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Complex Variables and Elliptic Equations
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Mountain pass and Ekeland's principle for eigenvalue problem with variable exponent
Khaled Benali a; Khaled Kefi b a Institut Préparatoire aux Etudes d'ingénieurs de Tunis, 1008 Montfleury-Tunis, Tunisia b Institut Supérieur du Transport et de la Logistique de Sousse, 4029-Sousse, Tunisia
To cite this Article Benali, Khaled and Kefi, Khaled(2009) 'Mountain pass and Ekeland's principle for eigenvalue problem
with variable exponent', Complex Variables and Elliptic Equations, 54: 8, 795 — 809 To link to this Article: DOI: 10.1080/17476930902999041 URL: http://dx.doi.org/10.1080/17476930902999041
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Complex Variables and Elliptic Equations Vol. 54, No. 8, August 2009, 795–809
Mountain pass and Ekeland’s principle for eigenvalue problem with variable exponenty Khaled Benalia and Khaled Kefib* a Institut Pre´paratoire aux Etudes d’inge´nieurs de Tunis, 2 Rue Jawaher Lel Nehru, 1008 Montfleury-Tunis, Tunisia; bInstitut Supe´rieur du Transport et de la Logistique de Sousse, 12 Rue Abdallah Ibn Zoube¨r, 4029-Sousse, Tunisia
Communicated by H. Begehr
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(Received 25 December 2008; final version received 3 April 2009) In this article, we study the boundary value problem div(jrujp(x)2ru) þ juj(x)2u ¼ jujq(x)2u, in , u ¼ 0 on @, where is a smooth bounded domain in RN and p, q, are continuous functions on . We show that for any 40 there exists infinitely many weak solutions (respectively, if 40 and small enough, then there exists a non-negative, non-trivial weak solution). Our approach relies on the variable exponent theory of generalized Lebesgue–Sobolev spaces, combined with a Z2 symmetric version for even functionals of the Mountain pass Theorem (respectively on simple variational arguments based on Ekeland’s variational principle). Keywords: p(x)-Laplace operator; Ekeland’s variational principle; generalized Sobolev spaces; Mountain pass Theorem; weak solution AMS Subject Classifications: 35D05; 35J60; 35J70; 58E05; 76A02
1. Introduction The study of differential equations and variational problems with non-standard p(x)-growth conditions have received more and more interest in recent years. The specific attention accorded to such kind of problems is due to their applications in mathematical physics. More precisely, such equations are used to model phenomenon which arise in elastic mechanics or electrorheological fluids (sometimes referred to as ‘smart fluids’). Miha˘ilescu and Ra˘dulescu [1] studied the following nonhomogeneous eigenvalue problem 8 qðxÞ2 > u, for x 2 < DpðxÞ u ¼ juj u 6 0, for x 2 > : u ¼ 0, for x 2 @ *Corresponding author. Email:
[email protected] yThis article is devoted to the special issue ‘Va¨xjo¨ Conference 2008’. ISSN 1747–6933 print/ISSN 1747–6941 online ß 2009 Taylor & Francis DOI: 10.1080/17476930902999041 http://www.informaworld.com
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K. Benali and K. Kefi
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where RN (N 3) is a bounded domain with smooth boundary, 40 is a real number and p, q are continuous functions in . They showed, under the basic assumption 15 min qðxÞ5 min pðxÞ5 max qðxÞ, that there exists *40 such that any 2 (0, *) is a eigenvalue for the above problem . The main arguments in their paper is related to the Ekeland’s variational principle [2]. Note that the above problem has been studied by Ambrosetti and Rabinowitz [3] in the particuliar case where p(x) ¼ 2 and q(x) ¼ q, is a constant. In this article we discuss the existence of the weak solutions for the following perturbed problem 8 ðxÞ2 > u ¼ jujqðxÞ2 u, for x 2 < DpðxÞ u þ juj ð1Þ u 6 0, for x 2 > : u ¼ 0, for x 2 @ where 40 is a real number, and p, q, : ! R are continuous. We show under appropriate conditions on p, q, and that for any 40 (respectively there exists a positive critical real number * under which) the problem (1) has infinitely many weak solutions (not necessarily positive) (respectively non-negative, non-trivial weak solution). We mention that problem (1) has not been studied in both cases when the exponents are constants and variables. We denoted by Dp(x) the p(x)-Laplace operator, i.e. DpðxÞ u :¼ div jrujpðxÞ2 ru , where p is a continuous non-constant function. This differential operator is a natural generalization of the p-Laplace operator Dpu ¼ div(jrujp2ru), where p41 is a real constant. However, the p(x)-Laplace operator possesses more complicated non-linearities than the p-Laplace operator, due to the fact that Dp(x) is not homogeneous. Recent qualitative propreties of solutions to quasilinear problems in Sobolev spaces with variable exponent have been obtained by Alves and Souto [4], Chabrowski and Fu [5] and Miha˘ilescu and Ra˘dulescu [6]. Equation (1) will be studied in the framework of the variable Lebesgue and Sobolev spaces Lp(x) and pðxÞ which will be briefly described in the next section. W1, 0
2. Preliminary results We recall in what follows some definitions and basic properties of the generalized Lebesgue spaces Lp(x)() and generalized Sobolev spaces W1,p(x)(), where is an open subset of RN. In that context we refer to the book of Musielak [7] and the papers of Kovacik and Rakosnik [8] and Fan et al. [9,10]. Set Cþ ðÞ ¼ fh; h 2 CðÞ, hðxÞ 4 1 for all x 2 g: For any h 2 Cþ ðÞ we define hþ ¼ sup hðxÞ and x2
h ¼ inf hðxÞ: x2
797
Complex Variables and Elliptic Equations For any pðxÞ 2 Cþ ðÞ, we define the variable exponent Lebesgue space Z LpðxÞ ðÞ ¼ fu : is a Borel real-valued function on : juðxÞjpðxÞ dx 5 1g: p(x)
, the so-called Luxemburg norm, by the formula Z uðxÞ pðxÞ jujpðxÞ :¼ inf 4 0; dx 1 :
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We define on L
Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are separable and Banach spaces [8, Theorem 2.5; Corollary 2.7] and the Ho¨lder inequality holds [8, Theorem 2.1]. The inclusion between Lebesgue spaces are also naturally generalized [8, Theorem 2.8]: if 05jj51 and r1, r2 are variable exponents so that r1(x) r2(x) almost everywhere in then there exists the 0 continuous embedding Lr2(x)() ,! Lr1(x)(). We denote by Lp (x)() the conjugate 0 space of Lp(x)(), where 1/p(x) þ 1/p0 (x) ¼ 1. For any u 2 Lp(x)() and v 2 Lp (x)() the Ho¨lder-type inequality Z uv dx 1 þ 1 jujpðxÞ jvjp0 ðxÞ ð2Þ p p0 holds. An important role in manipulating the generalized Lebesgue–Sobolev spaces is played by the modular of the Lp(x)() space, which is the mapping p(x) : Lp(x)() ! R defined by Z jujpðxÞ dx: pðxÞ ðuÞ ¼
The space W
1,p(x)
() is equipped by the following norm kuk ¼ jujpðxÞ þ jrujpðxÞ :
If (un), u, 2 W1,p(x)() and pþ51 then the following relations holds
þ
þ
minðjujppðxÞ , jujppðxÞ Þ pðxÞ ðuÞ maxðjujppðxÞ , jujppðxÞ Þ
þ
ð3Þ þ
minðjrujppðxÞ , jrujppðxÞ Þ pðxÞ ðjrujÞ maxðjrujppðxÞ , jrujppðxÞ Þ
ð4Þ
jujpðxÞ ! 0 , pðxÞ ðuÞ ! 0, lim jun ujpðxÞ ¼ 0 , lim pðxÞ ðun uÞ ¼ 0,
n!1
n!1
ð5Þ
jujpðxÞ ! 1 , pðxÞ ðuÞ ! 1: ðÞ as the closure of C1 We define also W1,pðxÞ 0 ðÞ under the norm 0 kuk ¼ jrujpðxÞ , using the result of Fan and Zhao [11], we remark that the norm kuk ¼ jujp(x) þ jrujp(x) pðxÞ is equivalent to kuk ¼ jrujp(x) in W1, ðÞ. 0 1, pðxÞ The space ðW0 ðÞ, k:kÞ is a separable and reflexive Banach space. Next, we remember some embedding results regarding variable exponent
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K. Benali and K. Kefi
Lebesgue–Sobolev spaces. We note that if sðxÞ 2 Cþ ðÞ and s(x)5p*(x) for all x 2 pðxÞ ðÞ ,! LsðxÞ ðÞ is compact and continuous, where then the embedding W1, 0 p*(x) ¼ Np(x)/(N p(x)) if p(x)5N or p*(x) ¼ þ1 if p(x) N. We refer to [8] for more properties of Lebesgue and Sobolev spaces with variable exponent. We also refer to the recent papers [1,6,9,10,12,13] for the treatment of non-linear boundary value problems in Lebesque–Sobolev spaces with variable exponent.
