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Multiple solutions for quasilinear Schrödinger equations involving critical exponent
Applied Mathematics and Computation 216 (2010) 849–856
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Multiple solutions for quasilinear Schrödinger equations involving critical exponent q Wang Youjun a,*, Zhang Yimin b, Shen Yaotian b a b
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China Department of Mathematics, South China University of Technology, Guangzhou 510640, PR China
a r t i c l e
i n f o
a b s t r a c t By using Lions’ second concentration–compactness principle and concentration–compactness principle at infinity to prove that the (PS) condition holds locally and by minimax methods and the Krasnoselski genus theory, we establish the multiplicity of solutions for a class of quasilinear Schrödinger equations arising from physics. Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.
Keywords: Quasilinear Schrödinger equations Critical exponents Concentration–compactness
1. Introduction This paper is concerned with the following quasilinear elliptic equations
Du þ VðxÞu ½Dðu2 Þu ¼ hðuÞ;
x 2 RN :
ð1:1Þ
Such equations arise in various branches of mathematical physics and they have been the subject of extensive study in recent years. Part of the interest is due to the fact that solutions of (1.1) are related to the existence of standing wave solutions for quasilinear Schrödinger equations of the form
@ t z ¼ Dz þ VðxÞz lðjzj2 Þz kDgðjzj2 Þg 0 ðjzj2 Þz;
x 2 RN ;
ð1:2Þ
where V is a given potential, k is a real constant, l and g are real functions. Quasilinear equations of the form (1.2) appear more naturally in mathematical physics and have been derived as models of several physical phenomena, such as superfluid film equations in plasma physics and the fluid mechanics in condensed matter theory. The related semilinear equations for k = 0 have been extensively studied (see e.g. [1,2,5,12–15], as well as their references). For k – 0, Poppenberg et al. [11] studied the existence of a positive ground state solution for the quasilinear Schrödinger equation in u00 þ VðxÞu ðu2 Þ00 u ¼ hjujp1 u in R. In [8], by a change of variables the quasilinear problem was transformed to a semilinear one and an Orlicz space framework was used as the working space, and they were able to prove the existence of positive solutions of (1.2) by the Mountain-Pass theorem. The same method of changing of variables was used in [10], but the usual Sobolev space Hl ðRN Þ framework was used as the working space and they studied a different class of nonlinearity. In [9], the existence of both one sign and nodal ground state type solutions were established by the Nehari method. The main purpose of the present paper is to show the existence of multiple solutions for quasilinear Schrödinger equations of the form
Du ½Dðu2 Þu ¼ akðxÞjujp2 u þ bu2ð2
where N P 3; 2 < p < 4; 2 ¼
2N N2
;
x 2 RN ;
r
N
Þ1
ð1:3Þ
and kðxÞ 2 L ðR Þ with r ¼ 2ð2 Þ=ð2ð2 Þ pÞ a and b are real parameters.
q
Rearch supported by the National Natural Science Foundation of China (No. 10771074). * Corresponding author. E-mail addresses: [email protected] (W. Youjun), [email protected] (Z. Yimin).
0096-3003/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.091
850
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
Note that 2ð2 Þ behaves like a critical exponent for the above equations [8, Remark 3.13]. For the subcritical case, the existence of solutions for Problem (1.1) was studied in [7–10] and it was left open for the critical exponent case [8, Remark 3.13]. To the best of our knowledge, the existence of nontrivial radial solutions for (1.1) with hðuÞ ¼ lu2ð2 Þ1 was firstly studied by Moameni [17], where the same Orlicz space as [8] was used. Our main result is the following: Theorem 1.1. Suppose that X :¼ fx 2 RN : kðxÞ > 0g is an open subset of RN and 0 < jXj < 1. Then (i) 8b > 0, there exists a1 > 0 such that if 0 < a < a1 , Problem (1.1) has a sequence of solutions. (ii) 8a > 0, there exists b1 > 0 such that if 0 < b < b1 , Problem (1.1) has a sequence of solutions. Notation. In this paper, we make use of the following notation: n o R 1=2 . D1;2 ðRN Þ :¼ u 2 L2 ðRN Þ : ru 2 L2 ðRN Þ endowed with the norm kuk ¼ RN jruj2 dx 1=r R . For 1 6 r < 1; Lr ðRN Þ denotes the usual Lebesgue space with norms kukr ¼ RN jujr dx c; c1 ; c2 denote positive (possibly different) constants. 2. Preliminaries It should be pointed out that the natural functional/associated with (1.3) given by
IðuÞ ¼
1 2
Z RN
ð1 þ 2u2 Þjruj2 dx
a p
Z
kðxÞjujp dx
RN
b 2ð2 Þ
Z
u2ð2 Þ dx RN
is not well defined in general, for instance, in D1;2 ðRN Þ. To overcome this difficulty, we employ an argument developed by Colin and Jeanjean [10]. We make the changing of variables v ¼ f 1 ðuÞ, where f is defined by:
1 f 0 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; and f ð0Þ ¼ 0 1 þ 2f 2 ðtÞ on ½0; 1Þ and by f ðtÞ ¼ f ðtÞ on ð1; 0. The following results is due to Colin and Jeanjean [10]. Lemma 2.1. f satisfies the following properties: ðf0 Þ f is uniquely defined, C 1 and invertible. ðf1 Þ jf 0 ðtÞ 6 1j for t 2 R. ! 1 as t ! 1. ðf2 Þ f ðtÞ t ðf3 Þ fpðtÞffit ! 21=4 as t ! 1. 0 ðf4 Þ 12 f ðtÞ 6 tf ðtÞ 6 f ðtÞ for all t P 0. So, after the change of variables, we can write IðuÞ as
