The Nehari manifold for indefinite semilinear elliptic systems involving ...
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Author's personal copy Applied Mathematics and Computation 218 (2012) 10817–10828
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The Nehari manifold for indefinite semilinear elliptic systems involving critical exponent Ching-yu Chen, Tsung-fang Wu ⇑ Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
a r t i c l e
i n f o
a b s t r a c t In this paper, we study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system ðEk;l Þ involving critical exponents and sign-changing weight functions. Using Nehari manifold, the system is proved to have at least two nontrivial nonnegative solutions when the pair of the parameters ðk; lÞ belongs to a certain subset of R2 . Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Indefinite semilinear elliptic systems Multiple positive solutions Critical Sobolev exponent Nehari manifold
1. Introduction In this paper we examine the multiplicity results derived from the nontrivial nonnegative solutions of the indefinite semilinear elliptic system below:
8 q2 a2 b a > > < Du ¼ fk ðxÞ j uj u þ aþb hðxÞ j uj u j v j b Dv ¼ g l ðxÞ j v jq2 v þ aþb hðxÞ j uja j v jb2 v > > : u¼v ¼0
in X; in X;
ðEk;l Þ
on @ X;
2N where X is a bounded domain in R with 0 2 X, a > 1; b > 1 satisfying a þ b ¼ 2 ¼ N2 ðN P 3Þ; 1 < q < 2, and the parameters k; l P 0. We assume that fk ðxÞ ¼ kfþ ðxÞ þ f ðxÞ and g l ðxÞ ¼ lg þ ðxÞ þ g ðxÞ where the weight functions f, g and h satisfy the following conditions: ðD1Þ f ; g 2 CðXÞ with kfþ k1 ¼ kg þ k1 ¼ 1; f ¼ maxff ; 0g X 0 and g ¼ maxfg; 0g X 0; ðD2Þ h 2 CðXÞ with the sets N
fx 2 X j hðxÞ P 0g \ fx 2 X j f ðxÞ > 0g and
fx 2 X j hðxÞ P 0g \ fx 2 X j gðxÞ > 0g having positive measures; ðD3Þ there exists a positive number q with q > 2 when N P 6; q > ðN 2Þ=2 when 3 6 N 6 5 such that
hð0Þ ¼ 1 ¼ maxhðxÞ x2X
and
hð0Þ hðxÞ ¼ Oðj xjq Þ as x ! z: ⇑ Corresponding author. E-mail addresses: [email protected] (C.-y. Chen), [email protected] (T.-f. Wu). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.026
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Remark 1.1. Let BN ð0; rÞ ¼ fx 2 RN jj x j< rg. Accordingly, by condition ðD3Þ, we may assume that there exist two positive constants D0 and r 0 such that
hðxÞ > 0 for all x 2 BN ðx0 ; r 0 Þ X and
hð0Þ hðxÞ 6 D0 j xjq
for all x 2 BN ðx0 ; r 0 Þ:
We propose to study the system ðEk;l Þ in the framework of the Sobolev space H ¼ H10 ðXÞ H10 ðXÞ using the standard norm
kðu; v ÞkH ¼
Z
j ruj2 þ
Z
X
j rv j2
12
X
and consider, as a weak solution of the system, a pair of functions ðu; v Þ 2 H if
Z
X
ruru1 þ
Z
X
rv ru2
Z
X
fk j ujq2 uu1
Z
X
g l j v jq2 v u2
a
Z
aþb
X
h j uja2 u j v jb u1
b aþb
Z X
h j uja j v jb2 v u2
¼ 0 8ðu1 ; u2 Þ 2 H: Subsequently, for ðu; v Þ 2 H, the associated energy functional is defined by
J k;l ðu; v Þ ¼
1 1 kðu; v Þk2H 2 q
Z
fk j ujq þ
X
Z
X
g l j v jq
1 aþb
Z
h j uja j v jb :
X
The corresponding scalar version of semilinear elliptic equations with concave–convex nonlinearities, namely,
(
Du ¼ kf ðxÞ j ujq2 u þ hðxÞ j ujp2 u in X; u¼0 in @ X:
ðEk Þ
has been widely studied with a plethora of results. For the case when the weight functions are taken to be constant, namely, f h 1 with 2 < p 6 2 in ðEk Þ, Ambrosetti et al. [3] demonstrated that there exists k0 > 0 such that at least two positive solutions are admitted for k 2 ð0; k0 Þ, a positive solution for k ¼ k0 while no positive solution exists if k > k0 . The problem was taken up by various authors for more general cases; the readers are referred to Ambrosetti et al. [2], Chen and Wu [14], de Figueiredo et al. [17], EL Hamidi [18], Lubyshev [24], Radulescu and Repovs [25] and Wu [29,30] for detailed results. In particular, extending the problem to consider sign-changing weight functions, the authors in [14,29,30] showed multiplicity results with respect to the parameter k via the extraction of Palais–Smale sequences in the Nehari manifold. For the systems of semilinear elliptic equations with concave–convex nonlinearities, various studies concerning the solutions structures have also been presented ([1,4,6,8,13,16,19,23,26,28]). Among these, Adriouch and EL Hamidi [4] considered the following system:
8 a2 b a > > < Du ¼ ku þ aþb j uj u j v j b j uja j v jb2 v Dv ¼ l j v jq2 v þ aþb > > : u¼v ¼0
in X; in X; on @ X;
which has been proved to permit at least two positive solutions when the pair of parameters ðk; lÞ belongs to a certain subset of R2 . Similar results were obtained by Hsu [21] of system ðEk;l Þ when constant weight functions f ¼ g ¼ h ¼ 1 were assumed. Further studies involving sign-changing weight functions were taken up by Hsu [22] and Wu [32], where the multiplicity results were obtained for the subcritical case 2 < a þ b < 2 in [32] while those for the critical case a þ b ¼ 2 were obtained in [22] for q P N; N P 3 with the constraint on one of the weight functions being positive. These results will be improved upon in this paper when we investigate further the multiplicity of nontrivial nonnegative solutions by considering the system given by ðEk;l Þ extending the approach that was previously developed in [29,30,32]. The results presented here include cases for q > 2 when N P 6 and q > ðN 2Þ=2 when 3 6 N 6 5 while permitting the sign-changing property of all weight functions. Taking S to be the best Sobolev constant for the embedding of H10 ðXÞ in Laþb ðXÞ, the main result is stated in the following theorem. Theorem 1.1. Suppose that the weight functions f, g, h satisfy the conditions (D1)–(D3). Then there exists an explicit number Cða; b; q; SÞ > 0 for which if the parameters k; l satisfy 2
2
0 < k2q þ l2q < Cða; b; q; SÞ;
þ then problem ðEk;l Þ has at least two solutions uþ k;l ; v k;l and uk;l ; v k;l such that uk;l P 0;
v k;l P 0 in X and uk;l – 0; v k;l – 0.
Note that the existence and multiplicity of solutions concerning other combined effects of nonlinearities or weight function profiles have also been studied by other authors, we refer the readers to [5,12,14,23] and the references therein.
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We will first provide a list of all the relevant notations and preliminaries in Section 2, the proof of Theorem 1.1 will be concluded in Section 3. 2. Notations and preliminaries
We denote the best Sobolev constant for the imbedding of D1;2 ðRN Þ into L2 ðRN Þ by S, which is given by
R S¼
inf
u2D
1;2
ðRN Þnf0g
R
RN
RN jruj2 dx
2=2 > 0: j uj2 dx
ð2:1Þ
It has been established that S is independent of X RN in the sense that if
R
SðXÞ ¼
j ruj2 dx R 2=2 > 0; 2 u2H10 ðXÞnf0g j uj dx X X
inf
then SðXÞ ¼ SðRN Þ ¼ S, and the function
eðN2Þ=2
ue ðxÞ ¼
2
e2 þ j xj
e > 0 and x 2 RN
ðN2Þ=2 ;
is an extremal function for the minimum problem (2.1). Moreover, for each
v e ðxÞ ¼
e > 0,
2 ðN2Þ4
½NðN 2Þe
ð2:2Þ
ðe2 þ j xj2 ÞðN2Þ=2
is a positive solution of critical problem:
Du ¼j uj
2 2
N
u in R
with
Z
2
N j rv e j dx ¼
R
Z
N
N j v e j2 dx ¼ S 2 :
R
Now, consider the Nehari minimization problem: for ðk; lÞ 2 R2 n fð0; 0Þg,
hk;l ¼ inffJ k;l ðu; v Þ j ðu; v Þ 2 Nk;l g;
D E where Nk;l ¼ fðu; v Þ 2 H n fð0; 0Þg j J 0k;l ðu; v Þ; ðu; v Þ ¼ 0g is the Nehari manifold and
D
Z Z Z E J 0k;l ðu; v Þ; ðu; v Þ ¼ kðu; v Þk2H fk j ujq þ g l j v jq h j uja j v jb : X
X
X
Note that Nk;l contains every nonzero solution of problem ðEk;l Þ. This indicates that ðu; v Þ 2 Nk;l if and only if
kðu; v
Þk2H
Z
q
fk j uj þ X
Z
X
gl j v j
q
Z
h j uja j v jb ¼ 0
X
and the following result is concluded. Lemma 2.1. The energy functional J k;l is coercive and bounded below on Nk;l . We next define
D
E
Uk;l ðu; v Þ ¼ J0k;l ðu; v Þ; ðu; v Þ ; such that for ðu; v Þ 2 Nk;l ,
D
E
Z
h j uja j v jb Z Z fk j ujq þ g l j v jq : ¼ ð2 a bÞkðu; v Þk2H ðq a bÞ
U0k;l ðu; v Þ; ðu; v Þ ¼ ð2 qÞkðu; v Þk2H ða þ b qÞ
X
X
Applying the method used in Tarantello [27], we split Nk;l into three parts:
n D E o Nþk;l ¼ ðu; v Þ 2 Nk;l j U0k;l ðu; v Þ; ðu; v Þ > 0 ; n D E o N0k;l ¼ ðu; v Þ 2 Nk;l j U0k;l ðu; v Þ; ðu; v Þ ¼ 0 ; n D E o Nk;l ¼ ðu; v Þ 2 Nk;l j U0k;l ðu; v Þ; ðu; v Þ < 0 :
The above thus leads to the following lemmas.