3. The main results and an auxiliary results In the first part of this article, we assume that (H1) 15inf p sup p5N; (H2) max(sup p, sup )5inf q, and qðxÞ 5 p ðxÞ, 8x 2 . Note that a weak solution of problem (1) satisfy the following definition
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Definition 1
pðxÞ We say that u 2 W1, ðÞ is a weak solution of (1) if 0 Z Z pðxÞ2 ðxÞ2 rurv þ juj uv dx ¼ jujqðxÞ2 uv dx, jruj
for any v 2
pðxÞ ðÞ. W1, 0
Remark 1 The assumptions (H1), (H2) and the Ho¨lder-type inequality (2) assures that the two terms cited in the definition (1) are finite. Our main result is given by the following theorem THEOREM 1 Suppose that assumptions (H1) and (H2) hold. Then for any 40 the problem (1) has infinitely many weak solutions. ðÞ. The energy functional Let E denote the generalized Sobolev space W1,pðxÞ 0 corresponding to problem (1) is defined as J : E ! R, Z Z Z 1 1 1 jrujpðxÞ dx þ jujðxÞ dx jujqðxÞ dx, J ðuÞ :¼ pðxÞ ðxÞ qðxÞ PROPOSITION 1 Proof
The functional J is well-defined on E.
Using the above hypothesis we get Z Z Z 1 1 J ðuÞ jrujpðxÞ dx þ jujðxÞ dx þ jujqðxÞ dx: p q
The proof, then holds by the Sobolev embeddings E ,! L(x) and E ,! Lq(x). We set Z
1 jrujpðxÞ dx þ J1 ðuÞ :¼ pðxÞ
Z
1 jujðxÞ dx ðxÞ
and Z J2 ðuÞ :¼
1 jujqðxÞ dx: qðxÞ
g
799
Complex Variables and Elliptic Equations We first prove an auxiliary result:
PROPOSITION 2 Under the assumptions (H1) and (H2) J 2 C1(E, R) and u 2 E is a critical point of J if and only if u is a weak solution for the problem (1). To show that J 2 C1(E, R), we show that for all ’ 2 E, J ðu þ t’Þ J ðuÞ ¼ hdJ ðuÞ, ’i, lim t!0þ t and dJ : E ! E* continuous, where we denote by E * the dual space of E. For all ’ 2 E we have Z J2 ðu þ t’Þ J2 ðuÞ d d ¼ J2 ðu þ t’Þjt¼0 ¼ ju þ t’jqðxÞ dxjt¼0 limþ t!0 t dt dt qðxÞ Z @ 1 qðxÞ ju þ t’j ¼ jt¼0 dx @t qðxÞ Z ¼ ju þ t’jqðxÞ1 sgnðu þ t’Þ’jt¼0 dx Z Z ¼ ju þ t’jqðxÞ2 ðu þ t’Þ’jt¼0 dx ¼ jujqðxÞ2 u’ dx
Proof
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¼ hdJ2 ðuÞ, ’i: The differentiation under the integral is allowed if for t close to zero, ju þ t’jq(x)2(u þ t’)’ can be dominated by one fixed function g 2 L1() which does not depend on t. Indeed we have, say if jtj51 that jju þ t’jqðxÞ2 ðu þ t’Þ’j ¼ ju þ t’jqðxÞ1 j’j ðjuj þ jtjj’jÞqðxÞ1 j’j ðjuj þ j’jÞqðxÞ1 j’j Set gðxÞ :¼ ðjuj þ j’jÞqðxÞ1 j’j We observe g 2 L1() because u, ’ 2 E imply juj, j’j 2 E ,! LqðxÞ ðÞ, and then qðxÞ
qðxÞ ðjuj þ j’jÞqðxÞ1 2 LqðxÞ1 ðÞ, j’j 2 LqðxÞ ðÞ ¼ LqðxÞ1 ðÞ : 0
For u 2 E chosen we show that dJ2(u) 2 W 1,p (x)() ¼ E *, where 1/p(x) þ 1/p (x) ¼ 1. It is easy to see that dJ2(u) is linear. Since there is a continuous embedding E ,! Lq(x)(), then there exists a constant M40 such that 0
jvjqðxÞ Mkvk,
for all v 2 E:
ð6Þ
Using (2) and (6) we obtain Z Z jhdJ2 ðuÞ, ’ij ¼ j jujqðxÞ2 u’ dxj jujqðxÞ1 j’jdx qðxÞ1
jjuj
j
qðxÞ qðxÞ1
j’jqðxÞ MjjujqðxÞ1 j
qðxÞ qðxÞ1
k’k:
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Hence there exists M1 ¼ MjjujqðxÞ1 j
qðxÞ qðxÞ1
4 0 such that
jhdJ2 ðuÞ; ’ij Mk’k: Using the linearity of dJ2(u) and the above inequality we deduce that 0 dJ2(u) 2 E* ¼ W1,p (x)(). For the Fre´chet differentiability we need the following Lemma 1 LEMMA 1 [14]
qðxÞ
The map u 2 LqðxÞ ðÞ ! jujqðxÞ2 u 2 LqðxÞ1 ðÞ is continuous.
We conclude that J2 is Fre´chet differentiable. Now we show that J1 is Fre´chet differentiable. For all ’ 2 E we have limþ
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t!0
J1 ðu þ t’Þ J1 ðuÞ d ¼ J1 ðu þ t’Þjt¼0 t dt Z d 1 1 jru þ tr’jpðxÞ þ ju þ t’jðxÞ dxjt¼0 ¼ dt pðxÞ ðxÞ Z @ 1 1 jru þ tr’jpðxÞ þ ju þ t’jðxÞ jt¼0 dx: ¼ ðxÞ @t pðxÞ
Since @ ju þ t’jðxÞ ¼ ðxÞju þ t’jðxÞ1 sgnðu þ t’Þ’ ¼ ðxÞju þ t’jðxÞ2 ðu þ t’Þ’, @t @ jru þ tr’jpðxÞ ¼ pðxÞjrðu þ t’ÞjpðxÞ2 rðu þ t’Þr’: @t So limþ
t!0
J1 ðu þ t’Þ J1 ðuÞ ¼ t
Z
Z ¼
ðjrðu þ t’ÞjpðxÞ2 rðu þ t’Þr’
þ ju þ t’jðxÞ2 ðu þ t’Þ’jt¼0 dx jrujpðxÞ2 rur’ þ jujðxÞ2 u’ dx:
The differentiation under the integral is allowed if for t close to zero jrðu þ t’ÞjpðxÞ2 rðu þ t’Þr’ þ ju þ t’jðxÞ2 ðu þ t’Þ’ can be dominated by one fixed function h 2 L1() which does not depend on t. Indeed we have, say if jtj51 that jjrðu þ t’ÞjpðxÞ2 rðu þ t’Þr’ þ ju þ t’jðxÞ2 ðu þ t’Þ’j jru þ tr’jpðxÞ1 jr’j þ ju þ t’jðxÞ1 j’j ðjruj þ jtjjr’jÞpðxÞ1 jr’j þ ðjuj þ jtjj’jÞðxÞ1 j’j ðjruj þ jr’jÞpðxÞ1 jr’j þ ðjuj þ j’jÞðxÞ1 j’j: Set hðxÞ :¼ ðjruj þ jr’jÞpðxÞ1 jr’j þ ðjuj þ j’jÞðxÞ1 j’j:
801
Complex Variables and Elliptic Equations We observe that h 2 L1() because u, ’ 2 E imply juj, j’j 2 E ,! LðxÞ ðÞ, so ðxÞ juj, j’j 2 LðxÞ , yield ðjuj þ j’jÞðxÞ1 2 LðxÞ1 ¼ LðxÞ and pðxÞ jruj, jr’j 2 LpðxÞ , yield ðjruj þ jr’jÞpðxÞ1 2 LpðxÞ1 ¼ LpðxÞ : For all u 2 E we have Z hdJ1 ðuÞ, ’i ¼
jrujpðxÞ2 rur’ þ jujðxÞ2 u’ dx:
It is easy to see that dJ1(u) is a linear function. We also have the continuous embedding E ,! L(x)(), which implies that there exists M0 40 such that, for all u 2 E,
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jujðxÞ M0 kuk:
ð7Þ
Using (2) and (7) we obtain Z jrujpðxÞ2 rur’ þ jujðxÞ2 u’ dx jhdJ1 ðuÞ, ’ij ¼ Z Z jrujpðxÞ1 jr’jdx þ jujðxÞ1 j’jdx
jjrujpðxÞ1 j jjrujpðxÞ1 j
pðxÞ pðxÞ1 pðxÞ pðxÞ1
Thus there exists M2 ¼ jjrujpðxÞ1 j
pðxÞ pðxÞ1
jr’jpðxÞ þ jjujðxÞ1 j
ðxÞ ðxÞ1
k’k þ M0 jjujðxÞ1 j
ðxÞ ðxÞ1
þ M0 jjujðxÞ1 j
jhdJ1 ðuÞ, ’ij M2 k’k,
ðxÞ ðxÞ1
j’jðxÞ
k’k:
4 0 such that
for all ’ 2 E:
The above relation and the linearity of dJ1(u) imply that dJ1(u) 2 E* and Z jrujpðxÞ2 rur’ þ jujðxÞ2 u’ dx: hdJ1 ðuÞ, ’i ¼
Using Lemma 1, we deduce that J1 is Fre´chet differentiable. We remark that J 2 C1(E, R) because J1, J2 2 C1(E, R). Moreover hdJ ðuÞ, vi ¼ hdJ1 ðuÞ dJ2 ðuÞ, vi ¼ hdJ1 ðuÞ, vi hdJ2 ðuÞ, vi Z Z pðxÞ2 ðxÞ2 ¼ rurv þ juj uv dx jujqðxÞ2 uv dx, jruj