Jðv Þ ¼
1 2
Z RN
jrv j2 dx
Z
a p
RN
kðxÞjf ðv Þjp dx
b 2ð2 Þ
Z RN
f 2ð2 Þ ðv Þ dx:
ð2:1Þ
Then, Jðv Þ is well defined on D1;2 ðRN Þ and J 2 C 1 . In fact, we have the following results. Lemma 2.2 R
(i) The functional Fðv Þ :¼
RN
kðxÞjf ðv Þjp dx is well defined and weakly continuous on D1;2 ðRN Þ. Moreover, FðuÞ is continuously
differentiable, its derivative F0 ðv Þ : D1;2 ðRN Þ ! ðD1;2 ðRN ÞÞ is given by
hF0 ðv Þ; hi ¼ p
Z RN
kðxÞjf ðv Þjp2 f ðv Þf 0 ðv Þh dx:
(ii) The functional Gðv Þ :¼ G ðv Þ :¼ D 0
1;2
R
1;2
N
ðR Þ ! ðD
hG0 ðv Þ; hi ¼ 2ð2 Þ
Z RN
f 2ð2
ð2:2Þ
RN
f 2ð2 Þ ðv Þ dx is well defined. Moreover, Gðv Þ is continuously differentiable, its derivative N
ðR ÞÞ is given by
Þ1
ðv Þf 0 ðv Þh dx:
ð2:3Þ
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
851
Proof. By ðf2 Þ and ðf3 Þ, it is clear that F and G are well defined on D1;2 ðRN Þ. In order to prove F; G 2 C 1 , it suffices to show both F and G have continuous Gateaux derivative on D1;2 ðRN Þ. We consider only F since the proof for G is simpler. Our argument is similar to that [18, Lemma 3.10]. Let u; h 2 D1;2 ðRN Þ. Given x 2 RN and 0 < jtj < 1, by the mean value theorem, there exists k 2 ð0; 1Þ such that
jf ðv þ thÞjp jf ðv Þjp p2 p2 p ¼ pjf ðv þ tkhÞjp1 jf 0 ðv þ tkhÞjjhj 6 cjv þ tkhj 2 jhj 6 cðjv j 2 jhj þ jhj2 Þ: jtj
By Hölder inequality, we have
Z
RN
kðxÞðjv j
p2 2
p
p2 2
jhj þ jhj2 Þ dx 6 kkðxÞkr khk2 ðkv k
p2
þ khk 2 Þ:
It follows from the Lebesgue Dominated Convergence theorem that F is Gateaux differentiable and
hF0 ðv Þ; hi ¼ p
Z
RN
kðxÞjf ðv Þjp2 f ðv Þf 0 ðv Þh dx:
Now, we proof continuity of Gateaux derivative. Assume that v n ! v , in D1;2 ðRN Þ, then f 2 ðv n Þ ! f 2 ðv Þ in D1;2 ðRN Þ. By the continuity of the embedding D1;2 ðRN Þ,!L2 ðRN Þ; f 2 ðv n Þ ! f 2 ðv Þ in L2 ðRN Þ. Let us define Hðv Þ :¼ pkðxÞjf ðv Þjp2 f ðv Þf 0 ðv Þ, then 2 2 N N 0 Hðv Þ 2 ðL ðR Þ; CðL ðR ÞÞ Þ [16, p. 30]. It follows that Hðv n Þ ! Hðv Þ in ðL2 ðRN ÞÞ0 . Using Hölder and Sobolev inequalities, we get
2 hF0 ðv n Þ F0 ðv Þ; hi 6 kHðv n Þ Hðv Þkð2 Þ0 khk 6 ckHðv n Þ Hðv Þkð2 Þ0 khk
Hence kF0 ðv n Þ F0 ðv Þk ! 0 and F 2 C 1 . As in [10], we observe that if f is a nontrivial critical point of J, then
Dv ¼ gðx; v Þ;
v is a nontrivial solution of problem
x 2 RN ;
ð2:4Þ
where
gðx; sÞ ¼ f 0 ðsÞðakðxÞÞjf ðsÞjp2 f ðsÞ þ bf 2ð2 Therefore, setting u ¼ f ðv Þ and since ðf problem
Þ1
ðsÞÞ:
1 0
Þ ðtÞ ¼ ½f 0 ðf 1 ðtÞÞ1 ¼
Du ½Dðu2 Þu ¼ akðxÞjujp2 u þ bu2ð2
Þ1
:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2t2 , we conclude that u is a nontrivial solution of the
Theorem 2.3. Suppose that X :¼ fx 2 RN : kðxÞ > 0g is an open subset of RN and 0 < jXj < þ1. Then, (i) 8b > 0, there exists a1 > 0 such that if 0 < a < a1 , Problem (2.4) has a sequence of solutions fv m g with Jðv m Þ < 0 and Jðv m Þ ! 0 as m ! þ1. (ii) 8a > 0, there exists b1 > 0 such that if 0 < b < b1 , Problem (2.4) has a sequence of solutions fv m g with Jðv m Þ < 0 and Jðv m Þ ! 0 as m ! þ1. We recall the second concentration–compactness principle of Lions [12].
Proposition 2.4. Let fum g be a weakly convergent sequence to u in D1;2 ðRN Þ such that jum j2 * m and jrum j2 * l in the sense of measures. Then, for some at most countable index set J, P (i) m ¼ juj2 þ j2J mj dxj ; mj > 0, P 2 (ii) l P jruj þ j2J lj dxj ; lj > 0, 2=2 6 lj , (iii) Smj
where S is the best Sobolev constant, i.e., S ¼ inf constants.
nR
RN
o jruj2 dx : kuk22 ¼ 1 ; xj 2 RN ; dxj are Dirac measures at xj and
The following result can be found in [4,17]. Proposition 2.5. Let fum g be a weakly convergent sequence to u in D1;2 ðRN Þ and define R
l1 ¼ limR!1 lim sup jxj>R jum j2 dx, m!1 R (ii) l1 ¼ limR!1 lim sup jxj>R jrum j2 dx. (i)
m!1
The quantities (iii) lim sup m!1 2=2 6 1
(iv) Sm
R
m1 and l1 exist and satisfy
RN
jum j2 dx ¼ juj22 þ kmk þ m1 ; lim sup
l1 .
m!1
RN R
jrum j2 dx ¼ jruj22 þ klk þ l1 .
lj mj are
852
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
We say that fv n g is a Palais–Smale sequence for the functional J at level c 2 R ((PS)c sequence in short), if Jðv n Þ ! c and J 0 ðv n Þ ! 0 as n ! 1. Lemma 2.6. Any ðPSÞc sequence for J is bounded. Proof. Let fv n g be a ðPSÞc sequence. Then, for n large,
Jðv n Þ ¼
1 2
Z
jrv n j2 dx
RN
a p
Z RN
kðxÞjf ðv n Þjp dx
b 2ð2 Þ
Z RN
f 2ð2 Þ ðv n Þ dx ¼ c þ oð1Þ;
ð2:5Þ
and for any x 2 D1;2 ðRN Þ,
hJ 0 ðv n Þ; xi ¼ Choose x ¼ xn ¼
Z RN
rv n rx dx a
Z RN
kðxÞjf ðv n Þjp2 f ðv n Þf 0 ðv n Þx dx b
Z
f 2ð2
RN
Þ1
ðv n Þf 0 ðv n Þx dx ¼ oðkv n kÞ:
ð2:6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þ, we have xn 2 D1;2 ðRN Þ. From ðf4 Þ and since
jrxn j ¼ 1 þ
2f 2 ðv n Þ jrv n j; 1 þ 2f 2 ðv n Þ
we deduce that kxn k 6 ckv n k. By (2.6),
hJ 0 ðv n Þ; xn i ¼
Z RN
1þ
Z Z 2f 2 ðv n Þ jrv n j2 dx a kðxÞjf ðv n Þjp dx b f 2ð2 Þ ðv n Þ dx ¼ oðkv n kÞ 2 1 þ 2f ðv n Þ RN RN
Then, by Hölder and Sobolev inequalities, since
Z RN
jrf 2 ðv n Þj2 dx ¼ 4
Z RN
f 2 ðv n Þ jrv n j2 dx 6 2 1 þ 2f 2 ðv n Þ
Z RN
jrv n j2 dx;
we have,
Z Z
1 1 1 2f 2 ðv n Þ a 2 0 j ð v Þ; x i ¼ r v j dx kðxÞjf ðv n Þjp dx hJ 1 þ n n n 2ð2 Þ 2ð2 Þ 1 þ 2f 2 ðv n Þ r RN RN 2 Z 1r Z p 2ð2 Þ a jrv n j2 dx jkðxÞjr dx jf ðv n Þj2ð2 Þ dx N N r R R Z 1r Z 4p a 1 jrv n j2 jkðxÞjr dx jrf 2 ðv n Þj2 dx P kv n k2 ckv n kp=2 ; N N N r R R
c þ oðkv n kÞ ¼ Jðv n Þ P P
1 N 1 N
Z RN
Z RN
we immediately deduce that fv n g is bounded since p < 4.
h
Proposition 2.7 (i) 8b > 0, there exists a1 > 0 such that if 0 < a < a1 and c < 0, then J satisfies (PS)c. (ii) 8a > 0, there exists b1 > 0 such that if 0 < b < b1 and c < 0, then J satisfies (PS)c
Proof. Let fv n g be a (PS)C sequence, by Lemma 2.6, fv n g is bounded in D1;2 ðRN Þ then ff 2 ðv n Þg is also bounded in D1;2 ðRN Þ. We can assume, going if necessary to a subsequence, that v n * v in D1;2 ðRN Þ; v n ! v a.e. in RN , since f 2 C 1 , then f 2 ðv n Þ ! f 2 ðv n Þ a.e. in RN and then f 2 ðv n Þ * f 2 ðv n Þ in D1;2 ðRN Þ. Thus, there exist measures l and m such that
N jrf 2 ðv n Þj2 * l; f 2ð2 Þ ðv n Þ * m. Let xk be a singular point of the measures l and m. We define a function / 2 C 1 0 ðR Þ such pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N N 2 ~ ~ that /ðxÞ ¼ 1 in Bðxj ; Þ; /ðxÞ ¼ 0 in R n Bðxj ; 2Þ and jr/j 6 2= in R . Let xn ¼ 1 þ 2f ðv n Þf ðv n Þ/, then fxn g is bounded
~ n i ¼ 0, that is, in D1;2 ðRN Þ. We have limn!1 hJ 0 ðv n Þ; x
lim
n!1
Z RN
Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þrv n r/dx ¼ lim n!1
Z Z 2f 2 ðv n Þ 1þ kðxÞjf ðv n Þjp /dx b f 2ð2 Þ ðv n Þ/dx /jrv n j2 dx a 2 1 þ 2f ðv n Þ RN RN RN
853
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
On the other hand, by ðf4 Þ, we have,
Z lim
n!1
RN
Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þrv n r/ dx 6 limn!1 c jv n rv n r/j dx RN "Z Z 1=2
6 c lim
n!1
6c
Z Bðxj ;2Þ
RN
jrv n j2 dx
RN
1=2 # Z c jv n r/j2 dx
RN
1=2 jv r/j2 dx
!1=2
jv j2 dx
! 0; as
! 0:
Then, by the weak continuity of Fðv Þ
0 ¼ lim
Z
n!1
RN
1þ
Z 1 P lim n!1 2 RN
Z Z 2f 2 ðv n Þ 2 p 2ð2 Þ /j r v j dx a kðxÞjf ð v Þj / dx b f ð v Þ/ dx n n n 1 þ 2f 2 ðv n Þ RN RN Z Z lj /jrf 2 ðv n Þj2 dx a kðxÞjf ðv n Þjp / dx b f 2ð2 Þ ðv n Þ/ dx ¼ bmj ; as N N 2 R R
! 0;
we get 2bmj P lj . By Proposition 2.4 (iii), we have that either
ðIÞ
mj ¼ 0; or ðIIÞ mj P ð21 b1 SÞN=2 :
To obtain the possible concentration of mass at infinity, similarly, we define a cut off function /R 2 /R ðxÞ ¼ 0 on jxj < R and /R ðxÞ ¼ 1 on jxj > R þ 1. Since f/R xn g is bounded in in D1;2 ðRN Þ, we have
ð2:7Þ N C1 0 ðR Þ
such that
Z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þrv n r/R dx RN
Z Z Z 2f 2 ðv n Þ 2 p 2ð2 Þ ¼ lim 1þ j r v j dx a kðxÞjf ð v Þj / dx b f ð v Þ/ dx : / n n n R R n!1 1 þ 2f 2 ðv n Þ R RN RN RN
lim
n!1
Notice that
Z lim lim
R!1 n!1
RN
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þrv n r/R dx ¼ 0
and using the weak continuity of F, we have,
Z lim
n!1
jxj>R
kðxÞjf ðv nÞjp /R dx ! 0; as R ! 1:
Therefore,
0 ¼ lim
n!1
Z RN
1þ
Z 1 P lim n!1 2 RN
Z Z 2f 2 ðv n Þ 2 p 2ð2 Þ / j r v j dx a kðxÞjf ð v Þj / dx b f ð v Þ/ dx n n n R R 1 þ 2f 2 ðv n Þ R RN RN Z Z l /R jrf 2 ðv n Þj2 dx a kðxÞjf ðv n Þjp /R dx b f 2ð2 Þ ðv n Þ/R dx ¼ 1 bm1 ; as R ! 1; 2 RN RN
Combining with (iv) of Proposition 2.5 we have that either
ðIIIÞ
m1 ¼ 0; or ðIVÞ m1 P ð21 b1 SÞN=2 :
ð2:8Þ
Next, we claim that (II) and (IV) cannot occur if a and b are chosen properly. Indeed, from the weak lower semicontinuity of the norm and the weak continuity of F,
0 > c ¼ lim Jðv n Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 2 ðv Þf ðv Þ ð v Þ; 1 þ 2f J n n n n!1 2ð2 Þ Z
Z 2 1 1 2f ðv n Þ a ¼ lim jrv n j2 dx kðxÞjf ðv n Þjp dx 1þ 2 n!1 2ð2 Þ 1 þ 2f ðv n Þ pr RN RN 2 Z Z 1 a 1 a p=2 p=2 P lim jrv n j2 dx kkðxÞkr kf 2 ðv n Þk2 P jrv j2 dx kkðxÞkr kf 2 ðv Þk2 n!1 N RN N RN pr pr Z 1 a S a p=2 p=2 kf 2 ðv Þk22 kkðxÞkr kf 2 ðv Þk2 ; P jrf 2 ðv Þj2 dx kkðxÞkr kf 2 ðv Þk2 P 2N RN 2N pr pr
this yields,
kf 2 ðv n Þk2 6 ca2=ð4pÞ :
854
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
Therefore,
0 > c ¼ lim Jðv n Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 J 0 ðv n Þ; 1 þ 2f 2 ðv n Þf ðv n Þ n!1 2ð2 Þ Z
Z 1 1 2f 2 ðv n Þ a 2 p ¼ lim r v j dx kðxÞjf ð v Þj dx 1þ j n n n!1 2ð2 Þ 1 þ 2f 2 ðv n Þ pr RN RN 2 Z 1 a 1 1 P lim lim jrv n j2 /R dx kkðxÞkr kf 2 ðv n Þkp=2 l cap=ð4pÞ P 2ð2NÞ=2 bð2NÞ=2 SN=2 cap=ð4pÞ : P 2 N R!1 n!1 RN 2N 1 2N pr
However, if a > 0 is given, we can choose b1 so small such that for every 0 < b < b1 , the last term on the right-hand side above is greater than 0, which is a contradiction. Similarly, if b > 0 is given, we can take a1 so small such that for every 0 < a < a1 , the last term on the right-hand side above is greater than 0. Similarly, we can prove that (II) cannot occur for each j. Thus,
lim
Z
n!1
and
lim
RN
Z
n!1
RN
f 2ð2 Þ ðv n Þ dx ¼
Z
n!1
RN
RN
f 2ð2 Þ ðv Þ dx;
ð2:9Þ
kðxÞ jf ðv n Þjp jf ðv Þjp dx 6 lim kkðxÞkr kjf ðv n Þjp jf ðv Þjp k2ð2 Þ ¼ 0: n!1
Since JðmÞ ¼ c, then,
lim
Z
jrv n j2 dx ¼
Z RN
p
jrv j2 dx:
ð2:10Þ
By Brézis–Lieb Lemma, we get,
lim
n!1
Z
RN
jrðv n v Þj2 dx ¼ 0:
ð2:11Þ
This completes the proof of Proposition 2.7.
h
3. Proof of theorems In this section, we use minimax procedure to prove Theorem 2.3. Let X be a Banach space, define the set
R ¼ fA X n f0g : A is closed in X and symmetric with respect to the origing: For A 2 R, define
cðAÞ :¼ inffm 2 N : 9u 2 CðA; Rm n f0gÞ; uðxÞ ¼ uðxÞg; if there is no mapping u as above for any m 2 N, then cðAÞ :¼ þ1. The properties of the genus can be found in [3,13]. Proposition 3.1. Let A; B 2 X, Then, (1) (2) (3) (4)
If there exists an odd map f 2 CðA; BÞ, then cðAÞ 6 cðBÞ. If A B, then cðAÞ 6 cðBÞ. c(A[B) 6c(A)+c(B). If A is compact, there exists a symmetric neighborhood N of A, such that cðNÞ ¼ cðAÞ.
Let J be the functional defined as in (2.1), then
Jðv Þ ¼
1 2
Z RN
jrv j2 dx
a p
Z RN
kðxÞjf ðv Þjp dx
b 2ð2 Þ
Z RN
f 2ð2 Þ ðv Þ dx
Z 1=r Z p=2ð2 Þ Z 1 a b 2 r 2ð2 Þ jkðxÞj dx f ðv Þ dx f 2ð2 Þ ðv Þ dx P kv k 2 2ð2 Þ RN p RN RN 1 1 P kv k2 ac1 kjf ðv Þjkp=2 bc2 kf ðv Þk2 P kv k2 ac1 kv kp=2 bc2 kv k2 : 2 2
We define the function Q ðtÞ :¼ 1=2t2 ac1 tp=2 bc2 t 2 . Then, it is easy to see that, given b > 0, there exists a2 > 0 so small that for every 0 < a < a2 , there exists 0 < T 0 < T 1 such that Q ðtÞ < 0 for 0 < t < T 0 ; Q ðtÞ > 0 for T 0 < t < T 1 ; Q ðtÞ < 0 for t > T 1 . Similarly, given a > 0, we can choose b2 > 0 with the property that T 0 ; T 1 as above exist for each 0 < b < b2 . Clearly, Q ðT 0 Þ ¼ Q ðT 1 Þ ¼ 0. Using the same idea as in [6], we consider the truncated functional
eJðv Þ ¼ 1 2
Z RN
jrv j2 dx
a p
Z RN
kðxÞjf ðv Þjp dx
b wðv Þ 2ð2 Þ
Z RN
f 2ð2 Þ ðv Þ dx;
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
855
where wðuÞ ¼ sðkukÞ and s : Rþ ! ½0; 1 is a nonincreasing C 1 function such that sðtÞ ¼ 1 if t 6 T 0 and sðtÞ ¼ 0 if t P T 1 . Thus,
e ðkv kÞ; eJðv Þ P Q
ð3:1Þ
e ðtÞ :¼ 1=2t2 ac1 t p=2 bc2 t2 wðtÞ. It is clear that eJðv Þ 2 C 1 and is bounded from below. where Q Moreover, we can follow from Proposition 2.7 and (3.1) that: Proposition 3.2 (i) If eJðv Þ < 0, then kv k 6 T 0 and eJðv Þ ¼ Jðv Þ; (ii) 8b > 0, there exists a3 > 0 such that if 0 < a < a3 and c < 0, then eJ satisfies (PS)c. (iii) 8a > 0, there exists b3 > 0 such that if 0 < b < b3 and c < 0, then eJ satisfies (PS)c.
Lemma 3.3. Denote K c :¼ fv 2 D1;2 ðRN Þ; eJ 0 ðv Þ ¼ 0; eJðv Þ ¼ cg. Then for any m 2 N, there is 1 Proof. Denote by D1;2 0 ðXÞ the closure of C 0 ðXÞ with respect to the norm kuk ¼
R X
em < 0 such that cðeJ em Þ P m.
1=2 jruj2 dx . Extending functions in D1;2 0 ð XÞ
1;2 by 0 outside X we can assume that D1;2 ðRN Þ. Let X m be a m-dimensional subspace of D1;2 0 ðXÞ D 0 ðXÞ. For any v 2 X m ; v –0 write v ¼ r m w with w 2 X m and kwk ¼ 1. From the assumptions of kðxÞ, it is easy to see for every w 2 X m with kwk ¼ 1 that there exist dm > 0 such that
Z
kðxÞjwjp=2 dx 6 dm :
X
Thus for 0 < rm < T 0 , since all the norms are equivalent and 0 < jXj < þ1, by ðf4 Þ we have,
Z Z Z a b eJðv Þ ¼ Jðv Þ ¼ 1 jrv j2 dx kðxÞjf ðv Þjp dx f 2ð2 Þ ðv Þ dx 2 X 2ð2 Þ X p X Z Z Z 1 a b 6 jrv j2 dx kðxÞðcjv jp=2 þ cÞ dx ðv 2 þ cÞ dx 2 X 2ð2 Þ X p X Z Z Z p 1 ac bc 1 2 6 jrv j2 dx kðxÞjv j2 dx v 2 dx c 6 r2m acdm rp=2 m bcr m c :¼ em : 2 X p X 2ð2 Þ X 2
Therefore, we can choose r m 2 ð0; T 0 Þ so small that eJðv Þ < em < 0. Let Srm ¼ fu 2 D1;2 ðRN Þ; kv k ¼ r m g. Then Srm \ X m eJ em . By Proposition 3.1 (2),cðeJ em Þ P cðSrm \ X m Þ P m. P Therefore, if we denote Cm ¼ fA 2 ; cðAÞ P mg and let
cm :¼ inf A2Cm supv 2AeJðv Þ;
ð3:2Þ
1 < cm 6 em < 0;
ð3:3Þ
then
m 2 N;
because eJ em 2 Cm and eJ is bounded from below. h Proposition 3.4. Let a; b be as in Proposition 3.2 (ii) or (iii). Then all cm given by (3.2) are critical values of eJ and cm ! 0 as m ! 1. Proof. It is clear that cm 6 cmþ1 , and by (3.3), cm < 0. Hence cm ! c 6 0. Moreover, since (PS)c is satisfied, it follows from a standard argument (e.g. [13]) that all cm are critical values of eJ. Now, we claim that c ¼ 0. If c < 0, because K c is compact and P K c 2 , it follows from that cðK c Þ ¼ mo < þ1 and there exists d > 0 such that cðK c Þ ¼ cðN d ðK c ÞÞ ¼ m0 . By the deformation lemma there exist > 0ðc þ < 0Þ and an odd homeomorphism g such that
gðeJ cþ n Nd ðK c ÞÞ eJ c :
ð3:4Þ
Since cm is increasing and converges to c, there exists m 2 N such that cm > c and C mþm0 6 c. There exists A 2 Cmþm0 such e < c þ . By Proposition 3.1, we have, that supu2A GðuÞ
cðA n Nd ðK c ÞÞ P cðAÞ cðNd ðK c ÞÞ P m; cðgðA n Nd ðK c ÞÞÞ P m:
ð3:5Þ
Therefore,
gðA n Nd ðK c ÞÞ 2 Cm : Consequently,
sup u2gðAnNd ðK c ÞÞ
e GðuÞ P cm > c :
ð3:6Þ
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W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
On the other hand, by (3.4) and (3.5),
gðA n Nd ðK c ÞÞ gðeJ cþ n Nd ðK c ÞÞ eJ c ; which contradicts (3.6). Hence cn ! 0.
ð3:7Þ
h
Proof of the Theorem 2.3. By Proposition 3.2 (i), eJðv Þ ¼ Jðv Þ if eJ < 0. This and Proposition 3.4 give the result. Proof of the Theorem 1.1. This follows from Theorem 2.3, since um ¼ f ðv m Þ – un ¼ f ðv n Þ if
vm – vn.