ð2:3Þ
X
ð2:4Þ
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Lemma 2.2. We have (i) if ðu; v Þ 2 Nþ k;l , then (ii) if ðu; v Þ 2 N k;l , then
R R
X fk
X
R
j ujq þ
X
g l j v jq > 0;
h j uja j v jb > 0.
Proof. The proof follows immediately from (2.3) and (2.4). h Lemma 2.3. There exists an explicit number Cða; b; q; SÞ > 0 such that if 2
2
0 < k2q þ l2q < Cða; b; q; SÞ; then N0k;l ¼ ;. Proof. Suppose the contrary, that is there exist k; 2
l P 0 with
2
0 < k2q þ l2q < Cða; b; q; SÞ; but N0k;l – ;. Using (2.3) and (2.4), we have, for ðu; v Þ 2 N0k;l ,
ð2 qÞkðu; v Þk2H ¼ ða þ b qÞ
Z
h j uja j v jb dx
X
and
ða þ b 2Þkðu; v Þk2H ¼ ða þ b qÞ
Z
fk j ujq þ
Z
X
X
g l j v jq :
Applying the Hölder inequality and the Sobolev imbedding theorem gives,
kðu; v ÞkH P
aþbq 2q
Saþb
1=ð2abÞ
and
kðu; v ÞkH 6
aþbq aþb2
1 2q
12 q 2 2 S2q j kj2q þ j lj2q :
This implies 2 2q
k
2 2q
þl
P Cða; b; q; SÞ ¼
aþbq 2q
S
aþb
2=ð2abÞ
a þ b 2 q S aþbq
2 2q
which contradicts the assumption. We, therefore, conclude that N0k;l ¼ ; if 2
2
0 < k2q þ l2q < Cða; b; q; SÞ: This completes the proof. h The result of Lemma 2.3 suggests the introduction of the set
2 2 qCða; b; q; SÞ 2 2q 2q H ¼ ðk; lÞ 2 R j k; l P 0; 0 < k þ l < 2 and for each ðk; lÞ 2 H, we write Nk;l ¼ Nþ k;l [ Nk;l and define
hþk;l ¼
inf
ðu;v Þ2Nþ k;l
J k;l ðu; v Þ and hk;l ¼
inf
ðu;v Þ2N k;l
J k;l ðu; v Þ:
Subsequently, we have the following result. Lemma 2.4. If the pair of parameters ðk; lÞ 2 H, then (i) hk;l ¼ hþ k;l < 0; (ii) h k;l > c0 for some c 0 > 0.
;
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Proof. The proof is a repeat of that in [10, Thorem 2.5] and therefore omitted here. h The following lemma shows that the minimizers on Nk;l are ‘‘usually’’ critical points for J k;l . Lemma 2.5. For each ðk; lÞ 2 H and ðu0 ; v 0 Þ a local minimizer for J k;l on Nk;l , we have J 0k;l ðu0 ; v 0 Þ ¼ 0 in H1 (the dual space of the Sobolev space H). Proof. If ðu0 ; v 0 Þ is a local minimizer for J k;l on Nk;l , then ðu0 ; v 0 Þ is a solution of the optimization problem
minimizeJ k;l ðu; v Þ subject to Uk;l ðu; v Þ ¼ 0: Using Lagrange multipliers, there exists K 2 R such that
J 0k;l ðu0 ; v 0 Þ ¼ KU0k;l ðu0 ; v 0 Þ in H1 and thus,
D
E D E J 0k;l ðu0 ; v 0 Þ; ðu0 ; v 0 Þ ¼ K U0k;l ðu0 ; v 0 Þ; ðu0 ; v 0 Þ ¼ 0: D
E However U0k;l ðu0 ; v 0 Þ; ðu0 ; v 0 Þ must be nonzero since ðu0 ; v 0 Þ R N0k;l , it follows that K ¼ 0. This completes the proof. R Before proceeding to the next lemma, we first define, for each ðu; v Þ 2 H with X h j uja j v jb > 0;,
tmax ¼
ð2 qÞkðu; v Þk2H R ða þ b qÞ X h j uja j v jb
while for each u 2 H with
R
ða þ b qÞ
R
tmax ¼
X fk
X fk
j ujq þ
R
j ujq þ
R
ð2:5Þ h
1 !aþb2
> 0;
X
g l j v jq > 0, we write
X
g l j v jq
1 !2q
ða þ b 2Þkðu; v Þk2H
> 0:
The following lemmas are presented without proof; the readers are referred to Brown and Wu [11, Lemmas 2.5 and 2.6, respectively] for similar proofs. Lemma 2.6. For each ðu; v Þ 2 H with (i) if (ii) if
R RX
fk j ujq þ
X fk
q
j uj þ
R RX X
J k;l ðt þ u; tþ v Þ ¼
R X
X
h j uja j v jb > 0, we have
g l j v jq 6 0, then J k;l ðu; v Þ ¼ suptP0 J k;l ðtu; t v Þ > 0; g l j v jq > 0, then there is a unique 0 < tþ ¼ tþ ðuÞ < tmax such that ðtþ u; tþ v Þ 2 Nþ k;l and
inf J k;l ðtu; t v Þ;
06t6t max
Lemma 2.7. For each u 2 H with (i) if
R
R
X fk
J k;l ðu; v Þ ¼ sup J k;l ðtu; t v Þ: tPt max
j ujq þ
R X
g l j v jq > 0, we have
h j uja j v jb 6 0, then there is a unique tþ < t max such that ðt þ u; t þ v Þ 2 Nþ k;l and
J k;l ðt þ u; tþ v Þ ¼ inf J k;l ðtu; t v Þ; tP0
(ii) if
R X
a
h j uj j v j > 0, then there are unique 0 < t þ < tmax < t such that ðt þ u; t þ v Þ 2 Nþ k;l ; t u 2 N and b
J k;l ðt þ u; tþ v Þ ¼
inf J k;l ðtu; t v Þ;
06t6t max
J k;l ðt u; t v Þ ¼ supJ k;l ðtu; t v Þ: tP0
Next, we write
R Sa;b ¼
inf
R j ruj2 dx þ X j rv j2 dx R 2=ðaþbÞ a b X h j uj j v j dx X
u;v 2H10 ðXÞnf0g
and using the result of Alves et al. [6, Theorem 5], we deduce that
Sa;b ¼
" b aþb
a b
a # aþb b S; þ
a
ð2:6Þ
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where S is the best Sobolev constant given by (2.1). The following proposition thus provides a precise description for the Palais–Smale sequence of J k;l . Proposition 2.8. Each sequence fðun ; v n Þg H that satisfies N (i) J k;l ðun ; v n Þ ¼ b þ oð1Þ with b 2 1; hk;l þ N1 Sa2 ;b and (ii) J 0k;l ðun ; v n Þ ¼ oð1Þ in H has a convergent subsequence.
Proof. By Lemma 2.4 and the compact imbedding theorem, there exist a subsequence fðun ; v n Þg and ðu0 ; v 0 Þ a solution of problem ðEk;l Þ satisfying
un * u0
and
un ! u0
and
weakly in H10 ðXÞ;
vn * v0 vn ! v0
strongly in Lr ðXÞ for 1 < r < 2 :
Moreover,
D E J 0k;l ðu0 ; v 0 Þ; ðu0 ; v 0 Þ ¼ 0 and
Z
q
fk j un j þ
X
Z X
q
gl j v n j !
Z
q
fk j u0 j þ
Z
X
X
g l j v 0 jq
as n ! 1;
it follows that, by Lemma 2.4, J k;l ðu0 ; v 0 Þ P hk;l . Given that un * u0 and
v n * v 0 weakly in H10 ðXÞ, we must have
kðun ; v n Þk2H ¼ ðu0 ; v 0 Þk2H þ kðun u0 ; v n v 0 Þk2H þ oð1Þ: Note that, by the Brezis–Lieb lemma [7],
Z
h j un ja j v n jb dx ¼
X
Z
h j u0 ja j v 0 jb dx þ
X
Z
h j un u0 ja j v n v 0 jb dx þ oð1Þ
X
and
Z h j un ja2 un j v n jb ju0 ja2 u0 j v 0 jb ðun u0 Þdx þ h jun ja jv n jb2 v n ju0 ja j v 0 jb2 v 0 ðv n v 0 Þdx X X Z a b ¼ hjun u0 j jv n v 0 j dx þ oð1Þ:
Z
X
As a result,
1 1 J k;l ðun ; v n Þ ¼ J k;l ðu0 ; v 0 Þ þ kðun u0 ; v n v 0 Þk2H 2 aþb
Z
hjun u0 ja j v n v 0 jb dx þ oð1Þ
X
and
D E D E oð1Þ ¼ ðun u0 ; v n v 0 Þ; J 0k;l ðun ; v n Þ ¼ ðun u0 ; v n v 0 Þ; J 0k;l ðun ; v n Þ J 0k;l ðu0 ; v 0 Þ Z ¼ kðun u0 ; v n v 0 Þk2H hjun u0 ja jv n v 0 jb dx þ oð1Þ:
ð2:7Þ
X
This implies
1 1 1 2 2 kðun u0 ; v n v 0 ÞkH ¼ kðun u0 ; v n v 0 ÞkH N 2 aþb
Z
hjun u0 ja jv n v 0 jb dx þ oð1Þ
X
¼ J k;l ðun ; v n Þ J k;l ðu0 ; v 0 Þ þ oð1Þ 6 J k;l ðun ; v n Þ hk; l þ oð1Þ and we can therefore conclude that N
2 kðun u0 ; v n v 0 ÞkH < Sa2 ;b ;
when n is sufficiently large. Using (2.6) and (2.7) leads to
Z N 4=ðN2Þ 2 2 6 u ð u u ; v v Þ ð u ; v v Þ hjun u0 ja jv n v 0 jb dx ¼ oð1Þ: kðun u0 ; v n v 0 ÞkH 1 S2N k k k k n 0 n 0 n 0 n 0 a;b H H X
ð2:8Þ Consequently, ðun ; v n Þ ! ðu0 ; v 0 Þ strongly in H. h
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3. Proof of Theorem 1 First, we establish the existence of a local minimum for J k;l on Nþ k;l . þ þ Theorem 3.1. If the pair of parameters ðk; lÞ 2 H, then J k;l has a minimizer uþ 0 ; v 0 in Nk;l satisfying (i) J k;l uþ ; v þ ¼ hk;l ¼ hþ k;l < 0; þ 0þ 0 (ii) u0 ; v 0 is a solution of problem ðEk;l Þ such that uþ 0 P 0;
v þ0 P 0 in X and uþ0 – 0; v þ0 – 0.