8v 2 E:
Let u be a critical point of J. Then we have dJ(u) ¼ 0E* that is hdJ ðuÞ, vi ¼ 0, which yields Z
for all 2 E
jrujpðxÞ2 rurv þ jujðxÞ2 uv dx
Z
jujqðxÞ2 uv dx, 8v 2 E:
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K. Benali and K. Kefi
It follows that u is a weak solution for the problem (1). Now we assume that u is a weak solution of the problem (1). By Definition 1 it results that Z Z pðxÞ2 ðxÞ2 rurv þ juj uv dx ¼ jujqðxÞ2 uv dx, 8v 2 E: jruj
that is hdJ(u), vi ¼ 0, 8v 2 E. We obtain dJ(u) ¼ 0E*. Hence u is a critical point of J. This completes the proof of Proposition 2. g Our idea is to prove Theorem 1 applying the following Z2-symmetric version (for even functionals) of the Mountain pass Theorem [15, Theorem 9.12]. Mountain pass Theorem. Let X be an infinite dimensional real Banach space and let I 2 C1(X, R) be even, satisfying the Palais–Smale condition (i.e. any sequence {xn} X such that {I(xn)} is bounded and dI(xn) ! 0 in X* has a convergent subsequence) and I(0) ¼ 0. Suppose that (I1) There exists two constants %, r40 such that I(x) %40 if kxk ¼ r. (I2) For each finite dimensional subspace X1 X, the set {x 2 X1; I(x) 0} is bounded.
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Then I has an unbounded sequence of critical values. LEMMA 2 Suppose we are under the hypotheses of Theorem 1. Then for all 40 there exist %40 and r40 such that for all u 2 E with kuk ¼ r, we have J ðuÞ % 4 0: Proof Let us assume that kuk5min(1, 1/M), where M is the positive constant from above. Then, we have jujq(x)51. Using relations (3), (4) and (6) we obtain: Z Z Z 1 1 pðxÞ ðxÞ J ðuÞ þ jruj dx þ þ juj dx jujqðxÞ dx p q Z Z 1 þ jrujpðxÞ dx jujqðxÞ dx p q Z 1 þ þ kukp jujqðxÞ dx p q 1 þ þ kukp Mq kukq : p q þ
Let h ðtÞ ¼ p1þ tp q Mq tq , t40. It is easy to see that h(t)40 for all t 2 (0, t1), 1 where t1 ¼ ðpþqMq Þq pþ . So for all 40 we can choose r40 and %40 such that J ðuÞ % 4 0
for all u 2 E with kuk ¼ r:
The proof of Lemma 2 is complete.
g
LEMMA 3 If E1 E is a finite dimensional subspace, the set S ¼ {u 2 E1; J(u) 0} is bounded in E. Proof
In order to prove Lemma 3, first, we show that Z 1 þ jrujpðxÞ dx K1 ðkukp þ kukp Þ, 8u 2 E pðxÞ
where K1 is positive constant.
ð8Þ
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Complex Variables and Elliptic Equations Indeed, using relation (4) we have Z þ þ jrujpðxÞ dx jrujppðxÞ þ jrujppðxÞ ¼ kukp þ kukp ,
8u 2 E
ð9Þ
So, thus (8) holds true. Using again (3) we have Z þ jujðxÞ dx jujðxÞ þ jujðxÞ ,
8u 2 E
ð10Þ
The fact that E is continuously embedded in L(x)() assures the existence of a positive constant M0 such that jujðxÞ M0 kuk, 8u 2 E
ð11Þ
The last two inequalities show that there exists a positive constant K2 such that Z 1 þ jujðxÞ dx K2 ðkuk þ kuk Þ, 8u 2 E ð12Þ ðxÞ
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By inequalities (8) and (12) we get J ðuÞ K1 ðkuk
p
pþ
þ kuk Þ þ K2 ðkuk
þ kuk Þ q þ
Z
jujqðxÞ dx,
for all u 2 E. Let u 2 E be arbitrary but fixed. We define 5 ¼ fx 2 ; juðxÞj 5 1g, ¼ n5: Then we have
þ
þ
þ
þ
Z jujqðxÞ dx qþ Z þ þ kuk Þ þ jujqðxÞ dx q Z þ þ kuk Þ þ jujq dx q Z Z þ q þ kuk Þ þ juj dx þ þ jujq dx: q q 5 þ
J ðuÞ K1 ðkukp þ kukp Þ þ K2 ðkuk þ kuk Þ K1 ðkukp þ kukp Þ þ K2 ðkuk K1 ðkukp þ kukp Þ þ K2 ðkuk K1 ðkukp þ kukp Þ þ K2 ðkuk
But for each 40 there exists a positive constant K3() such that Z jujq dx K3 ðÞ, 8u 2 E: þ q 5 The functional j.jq : E ! R defined by Z 1=q q juj dx jujq ¼
is a norm in E. In the finite dimensional subspace E1 the norms j.jq and k.k are equivalent, so there exists a positive constant K ¼ K(E1) such that kuk Kjujq , 8u 2 E1 :
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K. Benali and K. Kefi
So that there exists a positive constant K4() such that
þ
þ
J ðuÞ K1 ðkukp þ kukp Þ þ K2 ðkuk þ kuk Þ K4 ðÞkukq þ K3 ðÞ, 8u 2 E1 : Hence
þ
þ
K1 ðkukp þ kukp Þ þ K2 ðkuk þ kuk Þ K4 ðÞkukq þ K3 ðÞ 0, 8u 2 S, and since q4max(pþ, þ) we conclude that S is bounded in E. LEMMA 4
g
If {un} E is a sequence which satisfies the properties jJ ðun Þj5 M3 dJ ðun Þ ! 0
ð13Þ
as n ! 1
ð14Þ
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where M3 is a positive constant, then {un} possesses a convergent subsequence. Proof First, we show that {un} is bounded in E. Assume by contradiction the contrary. Then, passing eventually to a subsequence, still denoted by {un}, we may assume that kunk ! 1 as n ! 1. Thus, we may consider that kunk41 for any integer n. Using (14) it follows that there exists N140 such that for any n4N1 we have k dJ ðun Þ k 1: On the other hand, for any n4N1 fixed, the application E 3 v ! hdJ(un), vi is linear and continuous. The above information yields that jhdJ ðun Þ, vij k dJ ðun Þ k kvk kvk, 8v 2 E, n 4 N1 : Setting v ¼ un we have Z
jrun jpðxÞ dx þ
kun k
Z
jun jðxÞ dx
Z
jun jqðxÞ dx kun k,
for all n4N1. We obtain Z kun k
pðxÞ
jrun j
Z dx
jun j
ðxÞ
Z dx
jun jqðxÞ dx
ð15Þ
for any n4N1. Let l ¼ max( pþ, þ), provided that kunk41 relations (13), (15) and (3) imply Z Z 1 1 1 1 jrun jpðxÞ dx þ jun jðxÞ dx kun k l q l q q Z 1 1 jrun jpðxÞ dx kun k l q q 1 1 kun kp kun k: l q q
M3 4 J ðun Þ
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Complex Variables and Elliptic Equations Thus, M3 4
1 1 kun kp kun k: þ p q q
ð16Þ
Now dividing by kunk in (16) and passing to the limit as n ! 1 we obtain a contradiction. It follows that {un} is bounded in E. For the strong convergence of {un} in E, we need the following proposition: PROPOSITION 3
Let r 2 Cþ ðÞ such that rðxÞ 5 p ðxÞ, 8x 2 , then Z lim jun jrðxÞ2 un ðun uÞdx ¼ 0: n!1
Proof if jjun j
Using (2) we have rðxÞ2
un j
rðxÞ rðxÞ1
R
jun jrðxÞ2 un ðun uÞdx jjun jrðxÞ2 un j
rðxÞ rðxÞ1
jun ujrðxÞ , then
rðxÞ2
4 1, by (3), there exists C40 such that jjun j
un j
rðxÞ rðxÞ1
jun jC rðxÞ g
and this ends the proof. Since dJ(un) ! 0 and un is bounded in E we have Downloaded At: 18:33 7 January 2010
jhdJ ðun Þ, un uij jhdJ ðun Þ, un ij þ jhdJ ðun Þ, uij kdJ ðun Þkkun k þ kdJ ðun Þkkuk: So lim hdJ ðun Þ, un ui ¼ 0:
n!1
Using Proposition 3 and the last relation we deduce that Z lim jrun jpðxÞ2 run rðun uÞdx ¼ 0 n!1
Now, if we note L : E ! E*, such that Z jrujpðxÞ2 rurv dx hLðuÞ, vi ¼
ð17Þ
8u, v 2 E,
then jhLðun Þ LðuÞ, un uij jhLðun Þ, un uij þ jhLðuÞ, un uij jhLðun Þ, un uij þ jhLðuÞ, un i hLðuÞ, uij: From (17) and the fact that {un} converges weakly to u in E it follows that lim hLðun Þ LðuÞ, un ui ¼ 0
n!1
and by Theorem 3.1 in Fan and Zhang [12] we deduce that {un} converges strongly to u in E. The proof of Lemma 4 is complete. g PROOF OF THEOREM 1 It is clear that the functional J is even and verifies J(0) ¼ 0. Lemma 4 implies that J satisfies the Palais–Smale condition. On the other hand, Lemmas 2 and 3 show the conditions (I1) and (I2) are satisfied. The Mountain pass Theorem can be applied to the functional J. We conclude that the problem (1) has infinitely many weak solutions in E. The proof of Theorem 1 is complete.
806
K. Benali and K. Kefi
In the second part of the article we consider the following condition (H2)0 15inf q5min(inf p, inf ) and maxððxÞ, qðxÞÞ 5 p ðxÞ, 8x 2 . Note that, under the assumptions (H1) and (H2)0 the energy functional of the problem (1) is always well-defined and in C1(E, R). So, we show that there exists *40 such that for any 2 (0, *) the problem (1) has a non-trivial non-negative weak solution. Our main result is the following. THEOREM 2 Assume that (H1) and (H2)0 hold. Then there exists *40 such that for any 2 (0, *) the problem (1) has a non-negative, non-trivial weak solution. The key argument in the proof of Theorem 2 is related to Ekeland’s variational principle. LEMMA 5 Suppose we are under hypotheses of Theorem 2. Then for all 2 (0, 1), there exists *40 and a40 such that for all u 2 E with kuk ¼
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J ðuÞ a 4 0 for all 2 ð0, Þ: Proof Let us assume that kuk5min(1, 1/M), where M is the positive constant from above. Then, we have jujq(x)51. Using relations (3), (4) and (6) we deduce that for any u 2 E with kuk ¼ the following inequalities hold true Z Z Z 1 1 pðxÞ ðxÞ jruj dx þ juj dx jujqðxÞ dx pþ þ q Z Z 1 þ jrujpðxÞ dx jujqðxÞ dx p q 1 þ þ kukp Mq kukq p q 1 þ 1 pþ q q ¼ þ p Mq q ¼ q M : p q pþ q
J ðuÞ
By the above inequality we remark that if we define þ
¼
p q q pþ M q
ð18Þ
then for any 2 (0, *) and u 2 E with kuk ¼ there exists a40 such that J ðuÞ a 4 0: The proof of Lemma 5 is complete. LEMMA 6 enough.
g
There exists ’ 2 E such that ’ 0, ’ 6¼ 0 and J(t’)50, for t40 small
Proof Let h ¼ min( p, ). Since q5h, then let 040 be such that q þ 05h. On the other hand, since q 2 CðÞ it follows that there exists an open set 0 such that jq(x) qj50 for all x 2 0. Thus, we conclude that q(x) q þ 05h for all x 2 0 .
Complex Variables and Elliptic Equations
807
Let ’ 2 C1 0 ðÞ be such that suppð’Þ 0 , ’(x) ¼ 1 for all x 2 0 and 0 ’ 1 in . Then using the above information for any t 2 (0, 1) we have Z Z Z 1 1 1 jrðt’ÞjpðxÞ dx þ jt’jðxÞ dx jt’jqðxÞ dx J ðt’Þ ¼ pðxÞ ðxÞ qðxÞ Z Z Z tp t jr’jpðxÞ dx þ j’jðxÞ dx þ tqðxÞ j’jqðxÞ dx q p Z Z Z th jr’jpðxÞ dx þ j’jðxÞ dx þ tqðxÞ j’jqðxÞ dx q 0 h Z Z Z th :tq þ0 ¼ jr’jpðxÞ dx þ j’jðxÞ dx j’jqðxÞ dx: þ h q 0 Therefore J ðt’Þ 5 0 1/(hq0)
for t5
with
(
R
0 5 5 min 1, R Downloaded At: 18:33 7 January 2010
) j’jqðxÞ dx R : dx þ j’jðxÞ dx
h qþ 0 pðxÞ
jr’j
Finally, we point out that Z Z jr’jpðxÞ dx þ j’jðxÞ dx 4 0: R R R In fact, if jr’jp(x)dx þ j’j(x)dx ¼ 0, then j’j(x)dx ¼ 0. Using relation (5), we deduce that j’j(x) ¼ 0 and consequently ’ ¼ 0 in wish is a contradiction. The proof of Lemma 6 is complete. g PROOF OF THEOREM 2 Let *40 be defined as in (18) and 2 (0, *). By Lemma 5 it follows that on the boundary of the ball centred at the origin and of radius in E, denoted by B(0), we have inf J 4 0:
@B ð0Þ
ð19Þ
On the other hand, by Lemma 6, there exists ’ 2 E such that J(t’)50 for all t40 small enough. Moreover, relations (3), (4) and (6) imply that for any u 2 B(0) we have J ðuÞ
1 þ kukp Mq kukq : þ p q
It follows that 1 5 c :¼ inf J 5 0: B ð0Þ
We let now 055inf@B(0) J infB(0) J. Using the above information, then the functional J : B ð0Þ ! R, is lower bounded on B ð0Þ and J 2 C1 ðB ð0Þ, RÞ. Then by Ekeland’s variational principle there exists u 2 B ð0Þ such that
c J ðu Þ c þ 0 5 J ðuÞ J ðu Þ þ k u u k , u 6¼ u :
808
K. Benali and K. Kefi
Since J ðu Þ inf J þ inf J þ 5 inf J , B ð0Þ
B ð0Þ
@B ð0Þ
we deduce that u 2 B(0). Now, we define I : B ð0Þ ! R I(u) ¼ J(u) þ ku uk. It is clear that u is a minimum point of I and thus
by
I ðu þ t vÞ I ðu Þ 0 t for small t40 and any v 2 B1(0). The above relation yields J ðu þ t vÞ J ðu Þ þ k v k 0: t Letting t ! 0 it follows that hdJ(u), v i þ kvk 0 and we infer that kdJ(u)k . We deduce that there exists a sequence {wn} B(0) such that
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J ðwn Þ ! c
and dJ ðwn Þ ! 0E :
ð20Þ
It is clear that {wn} is bounded in E. Thus, there exists w in E such that, up to a subsequence, {wn} converges weakly to w in E. Since max((x), q(x))5p*(x) for all x 2 we deduce that there exists a compact embedding E ,! Lq(x)() and E ,! L(x)(). Then {wn} converges strongly respectively in Lq(x)() and L(x)(). Using Proposition 3 and by Theorem 3.1 in Fan and Zhang [12] we deduce that {wn} converges strongly to w in E. Since J 2 C1(E, R) we conclude dJ ðwn Þ ! dJ ðwÞ,
ð21Þ
as n ! 1. Relations (20) and (21) show that dJ(w) ¼ 0 and thus w is a weak solution for problem (1). Moreover, by relation (20) it follows that J(w)50 and thus, w is a nontrivial weak solution for (1), since J(jwj) ¼ J(w) then problem (1) have a nonnegative one. The proof of Theorem 2 is complete. Finally, we point out that, under the conditions (H1) and (H2) the Ekeland’s variational principle is not applied because Lemma 6 is not satisfied. Moreover, under the conditions (H1) and (H2)0 we cannot apply the Mountain pass Theorem [3] to prove, the existence of at least one non-trivial solution because Lemma 3 is not satisfied. This enables us to affirm that we cannot obtain a critical point for J by using this method.
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Complex Variables and Elliptic Equations
809
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