References [1] A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrodinger equation with a bounded potential, J. Funct. Anal. 69 (1986) 397–408. [2] A.M. Bertone, Joao Marcos Do Ó, On a class of semilinear Schrodinger equations involving critical growth and discontinuous nonlinearities, Nonlinear Anal. 54 (2003) 885–906. [3] J.Q. Chen, S.J. Li, On multiple solutions of a singular quasilinear equation on unbounded domain, J. Math. Anal. Appl. 275 (2002) 733–746. [4] J. Chabrowski, Concentration–compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. 3 (1995) 493–512. [5] J. Chabrowski, J.f. Yang, Existence theorems for the Schrodinger equation involving a critical exponent, Z. Angew. Math. Phys. 49 (1998) 276–293. [6] J.G. Azorero, I.P. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Am. Math. Soc. 323 (1991) 877–895. [7] J.Q. Liu, Z.Q. Wang, Soliton solutions for quasilinear Schrodinger equations, Proc. Am. Math. Soc. 131 (2003) 441–448. [8] J.Q. Liu, Y.Q. W ang, Z.Q. Wang, Soliton solutions to quasilinear Schrodinger equations II, J. Diff. Eqn. 187 (2003) 473–493. [9] J.Q. Liu, Y.Q. Wang, Z.Q. Wang, Solutions for quasilinear Schrodinge equations via the Nehari method, Commun. Partial Diff. Eqn. 29 (5/6) (2004) 879– 901. [10] M. Colin, L. Jeanjean, Solutions for a quasilinear Schrodinger equations: a dual approach, Nonlinear Anal. 56 (2004) 213–226. [11] M. Poppenberg, K. Schmitt, Z.Q. Wang, On the existence of solition solutions to quasilinear Schrodinger equations, Calc. Var. Partial Diff. Eqn. 14 (2002) 329–344. [12] P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincare Anal. Non. Lineaire. 1 (1984). pp. 109–145 and 223–283. [13] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in: CBMS Regional Conf. Ser. in Math. No. 65, Amer. Math. Soc, Providence, RI, 1986. [14] T. Bartsch, Z.Q. Wang, Existence and multiplicity results for some superlinear elliptic problem on RN , Commun. Partial Diff. Eqn. 20 (1995) 1725–1741. [15] T. Bartsch, Z.Q. Wang, Multiple positive solutions for a nonlinear Schrodinge equation, Z. Angew. Math. Phys. 51 (2000) 366–384. [16] M.A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Elmsford, NY, 1964. [17] A. Moameni, Existence of soliton solutions for a quasilinear Schrodinger quation involving critical exponent in RN , J. Diff. Eqn. 229 (2006) 570–587. [18] M. Willem, Minimax Theorems, Birkhauser, Boston, 1996.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Multiple solutions for quasilinear Schrödinger equations involving critical exponent q Wang Youjun a,*, Zhang Yimin b, Shen Yaotian b a b
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China Department of Mathematics, South China University of Technology, Guangzhou 510640, PR China
a r t i c l e
i n f o
a b s t r a c t By using Lions’ second concentration–compactness principle and concentration–compactness principle at infinity to prove that the (PS) condition holds locally and by minimax methods and the Krasnoselski genus theory, we establish the multiplicity of solutions for a class of quasilinear Schrödinger equations arising from physics. Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved.
Keywords: Quasilinear Schrödinger equations Critical exponents Concentration–compactness
1. Introduction This paper is concerned with the following quasilinear elliptic equations
Du þ VðxÞu ½Dðu2 Þu ¼ hðuÞ;
x 2 RN :
ð1:1Þ
Such equations arise in various branches of mathematical physics and they have been the subject of extensive study in recent years. Part of the interest is due to the fact that solutions of (1.1) are related to the existence of standing wave solutions for quasilinear Schrödinger equations of the form
@ t z ¼ Dz þ VðxÞz lðjzj2 Þz kDgðjzj2 Þg 0 ðjzj2 Þz;
x 2 RN ;
ð1:2Þ
where V is a given potential, k is a real constant, l and g are real functions. Quasilinear equations of the form (1.2) appear more naturally in mathematical physics and have been derived as models of several physical phenomena, such as superfluid film equations in plasma physics and the fluid mechanics in condensed matter theory. The related semilinear equations for k = 0 have been extensively studied (see e.g. [1,2,5,12–15], as well as their references). For k – 0, Poppenberg et al. [11] studied the existence of a positive ground state solution for the quasilinear Schrödinger equation in u00 þ VðxÞu ðu2 Þ00 u ¼ hjujp1 u in R. In [8], by a change of variables the quasilinear problem was transformed to a semilinear one and an Orlicz space framework was used as the working space, and they were able to prove the existence of positive solutions of (1.2) by the Mountain-Pass theorem. The same method of changing of variables was used in [10], but the usual Sobolev space Hl ðRN Þ framework was used as the working space and they studied a different class of nonlinearity. In [9], the existence of both one sign and nodal ground state type solutions were established by the Nehari method. The main purpose of the present paper is to show the existence of multiple solutions for quasilinear Schrödinger equations of the form
Du ½Dðu2 Þu ¼ akðxÞjujp2 u þ bu2ð2
where N P 3; 2 < p < 4; 2 ¼
2N N2
;
x 2 RN ;
r
N
Þ1
ð1:3Þ
and kðxÞ 2 L ðR Þ with r ¼ 2ð2 Þ=ð2ð2 Þ pÞ a and b are real parameters.
q
Rearch supported by the National Natural Science Foundation of China (No. 10771074). * Corresponding author. E-mail addresses: [email protected] (W. Youjun), [email protected] (Z. Yimin).
0096-3003/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.091
850
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
Note that 2ð2 Þ behaves like a critical exponent for the above equations [8, Remark 3.13]. For the subcritical case, the existence of solutions for Problem (1.1) was studied in [7–10] and it was left open for the critical exponent case [8, Remark 3.13]. To the best of our knowledge, the existence of nontrivial radial solutions for (1.1) with hðuÞ ¼ lu2ð2 Þ1 was firstly studied by Moameni [17], where the same Orlicz space as [8] was used. Our main result is the following: Theorem 1.1. Suppose that X :¼ fx 2 RN : kðxÞ > 0g is an open subset of RN and 0 < jXj < 1. Then (i) 8b > 0, there exists a1 > 0 such that if 0 < a < a1 , Problem (1.1) has a sequence of solutions. (ii) 8a > 0, there exists b1 > 0 such that if 0 < b < b1 , Problem (1.1) has a sequence of solutions. Notation. In this paper, we make use of the following notation: n o R 1=2 . D1;2 ðRN Þ :¼ u 2 L2 ðRN Þ : ru 2 L2 ðRN Þ endowed with the norm kuk ¼ RN jruj2 dx 1=r R . For 1 6 r < 1; Lr ðRN Þ denotes the usual Lebesgue space with norms kukr ¼ RN jujr dx c; c1 ; c2 denote positive (possibly different) constants. 2. Preliminaries It should be pointed out that the natural functional/associated with (1.3) given by
IðuÞ ¼
1 2
Z RN
ð1 þ 2u2 Þjruj2 dx
a p
Z
kðxÞjujp dx
RN
b 2ð2 Þ
Z
u2ð2 Þ dx RN
is not well defined in general, for instance, in D1;2 ðRN Þ. To overcome this difficulty, we employ an argument developed by Colin and Jeanjean [10]. We make the changing of variables v ¼ f 1 ðuÞ, where f is defined by:
1 f 0 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; and f ð0Þ ¼ 0 1 þ 2f 2 ðtÞ on ½0; 1Þ and by f ðtÞ ¼ f ðtÞ on ð1; 0. The following results is due to Colin and Jeanjean [10]. Lemma 2.1. f satisfies the following properties: ðf0 Þ f is uniquely defined, C 1 and invertible. ðf1 Þ jf 0 ðtÞ 6 1j for t 2 R. ! 1 as t ! 1. ðf2 Þ f ðtÞ t ðf3 Þ fpðtÞffit ! 21=4 as t ! 1. 0 ðf4 Þ 12 f ðtÞ 6 tf ðtÞ 6 f ðtÞ for all t P 0. So, after the change of variables, we can write IðuÞ as
Jðv Þ ¼
1 2
Z RN
jrv j2 dx
Z
a p
RN
kðxÞjf ðv Þjp dx
b 2ð2 Þ
Z RN
f 2ð2 Þ ðv Þ dx:
ð2:1Þ
Then, Jðv Þ is well defined on D1;2 ðRN Þ and J 2 C 1 . In fact, we have the following results. Lemma 2.2 R
(i) The functional Fðv Þ :¼
RN
kðxÞjf ðv Þjp dx is well defined and weakly continuous on D1;2 ðRN Þ. Moreover, FðuÞ is continuously
differentiable, its derivative F0 ðv Þ : D1;2 ðRN Þ ! ðD1;2 ðRN ÞÞ is given by
hF0 ðv Þ; hi ¼ p
Z RN
kðxÞjf ðv Þjp2 f ðv Þf 0 ðv Þh dx:
(ii) The functional Gðv Þ :¼ G ðv Þ :¼ D 0
1;2
R
1;2
N
ðR Þ ! ðD
hG0 ðv Þ; hi ¼ 2ð2 Þ
Z RN
f 2ð2
ð2:2Þ
RN
f 2ð2 Þ ðv Þ dx is well defined. Moreover, Gðv Þ is continuously differentiable, its derivative N
ðR ÞÞ is given by
Þ1
ðv Þf 0 ðv Þh dx:
ð2:3Þ
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
851
Proof. By ðf2 Þ and ðf3 Þ, it is clear that F and G are well defined on D1;2 ðRN Þ. In order to prove F; G 2 C 1 , it suffices to show both F and G have continuous Gateaux derivative on D1;2 ðRN Þ. We consider only F since the proof for G is simpler. Our argument is similar to that [18, Lemma 3.10]. Let u; h 2 D1;2 ðRN Þ. Given x 2 RN and 0 < jtj < 1, by the mean value theorem, there exists k 2 ð0; 1Þ such that
jf ðv þ thÞjp jf ðv Þjp p2 p2 p ¼ pjf ðv þ tkhÞjp1 jf 0 ðv þ tkhÞjjhj 6 cjv þ tkhj 2 jhj 6 cðjv j 2 jhj þ jhj2 Þ: jtj
By Hölder inequality, we have
Z
RN
kðxÞðjv j
p2 2
p
p2 2
jhj þ jhj2 Þ dx 6 kkðxÞkr khk2 ðkv k
p2
þ khk 2 Þ:
It follows from the Lebesgue Dominated Convergence theorem that F is Gateaux differentiable and
hF0 ðv Þ; hi ¼ p
Z
RN
kðxÞjf ðv Þjp2 f ðv Þf 0 ðv Þh dx:
Now, we proof continuity of Gateaux derivative. Assume that v n ! v , in D1;2 ðRN Þ, then f 2 ðv n Þ ! f 2 ðv Þ in D1;2 ðRN Þ. By the continuity of the embedding D1;2 ðRN Þ,!L2 ðRN Þ; f 2 ðv n Þ ! f 2 ðv Þ in L2 ðRN Þ. Let us define Hðv Þ :¼ pkðxÞjf ðv Þjp2 f ðv Þf 0 ðv Þ, then 2 2 N N 0 Hðv Þ 2 ðL ðR Þ; CðL ðR ÞÞ Þ [16, p. 30]. It follows that Hðv n Þ ! Hðv Þ in ðL2 ðRN ÞÞ0 . Using Hölder and Sobolev inequalities, we get
2 hF0 ðv n Þ F0 ðv Þ; hi 6 kHðv n Þ Hðv Þkð2 Þ0 khk 6 ckHðv n Þ Hðv Þkð2 Þ0 khk
Hence kF0 ðv n Þ F0 ðv Þk ! 0 and F 2 C 1 . As in [10], we observe that if f is a nontrivial critical point of J, then
Dv ¼ gðx; v Þ;
v is a nontrivial solution of problem
x 2 RN ;
ð2:4Þ
where
gðx; sÞ ¼ f 0 ðsÞðakðxÞÞjf ðsÞjp2 f ðsÞ þ bf 2ð2 Therefore, setting u ¼ f ðv Þ and since ðf problem
Þ1
ðsÞÞ:
1 0
Þ ðtÞ ¼ ½f 0 ðf 1 ðtÞÞ1 ¼
Du ½Dðu2 Þu ¼ akðxÞjujp2 u þ bu2ð2
Þ1
:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2t2 , we conclude that u is a nontrivial solution of the
Theorem 2.3. Suppose that X :¼ fx 2 RN : kðxÞ > 0g is an open subset of RN and 0 < jXj < þ1. Then, (i) 8b > 0, there exists a1 > 0 such that if 0 < a < a1 , Problem (2.4) has a sequence of solutions fv m g with Jðv m Þ < 0 and Jðv m Þ ! 0 as m ! þ1. (ii) 8a > 0, there exists b1 > 0 such that if 0 < b < b1 , Problem (2.4) has a sequence of solutions fv m g with Jðv m Þ < 0 and Jðv m Þ ! 0 as m ! þ1. We recall the second concentration–compactness principle of Lions [12].
Proposition 2.4. Let fum g be a weakly convergent sequence to u in D1;2 ðRN Þ such that jum j2 * m and jrum j2 * l in the sense of measures. Then, for some at most countable index set J, P (i) m ¼ juj2 þ j2J mj dxj ; mj > 0, P 2 (ii) l P jruj þ j2J lj dxj ; lj > 0, 2=2 6 lj , (iii) Smj
where S is the best Sobolev constant, i.e., S ¼ inf constants.
nR
RN
o jruj2 dx : kuk22 ¼ 1 ; xj 2 RN ; dxj are Dirac measures at xj and
The following result can be found in [4,17]. Proposition 2.5. Let fum g be a weakly convergent sequence to u in D1;2 ðRN Þ and define R
l1 ¼ limR!1 lim sup jxj>R jum j2 dx, m!1 R (ii) l1 ¼ limR!1 lim sup jxj>R jrum j2 dx. (i)
m!1
The quantities (iii) lim sup m!1 2=2 6 1
(iv) Sm
R
m1 and l1 exist and satisfy
RN
jum j2 dx ¼ juj22 þ kmk þ m1 ; lim sup
l1 .
m!1
RN R
jrum j2 dx ¼ jruj22 þ klk þ l1 .
lj mj are
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W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
We say that fv n g is a Palais–Smale sequence for the functional J at level c 2 R ((PS)c sequence in short), if Jðv n Þ ! c and J 0 ðv n Þ ! 0 as n ! 1. Lemma 2.6. Any ðPSÞc sequence for J is bounded. Proof. Let fv n g be a ðPSÞc sequence. Then, for n large,
Jðv n Þ ¼
1 2
Z
jrv n j2 dx
RN
a p
Z RN
kðxÞjf ðv n Þjp dx
b 2ð2 Þ
Z RN
f 2ð2 Þ ðv n Þ dx ¼ c þ oð1Þ;
ð2:5Þ
and for any x 2 D1;2 ðRN Þ,
hJ 0 ðv n Þ; xi ¼ Choose x ¼ xn ¼
Z RN
rv n rx dx a
Z RN
kðxÞjf ðv n Þjp2 f ðv n Þf 0 ðv n Þx dx b
Z
f 2ð2
RN
Þ1
ðv n Þf 0 ðv n Þx dx ¼ oðkv n kÞ:
ð2:6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þ, we have xn 2 D1;2 ðRN Þ. From ðf4 Þ and since
jrxn j ¼ 1 þ
2f 2 ðv n Þ jrv n j; 1 þ 2f 2 ðv n Þ
we deduce that kxn k 6 ckv n k. By (2.6),
hJ 0 ðv n Þ; xn i ¼
Z RN
1þ
Z Z 2f 2 ðv n Þ jrv n j2 dx a kðxÞjf ðv n Þjp dx b f 2ð2 Þ ðv n Þ dx ¼ oðkv n kÞ 2 1 þ 2f ðv n Þ RN RN
Then, by Hölder and Sobolev inequalities, since
Z RN
jrf 2 ðv n Þj2 dx ¼ 4
Z RN
f 2 ðv n Þ jrv n j2 dx 6 2 1 þ 2f 2 ðv n Þ
Z RN
jrv n j2 dx;
we have,
Z Z
1 1 1 2f 2 ðv n Þ a 2 0 j ð v Þ; x i ¼ r v j dx kðxÞjf ðv n Þjp dx hJ 1 þ n n n 2ð2 Þ 2ð2 Þ 1 þ 2f 2 ðv n Þ r RN RN 2 Z 1r Z p 2ð2 Þ a jrv n j2 dx jkðxÞjr dx jf ðv n Þj2ð2 Þ dx N N r R R Z 1r Z 4p a 1 jrv n j2 jkðxÞjr dx jrf 2 ðv n Þj2 dx P kv n k2 ckv n kp=2 ; N N N r R R
c þ oðkv n kÞ ¼ Jðv n Þ P P
1 N 1 N
Z RN
Z RN
we immediately deduce that fv n g is bounded since p < 4.
h
Proposition 2.7 (i) 8b > 0, there exists a1 > 0 such that if 0 < a < a1 and c < 0, then J satisfies (PS)c. (ii) 8a > 0, there exists b1 > 0 such that if 0 < b < b1 and c < 0, then J satisfies (PS)c
Proof. Let fv n g be a (PS)C sequence, by Lemma 2.6, fv n g is bounded in D1;2 ðRN Þ then ff 2 ðv n Þg is also bounded in D1;2 ðRN Þ. We can assume, going if necessary to a subsequence, that v n * v in D1;2 ðRN Þ; v n ! v a.e. in RN , since f 2 C 1 , then f 2 ðv n Þ ! f 2 ðv n Þ a.e. in RN and then f 2 ðv n Þ * f 2 ðv n Þ in D1;2 ðRN Þ. Thus, there exist measures l and m such that
N jrf 2 ðv n Þj2 * l; f 2ð2 Þ ðv n Þ * m. Let xk be a singular point of the measures l and m. We define a function / 2 C 1 0 ðR Þ such pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N N 2 ~ ~ that /ðxÞ ¼ 1 in Bðxj ; Þ; /ðxÞ ¼ 0 in R n Bðxj ; 2Þ and jr/j 6 2= in R . Let xn ¼ 1 þ 2f ðv n Þf ðv n Þ/, then fxn g is bounded
~ n i ¼ 0, that is, in D1;2 ðRN Þ. We have limn!1 hJ 0 ðv n Þ; x
lim
n!1
Z RN
Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þrv n r/dx ¼ lim n!1
Z Z 2f 2 ðv n Þ 1þ kðxÞjf ðv n Þjp /dx b f 2ð2 Þ ðv n Þ/dx /jrv n j2 dx a 2 1 þ 2f ðv n Þ RN RN RN
853
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
On the other hand, by ðf4 Þ, we have,
Z lim
n!1
RN
Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þrv n r/ dx 6 limn!1 c jv n rv n r/j dx RN "Z Z 1=2
6 c lim
n!1
6c
Z Bðxj ;2Þ
RN
jrv n j2 dx
RN
1=2 # Z c jv n r/j2 dx
RN
1=2 jv r/j2 dx
!1=2
jv j2 dx
! 0; as
! 0:
Then, by the weak continuity of Fðv Þ
0 ¼ lim
Z
n!1
RN
1þ
Z 1 P lim n!1 2 RN
Z Z 2f 2 ðv n Þ 2 p 2ð2 Þ /j r v j dx a kðxÞjf ð v Þj / dx b f ð v Þ/ dx n n n 1 þ 2f 2 ðv n Þ RN RN Z Z lj /jrf 2 ðv n Þj2 dx a kðxÞjf ðv n Þjp / dx b f 2ð2 Þ ðv n Þ/ dx ¼ bmj ; as N N 2 R R
! 0;
we get 2bmj P lj . By Proposition 2.4 (iii), we have that either
ðIÞ
mj ¼ 0; or ðIIÞ mj P ð21 b1 SÞN=2 :
To obtain the possible concentration of mass at infinity, similarly, we define a cut off function /R 2 /R ðxÞ ¼ 0 on jxj < R and /R ðxÞ ¼ 1 on jxj > R þ 1. Since f/R xn g is bounded in in D1;2 ðRN Þ, we have
ð2:7Þ N C1 0 ðR Þ
such that
Z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þrv n r/R dx RN
Z Z Z 2f 2 ðv n Þ 2 p 2ð2 Þ ¼ lim 1þ j r v j dx a kðxÞjf ð v Þj / dx b f ð v Þ/ dx : / n n n R R n!1 1 þ 2f 2 ðv n Þ R RN RN RN
lim
n!1
Notice that
Z lim lim
R!1 n!1
RN
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2f 2 ðv n Þf ðv n Þrv n r/R dx ¼ 0
and using the weak continuity of F, we have,
Z lim
n!1
jxj>R
kðxÞjf ðv nÞjp /R dx ! 0; as R ! 1:
Therefore,
0 ¼ lim
n!1
Z RN
1þ
Z 1 P lim n!1 2 RN
Z Z 2f 2 ðv n Þ 2 p 2ð2 Þ / j r v j dx a kðxÞjf ð v Þj / dx b f ð v Þ/ dx n n n R R 1 þ 2f 2 ðv n Þ R RN RN Z Z l /R jrf 2 ðv n Þj2 dx a kðxÞjf ðv n Þjp /R dx b f 2ð2 Þ ðv n Þ/R dx ¼ 1 bm1 ; as R ! 1; 2 RN RN
Combining with (iv) of Proposition 2.5 we have that either
ðIIIÞ
m1 ¼ 0; or ðIVÞ m1 P ð21 b1 SÞN=2 :
ð2:8Þ
Next, we claim that (II) and (IV) cannot occur if a and b are chosen properly. Indeed, from the weak lower semicontinuity of the norm and the weak continuity of F,
0 > c ¼ lim Jðv n Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 2 ðv Þf ðv Þ ð v Þ; 1 þ 2f J n n n n!1 2ð2 Þ Z
Z 2 1 1 2f ðv n Þ a ¼ lim jrv n j2 dx kðxÞjf ðv n Þjp dx 1þ 2 n!1 2ð2 Þ 1 þ 2f ðv n Þ pr RN RN 2 Z Z 1 a 1 a p=2 p=2 P lim jrv n j2 dx kkðxÞkr kf 2 ðv n Þk2 P jrv j2 dx kkðxÞkr kf 2 ðv Þk2 n!1 N RN N RN pr pr Z 1 a S a p=2 p=2 kf 2 ðv Þk22 kkðxÞkr kf 2 ðv Þk2 ; P jrf 2 ðv Þj2 dx kkðxÞkr kf 2 ðv Þk2 P 2N RN 2N pr pr
this yields,
kf 2 ðv n Þk2 6 ca2=ð4pÞ :
854
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
Therefore,
0 > c ¼ lim Jðv n Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 J 0 ðv n Þ; 1 þ 2f 2 ðv n Þf ðv n Þ n!1 2ð2 Þ Z
Z 1 1 2f 2 ðv n Þ a 2 p ¼ lim r v j dx kðxÞjf ð v Þj dx 1þ j n n n!1 2ð2 Þ 1 þ 2f 2 ðv n Þ pr RN RN 2 Z 1 a 1 1 P lim lim jrv n j2 /R dx kkðxÞkr kf 2 ðv n Þkp=2 l cap=ð4pÞ P 2ð2NÞ=2 bð2NÞ=2 SN=2 cap=ð4pÞ : P 2 N R!1 n!1 RN 2N 1 2N pr
However, if a > 0 is given, we can choose b1 so small such that for every 0 < b < b1 , the last term on the right-hand side above is greater than 0, which is a contradiction. Similarly, if b > 0 is given, we can take a1 so small such that for every 0 < a < a1 , the last term on the right-hand side above is greater than 0. Similarly, we can prove that (II) cannot occur for each j. Thus,
lim
Z
n!1
and
lim
RN
Z
n!1
RN
f 2ð2 Þ ðv n Þ dx ¼
Z
n!1
RN
RN
f 2ð2 Þ ðv Þ dx;
ð2:9Þ
kðxÞ jf ðv n Þjp jf ðv Þjp dx 6 lim kkðxÞkr kjf ðv n Þjp jf ðv Þjp k2ð2 Þ ¼ 0: n!1
Since JðmÞ ¼ c, then,
lim
Z
jrv n j2 dx ¼
Z RN
p
jrv j2 dx:
ð2:10Þ
By Brézis–Lieb Lemma, we get,
lim
n!1
Z
RN
jrðv n v Þj2 dx ¼ 0:
ð2:11Þ
This completes the proof of Proposition 2.7.
h
3. Proof of theorems In this section, we use minimax procedure to prove Theorem 2.3. Let X be a Banach space, define the set
R ¼ fA X n f0g : A is closed in X and symmetric with respect to the origing: For A 2 R, define
cðAÞ :¼ inffm 2 N : 9u 2 CðA; Rm n f0gÞ; uðxÞ ¼ uðxÞg; if there is no mapping u as above for any m 2 N, then cðAÞ :¼ þ1. The properties of the genus can be found in [3,13]. Proposition 3.1. Let A; B 2 X, Then, (1) (2) (3) (4)
If there exists an odd map f 2 CðA; BÞ, then cðAÞ 6 cðBÞ. If A B, then cðAÞ 6 cðBÞ. c(A[B) 6c(A)+c(B). If A is compact, there exists a symmetric neighborhood N of A, such that cðNÞ ¼ cðAÞ.
Let J be the functional defined as in (2.1), then
Jðv Þ ¼
1 2
Z RN
jrv j2 dx
a p
Z RN
kðxÞjf ðv Þjp dx
b 2ð2 Þ
Z RN
f 2ð2 Þ ðv Þ dx
Z 1=r Z p=2ð2 Þ Z 1 a b 2 r 2ð2 Þ jkðxÞj dx f ðv Þ dx f 2ð2 Þ ðv Þ dx P kv k 2 2ð2 Þ RN p RN RN 1 1 P kv k2 ac1 kjf ðv Þjkp=2 bc2 kf ðv Þk2 P kv k2 ac1 kv kp=2 bc2 kv k2 : 2 2
We define the function Q ðtÞ :¼ 1=2t2 ac1 tp=2 bc2 t 2 . Then, it is easy to see that, given b > 0, there exists a2 > 0 so small that for every 0 < a < a2 , there exists 0 < T 0 < T 1 such that Q ðtÞ < 0 for 0 < t < T 0 ; Q ðtÞ > 0 for T 0 < t < T 1 ; Q ðtÞ < 0 for t > T 1 . Similarly, given a > 0, we can choose b2 > 0 with the property that T 0 ; T 1 as above exist for each 0 < b < b2 . Clearly, Q ðT 0 Þ ¼ Q ðT 1 Þ ¼ 0. Using the same idea as in [6], we consider the truncated functional
eJðv Þ ¼ 1 2
Z RN
jrv j2 dx
a p
Z RN
kðxÞjf ðv Þjp dx
b wðv Þ 2ð2 Þ
Z RN
f 2ð2 Þ ðv Þ dx;
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
855
where wðuÞ ¼ sðkukÞ and s : Rþ ! ½0; 1 is a nonincreasing C 1 function such that sðtÞ ¼ 1 if t 6 T 0 and sðtÞ ¼ 0 if t P T 1 . Thus,
e ðkv kÞ; eJðv Þ P Q
ð3:1Þ
e ðtÞ :¼ 1=2t2 ac1 t p=2 bc2 t2 wðtÞ. It is clear that eJðv Þ 2 C 1 and is bounded from below. where Q Moreover, we can follow from Proposition 2.7 and (3.1) that: Proposition 3.2 (i) If eJðv Þ < 0, then kv k 6 T 0 and eJðv Þ ¼ Jðv Þ; (ii) 8b > 0, there exists a3 > 0 such that if 0 < a < a3 and c < 0, then eJ satisfies (PS)c. (iii) 8a > 0, there exists b3 > 0 such that if 0 < b < b3 and c < 0, then eJ satisfies (PS)c.
Lemma 3.3. Denote K c :¼ fv 2 D1;2 ðRN Þ; eJ 0 ðv Þ ¼ 0; eJðv Þ ¼ cg. Then for any m 2 N, there is 1 Proof. Denote by D1;2 0 ðXÞ the closure of C 0 ðXÞ with respect to the norm kuk ¼
R X
em < 0 such that cðeJ em Þ P m.
1=2 jruj2 dx . Extending functions in D1;2 0 ð XÞ
1;2 by 0 outside X we can assume that D1;2 ðRN Þ. Let X m be a m-dimensional subspace of D1;2 0 ðXÞ D 0 ðXÞ. For any v 2 X m ; v –0 write v ¼ r m w with w 2 X m and kwk ¼ 1. From the assumptions of kðxÞ, it is easy to see for every w 2 X m with kwk ¼ 1 that there exist dm > 0 such that
Z
kðxÞjwjp=2 dx 6 dm :
X
Thus for 0 < rm < T 0 , since all the norms are equivalent and 0 < jXj < þ1, by ðf4 Þ we have,
Z Z Z a b eJðv Þ ¼ Jðv Þ ¼ 1 jrv j2 dx kðxÞjf ðv Þjp dx f 2ð2 Þ ðv Þ dx 2 X 2ð2 Þ X p X Z Z Z 1 a b 6 jrv j2 dx kðxÞðcjv jp=2 þ cÞ dx ðv 2 þ cÞ dx 2 X 2ð2 Þ X p X Z Z Z p 1 ac bc 1 2 6 jrv j2 dx kðxÞjv j2 dx v 2 dx c 6 r2m acdm rp=2 m bcr m c :¼ em : 2 X p X 2ð2 Þ X 2
Therefore, we can choose r m 2 ð0; T 0 Þ so small that eJðv Þ < em < 0. Let Srm ¼ fu 2 D1;2 ðRN Þ; kv k ¼ r m g. Then Srm \ X m eJ em . By Proposition 3.1 (2),cðeJ em Þ P cðSrm \ X m Þ P m. P Therefore, if we denote Cm ¼ fA 2 ; cðAÞ P mg and let
cm :¼ inf A2Cm supv 2AeJðv Þ;
ð3:2Þ
1 < cm 6 em < 0;
ð3:3Þ
then
m 2 N;
because eJ em 2 Cm and eJ is bounded from below. h Proposition 3.4. Let a; b be as in Proposition 3.2 (ii) or (iii). Then all cm given by (3.2) are critical values of eJ and cm ! 0 as m ! 1. Proof. It is clear that cm 6 cmþ1 , and by (3.3), cm < 0. Hence cm ! c 6 0. Moreover, since (PS)c is satisfied, it follows from a standard argument (e.g. [13]) that all cm are critical values of eJ. Now, we claim that c ¼ 0. If c < 0, because K c is compact and P K c 2 , it follows from that cðK c Þ ¼ mo < þ1 and there exists d > 0 such that cðK c Þ ¼ cðN d ðK c ÞÞ ¼ m0 . By the deformation lemma there exist > 0ðc þ < 0Þ and an odd homeomorphism g such that
gðeJ cþ n Nd ðK c ÞÞ eJ c :
ð3:4Þ
Since cm is increasing and converges to c, there exists m 2 N such that cm > c and C mþm0 6 c. There exists A 2 Cmþm0 such e < c þ . By Proposition 3.1, we have, that supu2A GðuÞ
cðA n Nd ðK c ÞÞ P cðAÞ cðNd ðK c ÞÞ P m; cðgðA n Nd ðK c ÞÞÞ P m:
ð3:5Þ
Therefore,
gðA n Nd ðK c ÞÞ 2 Cm : Consequently,
sup u2gðAnNd ðK c ÞÞ
e GðuÞ P cm > c :
ð3:6Þ
856
W. Youjun et al. / Applied Mathematics and Computation 216 (2010) 849–856
On the other hand, by (3.4) and (3.5),
gðA n Nd ðK c ÞÞ gðeJ cþ n Nd ðK c ÞÞ eJ c ; which contradicts (3.6). Hence cn ! 0.
ð3:7Þ
h
Proof of the Theorem 2.3. By Proposition 3.2 (i), eJðv Þ ¼ Jðv Þ if eJ < 0. This and Proposition 3.4 give the result. Proof of the Theorem 1.1. This follows from Theorem 2.3, since um ¼ f ðv m Þ – un ¼ f ðv n Þ if
vm – vn.
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