Existence and multiplicity of weak solutions for ...

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Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition a

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R.A. Mashiyev , B. Cekic , M. Avci & Z. Yucedag

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Faculty of Science, Department of Mathematics, Dicle University, 21280-Diyarbakir, Turkey Version of record first published: 26 Aug 2011.

To cite this article: R.A. Mashiyev, B. Cekic, M. Avci & Z. Yucedag (2012): Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition, Complex Variables and Elliptic Equations: An International Journal, 57:5, 579-595 To link to this article: http://dx.doi.org/10.1080/17476933.2011.598928

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Complex Variables and Elliptic Equations Vol. 57, No. 5, May 2012, 579–595

Existence and multiplicity of weak solutions for nonuniformly elliptic equations with nonstandard growth condition R.A. Mashiyev*, B. Cekic, M. Avci and Z. Yucedag Faculty of Science, Department of Mathematics, Dicle University, 21280-Diyarbakir, Turkey Communicated by S. Leonardi

Downloaded by [Dicle University] at 08:53 22 October 2012

(Received 17 September 2010; final version received 3 June 2011) We discuss the problem – div(a(x, ru)) ¼ m(x)jujr(x)2u þ n(x)jujs(x)2u in , where is a bounded domain with smooth boundary in RN(N 2), and div(a(x, ru)) is a p(x)-Laplace type operator with 1 < r(x) < p(x) < s(x). We show the existence of infinitely many nontrivial weak solutions in W1,pðxÞ ðÞ. Our approach relies on the theory of the variable exponent 0 Lebesgue and Sobolev spaces combined with adequate variational methods and a variation of the Mountain Pass lemma and critical point theory. Keywords: p(x)-Laplace operator; nonuniform elliptic equations; critical point; multiple solutions; Ekeland’s variational principle; Mountain Pass theorem AMS Subject Classification: 35J60; 35B30; 35B40

1. Introduction and preliminaries The study of differential equations and variational problems involving p(x)-growth conditions has attracted a special interest because of the fact that there are some physical phenomena which can be modelled by such kind of equations, such as elastic mechanics, electrorheological fluids (sometimes referred to as ‘smart fluids’), image processing and flow in porous media. For more information we refer to the papers [1–9] and survey by Harjulehto et al. [10]. In that context it is accepted that the most convenient spaces for the mathematical modelling of such physical problems are variable exponent Lebesgue Lp(x) and Sobolev spaces W1,p(x), where p(x) is a real-valued function. These spaces were introduced firstly by Orlicz [11], and studied systematically by Nakano [12]. In this article, we study the existence and multiplicity of nontrivial weak solutions for the problem divðaðx, ruÞÞ ¼ mðxÞjujrðxÞ2 u þ nðxÞjujsðxÞ2 u,

*Corresponding author. Email: [email protected] ISSN 1747–6933 print/ISSN 1747–6941 online ß 2012 Taylor & Francis http://dx.doi.org/10.1080/17476933.2011.598928 http://www.tandfonline.com

ðPÞ

580

R.A. Mashiyev et al.


in , where is a bounded domain with smooth boundary in RN, m, n 2 C are weightpðxÞfunctions, ja(x, )j c0(h0(x) þ jjp(x)1) for all 2 RN and a.e. x 2 , h0 2 LpðxÞ1 ðÞ and r, p, s 2 C such that 8 < NpðxÞ , if pðxÞ 5 N : 1 5 rðxÞ 5 pðxÞ 5 sðxÞ 5 p ðxÞ ¼ N pðxÞ : 1, if pðxÞ N We remark that even if div(a(x, ru)) ¼ div(jrujp(x)2ru) ¼ Dp(x)u where DpðxÞ is defined as p(x)-Laplace operator, differs from p-Laplace operator, divðjrujp2 uÞ ¼ Dp u it is very troublesome to solve the equations involving p(x)-Laplace operator since it is not homogeneous [13]. This fact, i.e. nonhomogeneity, implies some difficulties; for example, we cannot use the lagrange multiplier theorem in many problems involving this operator. Problems of type (P) has been intensively studied in the past decades. As far as we know, such kind of equations were first studied by Duc and Vu in [14], and by Vu in [15]. In both papers, the authors discussed the problem divðaðx, ruÞÞ ¼ fðx, uÞ, for standard growth condition and assumed that the nonlinear term f verifies the Ambrosetti–Rabinowitz type condition. They showed the existence of weak solution by using a variation of the Mountain Pass theorem which was introduced by Duc in [16]. Moreover, the authors Toan and Ngoˆ [17] studied the similar problem for p-Laplacian operator. Besides the above mentioned papers, Miha˘ilescu and Ra˘dulescu studied the problem divðaðx, ruÞÞ ¼ u1 u1 , with 1 5 5 5 infx2 pðxÞ and u 0 in the elegant paper [6]. By using the critical point theory, they showed the existence of at least two distinct nonnegative, N inf pðxÞ nontrivial weak solutions provided that supx2 pðxÞ 5 min N, Ninfx2 pðxÞ . We x2 must point out that assumptions (A1)–(A5) established in the present article are similar to the assumptions accepted in [6], but not the same, for example, in the mentioned paper the authors assume that ðaðx, Þ aðx, ÞÞ ð Þ 0, for all x 2 and , 2 RN while we do not. We recall in what follows some definitions and basic properties of variable ðÞ. In this exponent Lebesgue–Sobolev spaces Lp(x)(), W1,p(x)() and W1,pðxÞ 0 context we refer to [18–23] for the fundamental properties of these spaces. For any h 2 C we denote 1 5 h :¼ min hðxÞ hðxÞ hþ :¼ max hðxÞ 5 1: x2

x2

Set Cþ ¼ p; p 2 C , p 4 1 8x 2 :

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Complex Variables and Elliptic Equations

We denote by U() the set of all measurable real functions defined on . For any p 2 Cþ , we define the variable exponent Lebesgue space by Z pðxÞ LpðxÞ ðÞ ¼ u 2 U ðÞ : uðxÞ dx 5 1 :

We define a norm, the so-called Luxemburg norm, on this space by the formula ( ) Z uðxÞpðxÞ jujpðxÞ ¼ inf 4 0 : dx 1 , and (Lp(x)(), jjp(x)) becomes a Banach space. Throughout this article, we will use c and ci as generic positive constants, i.e. their value may change from line to line.


0

PROPOSITION 1.1 [20,21] If p 2 L1(), the conjugate space of Lp(x)() is Lp ðxÞ ðÞ, 0 where p0 1ðxÞ þ pð1xÞ ¼ 1. For any u 2 Lp(x)() and v 2 Lp ðxÞ ðÞ, we have Z

uv dx 1 þ 1 jujpðxÞ jvjp0 ðxÞ : ð1:1Þ p ð p Þ 0 The modular of the Lp(x)() space, which is the mapping p(x) : Lp(x)() ! R is defined by Z pðxÞ pðxÞ ðuÞ ¼ uðxÞ dx 8u 2 LpðxÞ ðÞ:

PROPOSITION 1.2 [20,21]

If p < 1 and u 2 Lp(x)() , then we have þ

ðiÞ jujpðxÞ 5 1ð¼ 1; 4 1Þ , pðxÞ ðuÞ 5 1ð¼ 1; 4 1Þ;

þ

ðiiÞ jujpðxÞ 4 1 ) jujppðxÞ pðxÞ ðuÞ jujppðxÞ ; ðiiiÞ

þ

ð1:2Þ ð1:3Þ

jujpðxÞ 5 1 ) jujppðxÞ pðxÞ ðuÞ jujppðxÞ :

ð1:4Þ

PROPOSITION 1.3 [20,21] If pþ < 1 and u, un 2 Lp(x)(), n ¼ 1, 2, . . . , then the following statements are equivalent: ðiÞ ðiiÞ ðiiiÞ

lim jun ujpðxÞ ¼ 0;

n!1

lim pðxÞ ðun uÞ ¼ 0;

n!1

un ! u in measure in and lim pðxÞ ðun Þ ¼ pðxÞ ðuÞ: n!1

Define the variable exponent Sobolev space W1,p(x)() by W1,pðxÞ ðÞ ¼ fu 2 LpðxÞ ðÞ; jruj 2 LpðxÞ ðÞg, and it can be equipped with the norm kuk1,pðxÞ ¼ jujpðxÞ þ jrujpðxÞ

8u 2 W1,pðxÞ ðÞ:

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1,p(x) The space W1,pðxÞ ðÞ is denoted by the closure of C1 () . We will use 0 ðÞ in W 0 1,pðxÞ kuk ¼ jrujp(x) for u 2 W0 ðÞ in the following discussions. We also consider the weighted variable exponent Lebesgue space LpcððxxÞÞ ðÞ. Let c : RN ! R be a measurable real function such that c(x) > 0 a.e. x 2 . We define Z pðxÞ pðxÞ LcðxÞ ðÞ ¼ u 2 U ðÞ : cðxÞ uðxÞ dx 5 1; cðxÞ 4 0 ,

with the norm

(

jujLpðxÞ ðÞ ¼ jujcðxÞ,pðxÞ


cðxÞ

) uðxÞpðxÞ ¼ inf 4 0 : cðxÞ dx 1 , Z

ðx Þ then LpcðxÞ ðÞ is a Banach space which has similar properties with the usual variable ðx Þ exponent Lebesgue spaces. The modular of this space is pðxÞ,cðxÞ : LpcðxÞ ðÞ ! R defined by Z pðxÞ pðxÞ,cðxÞ ðuÞ ¼ cðxÞuðxÞ dx:

PROPOSITION 1.4 [20,21] ðiÞ

ðx Þ If p < 1 and un 2 LpcðxÞ ðÞ, n ¼ 1, 2, . . . , we have þ

lim jun jcðxÞ,pðxÞ ¼ 0 , lim pðxÞ,cðxÞ ðun Þ ¼ 0;

n!1

n!1

ðiiÞ jun jcðxÞ,pðxÞ ! 1 , pðxÞ,cðxÞ ðun Þ ! 1: ðx Þ ðÞ, PROPOSITION 1.5 [20,21] If p > 1 and pþ < 1 then, the spaces Lp(x)(), LpcðxÞ 1,pðxÞ 1,p(x) W (), and W0 ðÞ are separable and reflexive Banach spaces.

PROPOSITION 1.6 [19,21] Assume that is bounded, the boundary of possesses the cone property and p 2 Cþ ðÞ. If q 2 Cþ ðÞ and q(x) < p(x) for all x 2 then the embedding W1,p(x)() ,! Lq(x)() is compact and continuous, and there is a constant c > 0 such that jujqðxÞ ckuk

8u 2 W1,pðxÞ ðÞ: 0

PROPOSITION 1.7 [18] Let p and q be measurable functions such that p 2 L1(), and 1 p(x)q(x) 1, for a.e. x 2 . Let u 2 Lq(x)(), u 6¼ 0. Then þ jujpðxÞqðxÞ 5 1¼)jujppðxÞqðxÞ jujpðxÞ qðxÞ jujppðxÞqðxÞ , þ jujpðxÞqðxÞ 4 1¼)jujp jujpðxÞ jujp : pðxÞqðxÞ

q ðx Þ

pðxÞqðxÞ

In particular, if p(x) ¼ p ¼ const then jjujp jqðxÞ ¼ jujppqðxÞ : Let E denote the variable exponent Sobolev space W1,pðxÞ ðÞ. Define the energy 0 functional I : E ! R associated with (P) by Z

Z Aðx, ruÞdx

IðuÞ ¼

mðxÞ rðxÞ juj dx rðxÞ

Z

nðxÞ sðxÞ juj dx :¼ ðuÞ JðuÞ, sðxÞ


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where Z ð u Þ ¼

Aðx, ruÞdx,

and Z JðuÞ ¼

mðxÞ rðxÞ juj dx þ rðxÞ

Z

nðxÞ sðxÞ juj dx: sðxÞ

As we know, standard arguments imply that J 2 C1(E, R) and the derivative of J is Z Z hJ0 ðuÞ, i ¼ mðxÞjujrðxÞ2 u dx þ nðxÞjujsðxÞ2 u dx,


for all u, 2 E. We say that u 2 E is a weak solution of problem (P) if Z Z Z aðx, ruÞ r’ dx mðxÞjujrðxÞ2 u’ dx nðxÞjujsðxÞ2 u’ dx ¼ 0,

N

N

for all ’ 2 E, where a(x, ): R ! R is continuous derivative with respect to of the mapping A : RN ! R, A ¼ A(x, ), i.e. a(x, ) ¼ rA(x, ). In this article, we assume the following hypotheses: (A1)

The following inequality holds: aðx, Þ c0 ðh0 ðxÞ þ jjpðxÞ1 Þ,

for all 2 RN and a.e. x 2 ,

0

where h0 2 Lp ðxÞ ðÞ is a nonnegative measurable function. (A2) A is p(x)-uniformly convex: There exists a constant k > 0 such that

pðxÞ þ 1 1 A x, Aðx, Þ þ Aðx, Þ k , for all x 2 and , 2 RN : 2 2 2 (A3)

The following inequalities hold true: jjpðxÞ aðx, Þ pðxÞAðx, Þ,

for all 2 RN and a.e. x 2 .

(A4) A(x, 0) ¼ 0 for all x 2 . N (A5) A(x, ) ¼ A(x, ) for all 2 R and a.e. x 2 . NpðxÞ pðxÞ (x) (M) m 2 L () and 2 Cþ such that NpðxÞrðxÞðNpðxÞÞ 5 ðxÞ 5 pðxÞrðxÞ for all x 2 . pðxÞ NpðxÞ (N) n 2 L(x)() and 2 Cþ such that pðxÞsðxÞ 5 ðxÞ 5 NpðxÞsðxÞ ðNpðxÞÞ for all x 2 . Some examples for A: (i) Set A(x, ) ¼ (1/p(x))jjp(x), a(x, ) ¼ jjp(x)2, where p > 1. Then, we get the p(x)-Laplace operator div jrujpðxÞ2 ru : (ii) Set A(x, ) ¼ (1/p(x))[(1 þ jj2)p(x)/2 1], a(x, ) ¼ (1 þ jj2)(p(x)2)/2, where p > 1.

584


Then, we obtain the generalized mean curvature operator ð pðxÞ2Þ=2 divð 1 þ jruj2 ruÞ: The main results which we deal with are the following theorems. THEOREM 1.8 Let r, p, s 2 Cþ such that r rþ < p pþ < s sþ. Suppose that conditions (A1)–(A4), (M) and (N) hold. Then the problem (P) has at least two nontrivial weak solutions in E.


THEOREM 1.9 Suppose that assumptions of Theorem 1.8 hold. Additionally, if condition (A5) holds, then the problem (P) has infinitely many nontrivial weak solutions in E.

2. Auxiliary results LEMMA 2.1

Let p 2 Cþ .

(i) A verifies the growth condition Aðx, Þ c1 ðh0 ðxÞjj þ jjpðxÞ Þ, for all 2 RN and a.e. x 2 . (ii) A is p(x)-homogeneous, i.e. Aðx, zÞ Aðx, ÞzpðxÞ , for all z 1, 2 RN and x 2 . Proof (i) For any x 2 and 2 RN, we have Z Aðx, Þ ¼ 0

1

d Aðx, tÞdt ¼ dt

Z

1

aðx, tÞ dt: 0

By hypothesis (A1), we have Aðx, Þ

Z

Z

1

1

ðh0 ðxÞ þ jjpðxÞ1 jtjpðxÞ1 Þjjdt

jaðx, tÞjjjdt c0 0

0

Z

1

ðh0 ðxÞjj þ jjpðxÞ jtjpðxÞ1 Þdt

c0 0

c0 ðh0 ðxÞjj þ jjpðxÞ Þ: (ii) To see that, let us define g(t) ¼ A(x, t). Then, by (A3) 1 pðxÞ pðxÞ g0 ðtÞ ¼ aðx, tÞ ¼ aðx, tÞ t Aðx, tÞ ¼ gðtÞ, t t t g0 ðtÞ pðxