
2
2

ð2:30Þ

By Lemma 2.10, ak ! 0 as k ! 1 and uÃ0 > q > u0 > u0 > 1 we have that r k ! þ1 as k ! 1. Therefore, by (2.30) it follows
that ak ! þ1 as k ! 1.
ðiiÞ By (2.2), for any w 2 Y k with kwk ¼ 1 and t > 1, we have

JðtwÞ ¼

Z

X


0
kwku
¼t
u0

Z

0

Fðx; twÞdx 6 ktwku À M
X

Z
0
À M jwju dx þ C M jXj:

UðjrtwjÞdx À

Z

0

jtwju dx þ C M jXj
X

X

It is clear that we can choose M > 0 large enough such that

0

kwku À M

Z

0

jwju dx < 0:

X

For this choice, it follows from (2.31) that

lim JðtwÞ ¼ À1:

t!þ1

Hence, there exists t > r k > 1 large enough such that JðtwÞ 6 0 and thus, if we set qk ¼ t we conclude that

bk :¼

max

JðuÞ 6 0:

fu2Y k :kuk¼qk g

The proof of Lemma 2.11 is complete. h

ð2:31Þ


N.T. Chung, H.Q. Toan / Applied Mathematics and Computation 219 (2013) 7820–7829

7829

Proof. Proof of Theorem 2.4. By Lemma 2.11, the functional J satisfies all the assumptions of the fountain theorem. By
Lemma 2.9, the functional J satisfies the ðC c Þ condition for every c > 0. Moreover, from the condition ðf4 Þ; J is even. Hence,
we can apply the fountain theorem in order to obtain a sequence of critical points fuk g & X of J such that Jðuk Þ ! þ1 as
k ! 1. The proof of Theorem 2.4 is complete. h
Acknowledgments
The authors thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript. This work was supported by Vietnam National Foundation for Science and Technology Development
(NAFOSTED).
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