Proof. By the Ekeland variational principle [15] (or, alternatively, see Wu [29, Proposition 9]), there exists fðun ; v n Þg Nk;l which is a minimizing sequence for J k;l such that
J k;l ðun ; v n Þ ¼ hk;l þ oð1Þ and J0k;l ðun ; v n Þ ¼ oð1Þ in H1 : þ þ Using Proposition 3.1, it can be shown that a subsequence fðun ; v n Þg and uþ k;l ; v k;l 2 Nk;l exist such that ðun ; v n Þ !
þ þ
þ þ
þ þ þ þ þ þ þ uþ k;l ; v k;l strongly in H and J k;l uk;l ; v k;l ¼ hk;l . With J k;l uk;l ; v k;l ¼ J k;l uk;l ; v k;l and uk;l ; v k;l 2 Nk;l , by Lemma
þ
þ þ 2.5, it is possible to assume that uþ k;l ; v k;l is a solution of problem ðEk;l Þ such that uk;l P 0; v k;l P 0 in X. Next, we prove
v þk;l – 0. Assuming the contrary and, without loss of generality, we take v þk;l 0. By condition ðD2Þ, we may
that uþ k;l – 0; choose w 2
H10 ð
XÞ n f0g such that Z Z Z
a 2q
h uþk;l j wjb dx 6 j rwj2 dx ¼ g l j wjq dx: ð2 qÞða þ bÞ X X X
þ þ Applying Lemma 2.7 gives the result that a unique 0 < t þ < tmax exits satisfying t þ uþ k;l ; t w 2 Nk;l . Moreover,
0
tmax
1
q R
12q R
ða þ b qÞ X fk uþk;l dx þ X g l j wjq dx B C ¼@ A >1 2 þ ða þ b 2Þ uk;l ; w
H
and
J k;l tþ uþk;l ; t þ w ¼
inf J k;l tuþk;l ; tw :
06t6t max
This implies that
J k;l tþ uþk;l ; t þ w 6 J k;l uþk;l ; w 6 J k;l uþk;l ; 0 ¼ hþk;l ; þ which is a contradiction; accordingly, we must have uþ k;l – 0; v k;l – 0. We now proceed to consider the following critical problem:
(
Du ¼j uj2 u2
H10 ð
2
u in X;
h
ðb E0Þ
XÞ
and associated with the equation ð b E 0 Þ, the energy functional I1 in H10 ðXÞ is given by
I1 ðuÞ ¼
1 1 kuk2H1 2 2
Z
j uj2 dx:
X
It is established that
inf 1
u2M
ðRN Þ
I1 ðuÞ ¼
inf 1
u2M ðXÞ
I1 ðuÞ ¼
1 N2 S N
for all domain X RN ;
where
n o
M1 ðRN Þ ¼ u 2 D1;2 ðRN Þ n f0g j ðI1 Þ0 ðuÞ; u ¼ 0 and
n o
M1 ðXÞ ¼ u 2 H10 ðXÞ n f0g j ðI1 Þ0 ðuÞ; u ¼ 0
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are the Nehari manifolds. In fact, I1 ðuÞ is never attained on any domain X$RN . Repeating the method employed by [9] and N using Remark 1.1, we take g 2 C 1 0 ðR Þ as a radially symmetric function with 0 6 g 6 1; jrgj 6 C and
gðxÞ ¼
1 j x j6 r 0 =2; 0
j x jP r 0 :
Assume
we ðxÞ ¼ gðxÞv e ðxÞ; where
ð3:1Þ
v e ðxÞ is given by (2.2). Following the argument for the proof of Lemma 4.2 in [20] (or see Struwe [26]) gives N
kwe k2H1 ¼ S 2 þ O
eN2
and
Z
N
jwe j2 dx ¼ S 2 þ OðeN Þ:
ð3:2Þ
X
Moreover, we have the following results. Lemma 3.2. We have
8 N < S 2 þ o e2 ; if N P 6; N2 hjwe j dx ¼ N : S 2 þ o e 2 ; if 3 6 N 6 5: X
Z
2
Proof. The proof is almost identical to that of [14, Lemma 3.1] and is omitted here for brevity. h Next, we establish the existence of a local minimum for J k;l on N k;l , which requires the following result. Lemma 3.3. We have
pffiffiffi pffiffiffi 1 N supJ k;l uþk;l þ t awe ; v þk;l þ t bwe < hk;l þ Sa2 ;b : N tP0
Furthermore, there exists t 0 > 0 such that
pffiffiffi pffiffiffi uþk;l þ t0 awe ; v þk;l þ t0 bwe 2 Nk;l and
hk;l < hk;l þ
1 N2 S : N a;b
Proof. Since
pffiffiffi pffiffiffi J k;l ðuþk;l þ t awe ; v þk;l þ t bwe Þ Z
Z 2
pffiffiffi pffiffiffi
q pffiffiffi pffiffiffi
q 1 1
¼ ðuþk;l þ t awe ; v þk þ t bwe Þ 1 fk uþk;l þ t awe dx þ g l v þk;l þ t bwe dx 2 q X H X Z
pffiffiffi
b pffiffiffi
a
þ 1
þ h u þ t awe v k;l þ t bwe dx a þ b X k;l Z Z Z Z
pffiffiffi
2 pffiffiffi
q
pffiffiffi
2 pffiffiffi q 1
1
þ 1
fk uþk þ t awe dx þ g l v þk þ t bwe dx ¼
r uþk;l þ t awe dx þ
r v k;l þ t bwe dx 2 X 2 X q X X Z
p ffiffiffi a b ffiffiffi p 1
þ
þ
Xh uk;l þ t awe v k;l þ t bwe dx aþb Z
Z Z Z Z
2 pffiffiffi pffiffiffi 1 þ
2 1
aþb 2 þ
þ w ¼ r u dx þ r v dx þ r tw dx þ r u r t a rv þk;l r t bwe dx dx þ j j
k;l
e e k;l
k;l 2 X 2 X 2 X X X Z
Z Z
pffiffiffi
q pffiffiffi
b pffiffiffi
q pffiffiffi
a
þ 1 1
þ
þ
þ fk uk;l þ t awe dx þ g l v k;l þ t bwe dx h uk;l þ t awe v k;l þ t bwe dx q X aþb X X Z Z Z a b pffiffiffi b pffiffiffi a þ aþb 1 q h uþk;l þ t awe v k;l þ t bwe uþk;l v þk;l 6 J k;l ðuþk;l ; v þk;l Þ þ jrtwe j2 dx þ C 0 ðtwe Þ dx 2 aþb X X X a1 b pffiffiffi a b1 pffiffiffi v þk;l t awe b uþk;l v þk;l t bwe dx ¼ hk;l þ Kðtwe Þ; a uþk;l
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10825
where
Kðtwe Þ ¼
Z a b pffiffiffi b pffiffiffi a þ 1 h uþk;l þ t awe v k;l þ t bwe uþk;l v þk;l 2 2 X X X a1 b pffiffiffi a b1 pffiffiffi þ þ þ þ a uk;l v k;l t awe b uk;l v k;l t bwe dx;
aþb
Z
jrtwe j2 dx þ C 0
Z
ðtwe Þq dx
it is sufficient if we show that
supKðtwe Þ < tP0
Moreover,
R
1 N2 S : N a;b
X
gw2e dx > 0 and we have
" # 2 þ 2 Z 2 1 t2 uþk uk 2 ðuþk Þ we;z dx jrwe j C 0 t we dx g þ we 2 X t t t 2 1 X X p Z þ tp u ¼ Oðtq Þ g k þ we;z dx ! 1 as t ! 1: p X t Z
t2 Kðtwe Þ ¼ 2
2
q
It can be readily seen that for
Kðtwe Þ
0 such that
for all t 2 ½t 0 ; 1Þ;
which requires the following condition to be satisfied:
max Kðtwe Þ
< D1 e þ Oðe Þ 2 2 2 we dx ¼ D2 e j ln ej þ Oðe Þ if N ¼ 4; > X : D3 e þ Oðe2 Þ if N ¼ 3:
Z
ð3:3Þ
Additionally,
Z
q
X
we dx 6
Z
ðC N eÞ
BN ð0;eÞ
qðN2Þ 2
eqðN2Þ
þ
Z BN ð0;r 0 ÞnBN ð0;eÞ
qðN2Þ 2
ðC N eÞ
jxjqðN2Þ
ðN2Þð2qÞþ4 2
6 Ce
qðN2Þ 2
þ Ce
Z
r0
r ðq1Þð2NÞþ1 dr
e
and so
Z X
wqe dx 6
8 qðN2Þ < C eðN2Þð2qÞþ4 2 þ Ce 2 ;
if q –
N ; N2
qðN2Þ qðN2Þ : C eðN2Þð2qÞþ4 N 2 þ C e 2 þ C e 2 j ln ej; if q ¼ N2 :
As a result, we have
Z X
( q
we dx ¼
oðe2 Þ; N2 2
oðe
if N P 6; Þ; if 3 6 N 6 5
ð3:4Þ
and we proceed by considering the two cases separately: Case I ðN P 6Þ : Remark 1.1 implies we ðxÞ ¼ 0 and
a b a1 b pffiffiffi a b1 pffiffiffi pffiffiffi b pffiffiffi a þ uþk;l þ t awe v k;l þ t bwe uþk;l v þk;l a uþk;l v þk;l t awe b uþk;l v þk;l t bwe ¼ 0; for all x 2 X n BN ð0; r 0 Þ. Since hðxÞ > 0 for all x 2 BN ð0; r 0 Þ, using Lemma 4.1 in Appendix, we have
a1 b pffiffiffi a b1 pffiffiffi pffiffiffi b þ a þ b pffiffiffi a þ þ þ þ þ þ hðxÞ uk;l þ t awe v k;l þ t bwe uk;l v k;l a uk;l v k;l t awe b uk;l v k;l t bwe a1 b1 a b v þk;l ðtwe Þ2 ; P hðxÞ a2 b2 ðtwe Þaþb þ C 0 uþk;l ð3:5Þ
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which implies Z a b a1 b pffiffiffi a b1 pffiffiffi pffiffiffi b pffiffiffi a þ hðxÞ uþk;l þ t awe v k;l þ t bwe uþk;l v þk;l a uþk;l v þk;l t awe b uþk;l v þk;l t bwe dx X
Z
P
hðxÞ
h
i
b
a
a2 b2 ðtwe Þaþb þ Cðtwe Þ2 dx:
X
Using Lemma 3.2 and (3.2)–(3.4) leads to
Z Z a b a2 b2 aþb aþb w2e dx hwe þ C 0 t q wqe dx t 2 aþb X X X X 0 1N2 0 R 1N2 R N 2 2 2 2 2 Bða þ bÞ S C e þ oðe Þ C 1 Bða þ bÞ X jrwe j dx C X we dx C C þ oðe2 Þ 6 1 B C þ oðe2 Þ 6 B 2 2 aþb @ A A N N @ a b N2 a bR 2 aþb a2 b2 X hwe a2 b2 S þ oðe2 Þ
Kðtwe Þ ¼
aþb
1 6 N
t2
Z
" b aþb
a
N a #2 aþb b
þ
b
Z
jrwe j2 dx C
a
1 N2 1 N Sa;b C e2 þ oðe2 Þ < Sa2 ;b N N
N
S 2 C e2 þ oðe2 Þ ¼
for sufficiently small e and t 2 ½0; t 0 . Case II ð3 6 N 6 5Þ : Using Lemma 4.1 in Appendix, it follows that
Kðtwe Þ 6
aþb 2
a
t
2
b
a2 b2 aþb t aþb
kwe k2H1
Z
aþb
X
hwe dx C 0
t aþb1 aþb1
Z X
N2 wae þb1 dx þ o e 2 ;
2 1 R
Nþ2 N2 with some constant C 0 > 0. Note that X we;z
dx ¼ C 1 e 2 þ Oðe 2 Þ, for some C 1 > 0. Applying Lemma 3.2 and (3.2), we have
N2 Nþ2 a2 b2 aþb N2 t aþb1 N2 t S t S C0 Kðtwe Þ 6 e 2 þo e 2 þO e 2 2 aþb aþb1 a
aþb
¼
aþb 2
b
N 2
2
N2 a2 b2 aþb N2 taþb1 N2 e 2 þo e 2 : t S C0 aþb aþb1 a
N
t2 S2
b
Denoting by t e the value of t 2 ½0; t0 that gives rise to the maximum of the above right-hand side, then t e satisfies N
a
b
N
N2 2
ða þ bÞS 2 ¼ a2 b2 t ae þb2 S 2 þ tae þb3 C e
þo
N2 2
e
:
It can be deduced that 1 !aþb2
aþb te ¼ a b a2 b2
N2 2
Ce
t2e
3
N2 þo e 2
and thus
max Kðtwe Þ 6
t2½0;t0
aþb 2
N2 a2 b2 aþb N2 N2 te S Ct ae þb3 e 2 þ o e 2 aþb a
N
t 2e S 2
b
2 !aþb2
aþb aþb ¼ a b 2 a2 b2
a
b
aþb !aþb2
a2 b2 a þ b S a þ b aa2 b2b N 2
N
S 2 Ct2e
3
N2 2
e
þo
N2 2
e
N2 N2 1 N b a N 1 a aþb b aþb N2 ð Þ þð Þ S 2 Ct 2e 3 e 2 þ o e 2 < Sa2 ;b ¼ N b N a for
e sufficiently small. Using the result of Wu [31, Lemma 3.2], we therefore conclude that such t0 > 0 exists with pffiffiffi pffiffiffi uþk;l þ t0 awe ; v þk;l þ t0 bwe 2 Nk;l
and
pffiffiffi pffiffiffi 1 N J k;l uþk;l þ t 0 awe ; v þk;l þ t0 bwe < hk;l þ Sa2 ;b : N This completes the proof. h
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10827
Theorem 3.4. If the pair of parameters ðk; lÞ 2 H, then J k;l has a minimizer u in N 0 ; v0 k;l which satisfies (i) J k;l u ; v ¼ h k;l ; 0 0 (ii) u0 ; v 0 is a solution of problem ðEk;l Þ, such that u 0 P 0;
v 0 P 0 in X with u0 – 0 and v 0 – 0.
Proof. By the Ekeland variational principle [15] (or see Wu [29, Proposition 9]), there exists fðun ; v n Þg Nk;l which is a minimizing sequence for J k;l satisfying
J k;l ðun ; v n Þ ¼ hk;l þ oð1Þ and J 0k;l ðun ; v n Þ ¼ oð1Þ in H1 : Using Proposition 3.1 and Lemma 3.3, it follows that a subsequence fðun ; v n Þg and u k;l ; v k;l 2 Nk;l exist such that
strongly in H and J f ;g u and Since J k;l u ðun ; v n Þ ! u k;l ; v k;l k;l ; v k;l ¼ hk;l . k;l ; v k;l ¼ J k;l uk;l ; v k;l
uk;l ; v k;l 2 Nk;l , by Lemmas 2.2 and 2.5, we may assume that uk;l ; v k;l is a solution of problem ðEk;l Þ such that u k;l P 0; v k;l P 0 in X and uk;l – 0;
v k;l – 0.
h
We now complete the proof of Theorem 1.1. Applying Theorems 3.1 and 3.4 on problem ðEk;l Þ, we conclude that two solu þ þ tions uþ u k;l ; v k;l 2 Nk;l exist satisfying uk;l P 0; v k;l P 0 in Xwith uk;l – 0 and v k;l – 0. Since k;l ; v k;l 2 Nk;l and þ þ and u are distinct. Nþ k;l ; v k;l k;l \ Nk;l ¼ ;, this implies that uk;l ; v k;l 4. Appendix Lemma 4.1. There exist positive constants C 1 ðaÞ; C 2 ðbÞ > 0 such that a
a b
a
ða þ bÞa ðc þ dÞb P aa cb þ aa d þ b cb þ b d þ C 1 ðaÞaa1 bc þ C 1 ðaÞaa1 bd þ C 2 ðbÞb cb1 d þ C 2 ðbÞaa cb1 d b
þ C 1 ðaÞC 2 ðbÞaa1 bc
b1
b
b
d:
Proof. The result follows immediately from (4.3) in [3, p. 537]. h Acknowledgments The author is grateful for the referee’s valuable suggestions and helps. This research was supported in part by the National Science Council and the National Center for Theoretical Sciences (South), Taiwan. References [1] A. Ahammou, Positive radial solutions of nonlinear elliptic system, New York J. Math. 7 (2001) 267–280. [2] A. Ambrosetti, G.J. Azorero, I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996) 219–242. [3] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519– 543. [4] K. Adriouch, A. EL Hamidi, The Nehari manifold for systems of nonlinear elliptic equations, Nonlinear Anal. 64 (2006) 2149–2164. [5] C.O. Alves, G. Figueiredo, M. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Commun. Partial Differ. Equat. 36 (2011) 1565–1586. [6] C.O. Alves, D.C. de Morais Filho, M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000) 771–787. [7] H. Brézis, E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486–490. [8] Y. Bozhkov, E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differ. Equat. 190 (2003) 239–267. [9] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math. 16 (1983) 437–477. [10] K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336. [11] K.J. Brown, T.F. Wu, A fibrering map approach to a potential operator equation and its applications, Differ. Integral Equat. 22 (2009) 1097–1114. [12] A. Cañada, Nonlinear ordinary boundary value problems under a combined effect of periodic and attractive nonlinearities, J. Math. Anal. Appl. 243 (2000) 174–189. [13] P. Clément, D.G. de Figueiredo, E. Mitidieri, Positive solutions of semilinear elliptic systems, Commun. Partial Differ. Equat. 17 (1992) 923–940. [14] C.Y. Chen, T.F. Wu, Multiple positive solutions for indefinite semilinear elliptic problems involving critical Sobolev exponent, submitted for publication. [15] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 17 (1974) 324–353. [16] D.G. de Figueiredo, P. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994) 99–116. [17] D.G. de Figueiredo, J.P. Gossez, P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2003) 452– 467. [18] A. el Hamidi, Multiple solutions with changing sign energy to a nonlinear elliptic equation, Commun. Pure Appl. Anal. 3 (2004) 253–265. [19] A. el Hamidi, Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl. 300 (2004) 30–42. [20] H. He, J. Yang, Positive solutions for critical inhomogeneous elliptic problems in non-contractible domains, Nonlinear Anal. 70 (2009) 952–973.
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[21] T.S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave–convex nonlinearities, Nonlinear Anal. 71 (2009) 2688– 2698. [22] T.S. Hsu, Multiple positive solutions for a quasilinear elliptic system involving concave–convex nonlinearities and sign-changing weight functions, Int. J. Math. Math. Sci. (2012) (Article ID 109214, 21pp), doi:10.1155/2012/109214. [23] A. Kristály, V. Radulescu, C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, no. 136, Cambridge University Press, Cambridge, 2010. [24] V.F. Lubyshev, Multiple positive solutions of an elliptic equation with a convex–concave nonlinearity containing a sign-changing term, Tr. Mat. Inst. Steklova (Russian) 269 (2010). Teoriya Funktsii i Differentsialnye Uravneniya, 167–180; translation in Proc. Steklov Inst. Math. 269 (2010) 160–173. [25] V. Radulescu, D. Repovs, Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal. 75 (2012) 1524–1530. [26] M. Squassina, An eigenvalue problem for elliptic systems, New York J. Math. 6 (2000) 95–106. [27] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Lineairé 9 (3) (1992) 281– 304. [28] J. Vélin, Existence results for some nonlinear elliptic system with lack of compactness, Nonlinear Anal. 52 (2003) 1017–1034. [29] T.F. Wu, On semilinear elliptic equations involving concave–convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. 318 (2006) 253–270. [30] T.F. Wu, Multiplicity results for a semilinear elliptic equation involving sign-changing weight function, Rocky Mount. J. Math. 39 (2009) 995–1012. [31] T.F. Wu, Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight, J. Differ. Equat. 249 (2010) 1459–1578. [32] T.F. Wu, The Nehari manifold for a elliptic system involving sing-changing weight functions, Nonlinear Anal. 68 (2008) 1733–1745.
Author's personal copy Applied Mathematics and Computation 218 (2012) 10817–10828
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The Nehari manifold for indefinite semilinear elliptic systems involving critical exponent Ching-yu Chen, Tsung-fang Wu ⇑ Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
a r t i c l e
i n f o
a b s t r a c t In this paper, we study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system ðEk;l Þ involving critical exponents and sign-changing weight functions. Using Nehari manifold, the system is proved to have at least two nontrivial nonnegative solutions when the pair of the parameters ðk; lÞ belongs to a certain subset of R2 . Ó 2012 Elsevier Inc. All rights reserved.
Keywords: Indefinite semilinear elliptic systems Multiple positive solutions Critical Sobolev exponent Nehari manifold
1. Introduction In this paper we examine the multiplicity results derived from the nontrivial nonnegative solutions of the indefinite semilinear elliptic system below:
8 q2 a2 b a > > < Du ¼ fk ðxÞ j uj u þ aþb hðxÞ j uj u j v j b Dv ¼ g l ðxÞ j v jq2 v þ aþb hðxÞ j uja j v jb2 v > > : u¼v ¼0
in X; in X;
ðEk;l Þ
on @ X;
2N where X is a bounded domain in R with 0 2 X, a > 1; b > 1 satisfying a þ b ¼ 2 ¼ N2 ðN P 3Þ; 1 < q < 2, and the parameters k; l P 0. We assume that fk ðxÞ ¼ kfþ ðxÞ þ f ðxÞ and g l ðxÞ ¼ lg þ ðxÞ þ g ðxÞ where the weight functions f, g and h satisfy the following conditions: ðD1Þ f ; g 2 CðXÞ with kfþ k1 ¼ kg þ k1 ¼ 1; f ¼ maxff ; 0g X 0 and g ¼ maxfg; 0g X 0; ðD2Þ h 2 CðXÞ with the sets N
fx 2 X j hðxÞ P 0g \ fx 2 X j f ðxÞ > 0g and
fx 2 X j hðxÞ P 0g \ fx 2 X j gðxÞ > 0g having positive measures; ðD3Þ there exists a positive number q with q > 2 when N P 6; q > ðN 2Þ=2 when 3 6 N 6 5 such that
hð0Þ ¼ 1 ¼ maxhðxÞ x2X
and
hð0Þ hðxÞ ¼ Oðj xjq Þ as x ! z: ⇑ Corresponding author. E-mail addresses: [email protected] (C.-y. Chen), [email protected] (T.-f. Wu). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.04.026
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Remark 1.1. Let BN ð0; rÞ ¼ fx 2 RN jj x j< rg. Accordingly, by condition ðD3Þ, we may assume that there exist two positive constants D0 and r 0 such that
hðxÞ > 0 for all x 2 BN ðx0 ; r 0 Þ X and
hð0Þ hðxÞ 6 D0 j xjq
for all x 2 BN ðx0 ; r 0 Þ:
We propose to study the system ðEk;l Þ in the framework of the Sobolev space H ¼ H10 ðXÞ H10 ðXÞ using the standard norm
kðu; v ÞkH ¼
Z
j ruj2 þ
Z
X
j rv j2
12
X
and consider, as a weak solution of the system, a pair of functions ðu; v Þ 2 H if
Z
X
ruru1 þ
Z
X
rv ru2
Z
X
fk j ujq2 uu1
Z
X
g l j v jq2 v u2
a
Z
aþb
X
h j uja2 u j v jb u1
b aþb
Z X
h j uja j v jb2 v u2
¼ 0 8ðu1 ; u2 Þ 2 H: Subsequently, for ðu; v Þ 2 H, the associated energy functional is defined by
J k;l ðu; v Þ ¼
1 1 kðu; v Þk2H 2 q
Z
fk j ujq þ
X
Z
X
g l j v jq
1 aþb
Z
h j uja j v jb :
X
The corresponding scalar version of semilinear elliptic equations with concave–convex nonlinearities, namely,
(
Du ¼ kf ðxÞ j ujq2 u þ hðxÞ j ujp2 u in X; u¼0 in @ X:
ðEk Þ
has been widely studied with a plethora of results. For the case when the weight functions are taken to be constant, namely, f h 1 with 2 < p 6 2 in ðEk Þ, Ambrosetti et al. [3] demonstrated that there exists k0 > 0 such that at least two positive solutions are admitted for k 2 ð0; k0 Þ, a positive solution for k ¼ k0 while no positive solution exists if k > k0 . The problem was taken up by various authors for more general cases; the readers are referred to Ambrosetti et al. [2], Chen and Wu [14], de Figueiredo et al. [17], EL Hamidi [18], Lubyshev [24], Radulescu and Repovs [25] and Wu [29,30] for detailed results. In particular, extending the problem to consider sign-changing weight functions, the authors in [14,29,30] showed multiplicity results with respect to the parameter k via the extraction of Palais–Smale sequences in the Nehari manifold. For the systems of semilinear elliptic equations with concave–convex nonlinearities, various studies concerning the solutions structures have also been presented ([1,4,6,8,13,16,19,23,26,28]). Among these, Adriouch and EL Hamidi [4] considered the following system:
8 a2 b a > > < Du ¼ ku þ aþb j uj u j v j b j uja j v jb2 v Dv ¼ l j v jq2 v þ aþb > > : u¼v ¼0
in X; in X; on @ X;
which has been proved to permit at least two positive solutions when the pair of parameters ðk; lÞ belongs to a certain subset of R2 . Similar results were obtained by Hsu [21] of system ðEk;l Þ when constant weight functions f ¼ g ¼ h ¼ 1 were assumed. Further studies involving sign-changing weight functions were taken up by Hsu [22] and Wu [32], where the multiplicity results were obtained for the subcritical case 2 < a þ b < 2 in [32] while those for the critical case a þ b ¼ 2 were obtained in [22] for q P N; N P 3 with the constraint on one of the weight functions being positive. These results will be improved upon in this paper when we investigate further the multiplicity of nontrivial nonnegative solutions by considering the system given by ðEk;l Þ extending the approach that was previously developed in [29,30,32]. The results presented here include cases for q > 2 when N P 6 and q > ðN 2Þ=2 when 3 6 N 6 5 while permitting the sign-changing property of all weight functions. Taking S to be the best Sobolev constant for the embedding of H10 ðXÞ in Laþb ðXÞ, the main result is stated in the following theorem. Theorem 1.1. Suppose that the weight functions f, g, h satisfy the conditions (D1)–(D3). Then there exists an explicit number Cða; b; q; SÞ > 0 for which if the parameters k; l satisfy 2
2
0 < k2q þ l2q < Cða; b; q; SÞ;
þ then problem ðEk;l Þ has at least two solutions uþ k;l ; v k;l and uk;l ; v k;l such that uk;l P 0;
v k;l P 0 in X and uk;l – 0; v k;l – 0.
Note that the existence and multiplicity of solutions concerning other combined effects of nonlinearities or weight function profiles have also been studied by other authors, we refer the readers to [5,12,14,23] and the references therein.
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We will first provide a list of all the relevant notations and preliminaries in Section 2, the proof of Theorem 1.1 will be concluded in Section 3. 2. Notations and preliminaries
We denote the best Sobolev constant for the imbedding of D1;2 ðRN Þ into L2 ðRN Þ by S, which is given by
R S¼
inf
u2D
1;2
ðRN Þnf0g
R
RN
RN jruj2 dx
2=2 > 0: j uj2 dx
ð2:1Þ
It has been established that S is independent of X RN in the sense that if
R
SðXÞ ¼
j ruj2 dx R 2=2 > 0; 2 u2H10 ðXÞnf0g j uj dx X X
inf
then SðXÞ ¼ SðRN Þ ¼ S, and the function
eðN2Þ=2
ue ðxÞ ¼
2
e2 þ j xj
e > 0 and x 2 RN
ðN2Þ=2 ;
is an extremal function for the minimum problem (2.1). Moreover, for each
v e ðxÞ ¼
e > 0,
2 ðN2Þ4
½NðN 2Þe
ð2:2Þ
ðe2 þ j xj2 ÞðN2Þ=2
is a positive solution of critical problem:
Du ¼j uj
2 2
N
u in R
with
Z
2
N j rv e j dx ¼
R
Z
N
N j v e j2 dx ¼ S 2 :
R
Now, consider the Nehari minimization problem: for ðk; lÞ 2 R2 n fð0; 0Þg,
hk;l ¼ inffJ k;l ðu; v Þ j ðu; v Þ 2 Nk;l g;
D E where Nk;l ¼ fðu; v Þ 2 H n fð0; 0Þg j J 0k;l ðu; v Þ; ðu; v Þ ¼ 0g is the Nehari manifold and
D
Z Z Z E J 0k;l ðu; v Þ; ðu; v Þ ¼ kðu; v Þk2H fk j ujq þ g l j v jq h j uja j v jb : X
X
X
Note that Nk;l contains every nonzero solution of problem ðEk;l Þ. This indicates that ðu; v Þ 2 Nk;l if and only if
kðu; v
Þk2H
Z
q
fk j uj þ X
Z
X
gl j v j
q
Z
h j uja j v jb ¼ 0
X
and the following result is concluded. Lemma 2.1. The energy functional J k;l is coercive and bounded below on Nk;l . We next define
D
E
Uk;l ðu; v Þ ¼ J0k;l ðu; v Þ; ðu; v Þ ; such that for ðu; v Þ 2 Nk;l ,
D
E
Z
h j uja j v jb Z Z fk j ujq þ g l j v jq : ¼ ð2 a bÞkðu; v Þk2H ðq a bÞ
U0k;l ðu; v Þ; ðu; v Þ ¼ ð2 qÞkðu; v Þk2H ða þ b qÞ
X
X
Applying the method used in Tarantello [27], we split Nk;l into three parts:
n D E o Nþk;l ¼ ðu; v Þ 2 Nk;l j U0k;l ðu; v Þ; ðu; v Þ > 0 ; n D E o N0k;l ¼ ðu; v Þ 2 Nk;l j U0k;l ðu; v Þ; ðu; v Þ ¼ 0 ; n D E o Nk;l ¼ ðu; v Þ 2 Nk;l j U0k;l ðu; v Þ; ðu; v Þ < 0 :
The above thus leads to the following lemmas.
ð2:3Þ
X
ð2:4Þ
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Lemma 2.2. We have (i) if ðu; v Þ 2 Nþ k;l , then (ii) if ðu; v Þ 2 N k;l , then
R R
X fk
X
R
j ujq þ
X
g l j v jq > 0;
h j uja j v jb > 0.
Proof. The proof follows immediately from (2.3) and (2.4). h Lemma 2.3. There exists an explicit number Cða; b; q; SÞ > 0 such that if 2
2
0 < k2q þ l2q < Cða; b; q; SÞ; then N0k;l ¼ ;. Proof. Suppose the contrary, that is there exist k; 2
l P 0 with
2
0 < k2q þ l2q < Cða; b; q; SÞ; but N0k;l – ;. Using (2.3) and (2.4), we have, for ðu; v Þ 2 N0k;l ,
ð2 qÞkðu; v Þk2H ¼ ða þ b qÞ
Z
h j uja j v jb dx
X
and
ða þ b 2Þkðu; v Þk2H ¼ ða þ b qÞ
Z
fk j ujq þ
Z
X
X
g l j v jq :
Applying the Hölder inequality and the Sobolev imbedding theorem gives,
kðu; v ÞkH P
aþbq 2q
Saþb
1=ð2abÞ
and
kðu; v ÞkH 6
aþbq aþb2
1 2q
12 q 2 2 S2q j kj2q þ j lj2q :
This implies 2 2q
k
2 2q
þl
P Cða; b; q; SÞ ¼
aþbq 2q
S
aþb
2=ð2abÞ
a þ b 2 q S aþbq
2 2q
which contradicts the assumption. We, therefore, conclude that N0k;l ¼ ; if 2
2
0 < k2q þ l2q < Cða; b; q; SÞ: This completes the proof. h The result of Lemma 2.3 suggests the introduction of the set
2 2 qCða; b; q; SÞ 2 2q 2q H ¼ ðk; lÞ 2 R j k; l P 0; 0 < k þ l < 2 and for each ðk; lÞ 2 H, we write Nk;l ¼ Nþ k;l [ Nk;l and define
hþk;l ¼
inf
ðu;v Þ2Nþ k;l
J k;l ðu; v Þ and hk;l ¼
inf
ðu;v Þ2N k;l
J k;l ðu; v Þ:
Subsequently, we have the following result. Lemma 2.4. If the pair of parameters ðk; lÞ 2 H, then (i) hk;l ¼ hþ k;l < 0; (ii) h k;l > c0 for some c 0 > 0.
;
Author's personal copy C.-y. Chen, T.-f. Wu / Applied Mathematics and Computation 218 (2012) 10817–10828
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Proof. The proof is a repeat of that in [10, Thorem 2.5] and therefore omitted here. h The following lemma shows that the minimizers on Nk;l are ‘‘usually’’ critical points for J k;l . Lemma 2.5. For each ðk; lÞ 2 H and ðu0 ; v 0 Þ a local minimizer for J k;l on Nk;l , we have J 0k;l ðu0 ; v 0 Þ ¼ 0 in H1 (the dual space of the Sobolev space H). Proof. If ðu0 ; v 0 Þ is a local minimizer for J k;l on Nk;l , then ðu0 ; v 0 Þ is a solution of the optimization problem
minimizeJ k;l ðu; v Þ subject to Uk;l ðu; v Þ ¼ 0: Using Lagrange multipliers, there exists K 2 R such that
J 0k;l ðu0 ; v 0 Þ ¼ KU0k;l ðu0 ; v 0 Þ in H1 and thus,
D
E D E J 0k;l ðu0 ; v 0 Þ; ðu0 ; v 0 Þ ¼ K U0k;l ðu0 ; v 0 Þ; ðu0 ; v 0 Þ ¼ 0: D
E However U0k;l ðu0 ; v 0 Þ; ðu0 ; v 0 Þ must be nonzero since ðu0 ; v 0 Þ R N0k;l , it follows that K ¼ 0. This completes the proof. R Before proceeding to the next lemma, we first define, for each ðu; v Þ 2 H with X h j uja j v jb > 0;,
tmax ¼
ð2 qÞkðu; v Þk2H R ða þ b qÞ X h j uja j v jb
while for each u 2 H with
R
ða þ b qÞ
R
tmax ¼
X fk
X fk
j ujq þ
R
j ujq þ
R
ð2:5Þ h
1 !aþb2
> 0;
X
g l j v jq > 0, we write
X
g l j v jq
1 !2q
ða þ b 2Þkðu; v Þk2H
> 0:
The following lemmas are presented without proof; the readers are referred to Brown and Wu [11, Lemmas 2.5 and 2.6, respectively] for similar proofs. Lemma 2.6. For each ðu; v Þ 2 H with (i) if (ii) if
R RX
fk j ujq þ
X fk
q
j uj þ
R RX X
J k;l ðt þ u; tþ v Þ ¼
R X
X
h j uja j v jb > 0, we have
g l j v jq 6 0, then J k;l ðu; v Þ ¼ suptP0 J k;l ðtu; t v Þ > 0; g l j v jq > 0, then there is a unique 0 < tþ ¼ tþ ðuÞ < tmax such that ðtþ u; tþ v Þ 2 Nþ k;l and
inf J k;l ðtu; t v Þ;
06t6t max
Lemma 2.7. For each u 2 H with (i) if
R
R
X fk
J k;l ðu; v Þ ¼ sup J k;l ðtu; t v Þ: tPt max
j ujq þ
R X
g l j v jq > 0, we have
h j uja j v jb 6 0, then there is a unique tþ < t max such that ðt þ u; t þ v Þ 2 Nþ k;l and
J k;l ðt þ u; tþ v Þ ¼ inf J k;l ðtu; t v Þ; tP0
(ii) if
R X
a
h j uj j v j > 0, then there are unique 0 < t þ < tmax < t such that ðt þ u; t þ v Þ 2 Nþ k;l ; t u 2 N and b
J k;l ðt þ u; tþ v Þ ¼
inf J k;l ðtu; t v Þ;
06t6t max
J k;l ðt u; t v Þ ¼ supJ k;l ðtu; t v Þ: tP0
Next, we write
R Sa;b ¼
inf
R j ruj2 dx þ X j rv j2 dx R 2=ðaþbÞ a b X h j uj j v j dx X
u;v 2H10 ðXÞnf0g
and using the result of Alves et al. [6, Theorem 5], we deduce that
Sa;b ¼
" b aþb
a b
a # aþb b S; þ
a
ð2:6Þ
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where S is the best Sobolev constant given by (2.1). The following proposition thus provides a precise description for the Palais–Smale sequence of J k;l . Proposition 2.8. Each sequence fðun ; v n Þg H that satisfies N (i) J k;l ðun ; v n Þ ¼ b þ oð1Þ with b 2 1; hk;l þ N1 Sa2 ;b and (ii) J 0k;l ðun ; v n Þ ¼ oð1Þ in H has a convergent subsequence.
Proof. By Lemma 2.4 and the compact imbedding theorem, there exist a subsequence fðun ; v n Þg and ðu0 ; v 0 Þ a solution of problem ðEk;l Þ satisfying
un * u0
and
un ! u0
and
weakly in H10 ðXÞ;
vn * v0 vn ! v0
strongly in Lr ðXÞ for 1 < r < 2 :
Moreover,
D E J 0k;l ðu0 ; v 0 Þ; ðu0 ; v 0 Þ ¼ 0 and
Z
q
fk j un j þ
X
Z X
q
gl j v n j !
Z
q
fk j u0 j þ
Z
X
X
g l j v 0 jq
as n ! 1;
it follows that, by Lemma 2.4, J k;l ðu0 ; v 0 Þ P hk;l . Given that un * u0 and
v n * v 0 weakly in H10 ðXÞ, we must have
kðun ; v n Þk2H ¼ ðu0 ; v 0 Þk2H þ kðun u0 ; v n v 0 Þk2H þ oð1Þ: Note that, by the Brezis–Lieb lemma [7],
Z
h j un ja j v n jb dx ¼
X
Z
h j u0 ja j v 0 jb dx þ
X
Z
h j un u0 ja j v n v 0 jb dx þ oð1Þ
X
and
Z h j un ja2 un j v n jb ju0 ja2 u0 j v 0 jb ðun u0 Þdx þ h jun ja jv n jb2 v n ju0 ja j v 0 jb2 v 0 ðv n v 0 Þdx X X Z a b ¼ hjun u0 j jv n v 0 j dx þ oð1Þ:
Z
X
As a result,
1 1 J k;l ðun ; v n Þ ¼ J k;l ðu0 ; v 0 Þ þ kðun u0 ; v n v 0 Þk2H 2 aþb
Z
hjun u0 ja j v n v 0 jb dx þ oð1Þ
X
and
D E D E oð1Þ ¼ ðun u0 ; v n v 0 Þ; J 0k;l ðun ; v n Þ ¼ ðun u0 ; v n v 0 Þ; J 0k;l ðun ; v n Þ J 0k;l ðu0 ; v 0 Þ Z ¼ kðun u0 ; v n v 0 Þk2H hjun u0 ja jv n v 0 jb dx þ oð1Þ:
ð2:7Þ
X
This implies
1 1 1 2 2 kðun u0 ; v n v 0 ÞkH ¼ kðun u0 ; v n v 0 ÞkH N 2 aþb
Z
hjun u0 ja jv n v 0 jb dx þ oð1Þ
X
¼ J k;l ðun ; v n Þ J k;l ðu0 ; v 0 Þ þ oð1Þ 6 J k;l ðun ; v n Þ hk; l þ oð1Þ and we can therefore conclude that N
2 kðun u0 ; v n v 0 ÞkH < Sa2 ;b ;
when n is sufficiently large. Using (2.6) and (2.7) leads to
Z N 4=ðN2Þ 2 2 6 u ð u u ; v v Þ ð u ; v v Þ hjun u0 ja jv n v 0 jb dx ¼ oð1Þ: kðun u0 ; v n v 0 ÞkH 1 S2N k k k k n 0 n 0 n 0 n 0 a;b H H X
ð2:8Þ Consequently, ðun ; v n Þ ! ðu0 ; v 0 Þ strongly in H. h
Author's personal copy C.-y. Chen, T.-f. Wu / Applied Mathematics and Computation 218 (2012) 10817–10828
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3. Proof of Theorem 1 First, we establish the existence of a local minimum for J k;l on Nþ k;l . þ þ Theorem 3.1. If the pair of parameters ðk; lÞ 2 H, then J k;l has a minimizer uþ 0 ; v 0 in Nk;l satisfying (i) J k;l uþ ; v þ ¼ hk;l ¼ hþ k;l < 0; þ 0þ 0 (ii) u0 ; v 0 is a solution of problem ðEk;l Þ such that uþ 0 P 0;
v þ0 P 0 in X and uþ0 – 0; v þ0 – 0.
Proof. By the Ekeland variational principle [15] (or, alternatively, see Wu [29, Proposition 9]), there exists fðun ; v n Þg Nk;l which is a minimizing sequence for J k;l such that
J k;l ðun ; v n Þ ¼ hk;l þ oð1Þ and J0k;l ðun ; v n Þ ¼ oð1Þ in H1 : þ þ Using Proposition 3.1, it can be shown that a subsequence fðun ; v n Þg and uþ k;l ; v k;l 2 Nk;l exist such that ðun ; v n Þ !
þ þ
þ þ
þ þ þ þ þ þ þ uþ k;l ; v k;l strongly in H and J k;l uk;l ; v k;l ¼ hk;l . With J k;l uk;l ; v k;l ¼ J k;l uk;l ; v k;l and uk;l ; v k;l 2 Nk;l , by Lemma
þ
þ þ 2.5, it is possible to assume that uþ k;l ; v k;l is a solution of problem ðEk;l Þ such that uk;l P 0; v k;l P 0 in X. Next, we prove
v þk;l – 0. Assuming the contrary and, without loss of generality, we take v þk;l 0. By condition ðD2Þ, we may
that uþ k;l – 0; choose w 2
H10 ð
XÞ n f0g such that Z Z Z
a 2q
h uþk;l j wjb dx 6 j rwj2 dx ¼ g l j wjq dx: ð2 qÞða þ bÞ X X X
þ þ Applying Lemma 2.7 gives the result that a unique 0 < t þ < tmax exits satisfying t þ uþ k;l ; t w 2 Nk;l . Moreover,
0
tmax
1
q R
12q R
ða þ b qÞ X fk uþk;l dx þ X g l j wjq dx B C ¼@ A >1 2 þ ða þ b 2Þ uk;l ; w
H
and
J k;l tþ uþk;l ; t þ w ¼
inf J k;l tuþk;l ; tw :
06t6t max
This implies that
J k;l tþ uþk;l ; t þ w 6 J k;l uþk;l ; w 6 J k;l uþk;l ; 0 ¼ hþk;l ; þ which is a contradiction; accordingly, we must have uþ k;l – 0; v k;l – 0. We now proceed to consider the following critical problem:
(
Du ¼j uj2 u2
H10 ð
2
u in X;
h
ðb E0Þ
XÞ
and associated with the equation ð b E 0 Þ, the energy functional I1 in H10 ðXÞ is given by
I1 ðuÞ ¼
1 1 kuk2H1 2 2
Z
j uj2 dx:
X
It is established that
inf 1
u2M
ðRN Þ
I1 ðuÞ ¼
inf 1
u2M ðXÞ
I1 ðuÞ ¼
1 N2 S N
for all domain X RN ;
where
n o
M1 ðRN Þ ¼ u 2 D1;2 ðRN Þ n f0g j ðI1 Þ0 ðuÞ; u ¼ 0 and
n o
M1 ðXÞ ¼ u 2 H10 ðXÞ n f0g j ðI1 Þ0 ðuÞ; u ¼ 0
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are the Nehari manifolds. In fact, I1 ðuÞ is never attained on any domain X$RN . Repeating the method employed by [9] and N using Remark 1.1, we take g 2 C 1 0 ðR Þ as a radially symmetric function with 0 6 g 6 1; jrgj 6 C and
gðxÞ ¼
1 j x j6 r 0 =2; 0
j x jP r 0 :
Assume
we ðxÞ ¼ gðxÞv e ðxÞ; where
ð3:1Þ
v e ðxÞ is given by (2.2). Following the argument for the proof of Lemma 4.2 in [20] (or see Struwe [26]) gives N
kwe k2H1 ¼ S 2 þ O
eN2
and
Z
N
jwe j2 dx ¼ S 2 þ OðeN Þ:
ð3:2Þ
X
Moreover, we have the following results. Lemma 3.2. We have
8 N < S 2 þ o e2 ; if N P 6; N2 hjwe j dx ¼ N : S 2 þ o e 2 ; if 3 6 N 6 5: X
Z
2
Proof. The proof is almost identical to that of [14, Lemma 3.1] and is omitted here for brevity. h Next, we establish the existence of a local minimum for J k;l on N k;l , which requires the following result. Lemma 3.3. We have
pffiffiffi pffiffiffi 1 N supJ k;l uþk;l þ t awe ; v þk;l þ t bwe < hk;l þ Sa2 ;b : N tP0
Furthermore, there exists t 0 > 0 such that
pffiffiffi pffiffiffi uþk;l þ t0 awe ; v þk;l þ t0 bwe 2 Nk;l and
hk;l < hk;l þ
1 N2 S : N a;b
Proof. Since
pffiffiffi pffiffiffi J k;l ðuþk;l þ t awe ; v þk;l þ t bwe Þ Z
Z 2
pffiffiffi pffiffiffi
q pffiffiffi pffiffiffi
q 1 1
¼ ðuþk;l þ t awe ; v þk þ t bwe Þ 1 fk uþk;l þ t awe dx þ g l v þk;l þ t bwe dx 2 q X H X Z
pffiffiffi
b pffiffiffi
a
þ 1
þ h u þ t awe v k;l þ t bwe dx a þ b X k;l Z Z Z Z
pffiffiffi
2 pffiffiffi
q
pffiffiffi
2 pffiffiffi q 1
1
þ 1
fk uþk þ t awe dx þ g l v þk þ t bwe dx ¼
r uþk;l þ t awe dx þ
r v k;l þ t bwe dx 2 X 2 X q X X Z
p ffiffiffi a b ffiffiffi p 1
þ
þ
Xh uk;l þ t awe v k;l þ t bwe dx aþb Z
Z Z Z Z
2 pffiffiffi pffiffiffi 1 þ
2 1
aþb 2 þ
þ w ¼ r u dx þ r v dx þ r tw dx þ r u r t a rv þk;l r t bwe dx dx þ j j
k;l
e e k;l
k;l 2 X 2 X 2 X X X Z
Z Z
pffiffiffi
q pffiffiffi
b pffiffiffi
q pffiffiffi
a
þ 1 1
þ
þ
þ fk uk;l þ t awe dx þ g l v k;l þ t bwe dx h uk;l þ t awe v k;l þ t bwe dx q X aþb X X Z Z Z a b pffiffiffi b pffiffiffi a þ aþb 1 q h uþk;l þ t awe v k;l þ t bwe uþk;l v þk;l 6 J k;l ðuþk;l ; v þk;l Þ þ jrtwe j2 dx þ C 0 ðtwe Þ dx 2 aþb X X X a1 b pffiffiffi a b1 pffiffiffi v þk;l t awe b uþk;l v þk;l t bwe dx ¼ hk;l þ Kðtwe Þ; a uþk;l
Author's personal copy C.-y. Chen, T.-f. Wu / Applied Mathematics and Computation 218 (2012) 10817–10828
10825
where
Kðtwe Þ ¼
Z a b pffiffiffi b pffiffiffi a þ 1 h uþk;l þ t awe v k;l þ t bwe uþk;l v þk;l 2 2 X X X a1 b pffiffiffi a b1 pffiffiffi þ þ þ þ a uk;l v k;l t awe b uk;l v k;l t bwe dx;
aþb
Z
jrtwe j2 dx þ C 0
Z
ðtwe Þq dx
it is sufficient if we show that
supKðtwe Þ < tP0
Moreover,
R
1 N2 S : N a;b
X
gw2e dx > 0 and we have
" # 2 þ 2 Z 2 1 t2 uþk uk 2 ðuþk Þ we;z dx jrwe j C 0 t we dx g þ we 2 X t t t 2 1 X X p Z þ tp u ¼ Oðtq Þ g k þ we;z dx ! 1 as t ! 1: p X t Z
t2 Kðtwe Þ ¼ 2
2
q
It can be readily seen that for
Kðtwe Þ
0 such that
for all t 2 ½t 0 ; 1Þ;
which requires the following condition to be satisfied:
max Kðtwe Þ
< D1 e þ Oðe Þ 2 2 2 we dx ¼ D2 e j ln ej þ Oðe Þ if N ¼ 4; > X : D3 e þ Oðe2 Þ if N ¼ 3:
Z
ð3:3Þ
Additionally,
Z
q
X
we dx 6
Z
ðC N eÞ
BN ð0;eÞ
qðN2Þ 2
eqðN2Þ
þ
Z BN ð0;r 0 ÞnBN ð0;eÞ
qðN2Þ 2
ðC N eÞ
jxjqðN2Þ
ðN2Þð2qÞþ4 2
6 Ce
qðN2Þ 2
þ Ce
Z
r0
r ðq1Þð2NÞþ1 dr
e
and so
Z X
wqe dx 6
8 qðN2Þ < C eðN2Þð2qÞþ4 2 þ Ce 2 ;
if q –
N ; N2
qðN2Þ qðN2Þ : C eðN2Þð2qÞþ4 N 2 þ C e 2 þ C e 2 j ln ej; if q ¼ N2 :
As a result, we have
Z X
( q
we dx ¼
oðe2 Þ; N2 2
oðe
if N P 6; Þ; if 3 6 N 6 5
ð3:4Þ
and we proceed by considering the two cases separately: Case I ðN P 6Þ : Remark 1.1 implies we ðxÞ ¼ 0 and
a b a1 b pffiffiffi a b1 pffiffiffi pffiffiffi b pffiffiffi a þ uþk;l þ t awe v k;l þ t bwe uþk;l v þk;l a uþk;l v þk;l t awe b uþk;l v þk;l t bwe ¼ 0; for all x 2 X n BN ð0; r 0 Þ. Since hðxÞ > 0 for all x 2 BN ð0; r 0 Þ, using Lemma 4.1 in Appendix, we have
a1 b pffiffiffi a b1 pffiffiffi pffiffiffi b þ a þ b pffiffiffi a þ þ þ þ þ þ hðxÞ uk;l þ t awe v k;l þ t bwe uk;l v k;l a uk;l v k;l t awe b uk;l v k;l t bwe a1 b1 a b v þk;l ðtwe Þ2 ; P hðxÞ a2 b2 ðtwe Þaþb þ C 0 uþk;l ð3:5Þ
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C.-y. Chen, T.-f. Wu / Applied Mathematics and Computation 218 (2012) 10817–10828
which implies Z a b a1 b pffiffiffi a b1 pffiffiffi pffiffiffi b pffiffiffi a þ hðxÞ uþk;l þ t awe v k;l þ t bwe uþk;l v þk;l a uþk;l v þk;l t awe b uþk;l v þk;l t bwe dx X
Z
P
hðxÞ
h
i
b
a
a2 b2 ðtwe Þaþb þ Cðtwe Þ2 dx:
X
Using Lemma 3.2 and (3.2)–(3.4) leads to
Z Z a b a2 b2 aþb aþb w2e dx hwe þ C 0 t q wqe dx t 2 aþb X X X X 0 1N2 0 R 1N2 R N 2 2 2 2 2 Bða þ bÞ S C e þ oðe Þ C 1 Bða þ bÞ X jrwe j dx C X we dx C C þ oðe2 Þ 6 1 B C þ oðe2 Þ 6 B 2 2 aþb @ A A N N @ a b N2 a bR 2 aþb a2 b2 X hwe a2 b2 S þ oðe2 Þ
Kðtwe Þ ¼
aþb
1 6 N
t2
Z
" b aþb
a
N a #2 aþb b
þ
b
Z
jrwe j2 dx C
a
1 N2 1 N Sa;b C e2 þ oðe2 Þ < Sa2 ;b N N
N
S 2 C e2 þ oðe2 Þ ¼
for sufficiently small e and t 2 ½0; t 0 . Case II ð3 6 N 6 5Þ : Using Lemma 4.1 in Appendix, it follows that
Kðtwe Þ 6
aþb 2
a
t
2
b
a2 b2 aþb t aþb
kwe k2H1
Z
aþb
X
hwe dx C 0
t aþb1 aþb1
Z X
N2 wae þb1 dx þ o e 2 ;
2 1 R
Nþ2 N2 with some constant C 0 > 0. Note that X we;z
dx ¼ C 1 e 2 þ Oðe 2 Þ, for some C 1 > 0. Applying Lemma 3.2 and (3.2), we have
N2 Nþ2 a2 b2 aþb N2 t aþb1 N2 t S t S C0 Kðtwe Þ 6 e 2 þo e 2 þO e 2 2 aþb aþb1 a
aþb
¼
aþb 2
b
N 2
2
N2 a2 b2 aþb N2 taþb1 N2 e 2 þo e 2 : t S C0 aþb aþb1 a
N
t2 S2
b
Denoting by t e the value of t 2 ½0; t0 that gives rise to the maximum of the above right-hand side, then t e satisfies N
a
b
N
N2 2
ða þ bÞS 2 ¼ a2 b2 t ae þb2 S 2 þ tae þb3 C e
þo
N2 2
e
:
It can be deduced that 1 !aþb2
aþb te ¼ a b a2 b2
N2 2
Ce
t2e
3
N2 þo e 2
and thus
max Kðtwe Þ 6
t2½0;t0
aþb 2
N2 a2 b2 aþb N2 N2 te S Ct ae þb3 e 2 þ o e 2 aþb a
N
t 2e S 2
b
2 !aþb2
aþb aþb ¼ a b 2 a2 b2
a
b
aþb !aþb2
a2 b2 a þ b S a þ b aa2 b2b N 2
N
S 2 Ct2e
3
N2 2
e
þo
N2 2
e
N2 N2 1 N b a N 1 a aþb b aþb N2 ð Þ þð Þ S 2 Ct 2e 3 e 2 þ o e 2 < Sa2 ;b ¼ N b N a for
e sufficiently small. Using the result of Wu [31, Lemma 3.2], we therefore conclude that such t0 > 0 exists with pffiffiffi pffiffiffi uþk;l þ t0 awe ; v þk;l þ t0 bwe 2 Nk;l
and
pffiffiffi pffiffiffi 1 N J k;l uþk;l þ t 0 awe ; v þk;l þ t0 bwe < hk;l þ Sa2 ;b : N This completes the proof. h
Author's personal copy C.-y. Chen, T.-f. Wu / Applied Mathematics and Computation 218 (2012) 10817–10828
10827
Theorem 3.4. If the pair of parameters ðk; lÞ 2 H, then J k;l has a minimizer u in N 0 ; v0 k;l which satisfies (i) J k;l u ; v ¼ h k;l ; 0 0 (ii) u0 ; v 0 is a solution of problem ðEk;l Þ, such that u 0 P 0;
v 0 P 0 in X with u0 – 0 and v 0 – 0.
Proof. By the Ekeland variational principle [15] (or see Wu [29, Proposition 9]), there exists fðun ; v n Þg Nk;l which is a minimizing sequence for J k;l satisfying
J k;l ðun ; v n Þ ¼ hk;l þ oð1Þ and J 0k;l ðun ; v n Þ ¼ oð1Þ in H1 : Using Proposition 3.1 and Lemma 3.3, it follows that a subsequence fðun ; v n Þg and u k;l ; v k;l 2 Nk;l exist such that
strongly in H and J f ;g u and Since J k;l u ðun ; v n Þ ! u k;l ; v k;l k;l ; v k;l ¼ hk;l . k;l ; v k;l ¼ J k;l uk;l ; v k;l
uk;l ; v k;l 2 Nk;l , by Lemmas 2.2 and 2.5, we may assume that uk;l ; v k;l is a solution of problem ðEk;l Þ such that u k;l P 0; v k;l P 0 in X and uk;l – 0;
v k;l – 0.
h
We now complete the proof of Theorem 1.1. Applying Theorems 3.1 and 3.4 on problem ðEk;l Þ, we conclude that two solu þ þ tions uþ u k;l ; v k;l 2 Nk;l exist satisfying uk;l P 0; v k;l P 0 in Xwith uk;l – 0 and v k;l – 0. Since k;l ; v k;l 2 Nk;l and þ þ and u are distinct. Nþ k;l ; v k;l k;l \ Nk;l ¼ ;, this implies that uk;l ; v k;l 4. Appendix Lemma 4.1. There exist positive constants C 1 ðaÞ; C 2 ðbÞ > 0 such that a
a b
a
ða þ bÞa ðc þ dÞb P aa cb þ aa d þ b cb þ b d þ C 1 ðaÞaa1 bc þ C 1 ðaÞaa1 bd þ C 2 ðbÞb cb1 d þ C 2 ðbÞaa cb1 d b
þ C 1 ðaÞC 2 ðbÞaa1 bc
b1
b
b
d:
Proof. The result follows immediately from (4.3) in [3, p. 537]. h Acknowledgments The author is grateful for the referee’s valuable suggestions and helps. This research was supported in part by the National Science Council and the National Center for Theoretical Sciences (South), Taiwan. References [1] A. Ahammou, Positive radial solutions of nonlinear elliptic system, New York J. Math. 7 (2001) 267–280. [2] A. Ambrosetti, G.J. Azorero, I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996) 219–242. [3] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519– 543. [4] K. Adriouch, A. EL Hamidi, The Nehari manifold for systems of nonlinear elliptic equations, Nonlinear Anal. 64 (2006) 2149–2164. [5] C.O. Alves, G. Figueiredo, M. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Commun. Partial Differ. Equat. 36 (2011) 1565–1586. [6] C.O. Alves, D.C. de Morais Filho, M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000) 771–787. [7] H. Brézis, E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486–490. [8] Y. Bozhkov, E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differ. Equat. 190 (2003) 239–267. [9] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math. 16 (1983) 437–477. [10] K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336. [11] K.J. Brown, T.F. Wu, A fibrering map approach to a potential operator equation and its applications, Differ. Integral Equat. 22 (2009) 1097–1114. [12] A. Cañada, Nonlinear ordinary boundary value problems under a combined effect of periodic and attractive nonlinearities, J. Math. Anal. Appl. 243 (2000) 174–189. [13] P. Clément, D.G. de Figueiredo, E. Mitidieri, Positive solutions of semilinear elliptic systems, Commun. Partial Differ. Equat. 17 (1992) 923–940. [14] C.Y. Chen, T.F. Wu, Multiple positive solutions for indefinite semilinear elliptic problems involving critical Sobolev exponent, submitted for publication. [15] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 17 (1974) 324–353. [16] D.G. de Figueiredo, P. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343 (1994) 99–116. [17] D.G. de Figueiredo, J.P. Gossez, P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2003) 452– 467. [18] A. el Hamidi, Multiple solutions with changing sign energy to a nonlinear elliptic equation, Commun. Pure Appl. Anal. 3 (2004) 253–265. [19] A. el Hamidi, Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl. 300 (2004) 30–42. [20] H. He, J. Yang, Positive solutions for critical inhomogeneous elliptic problems in non-contractible domains, Nonlinear Anal. 70 (2009) 952–973.
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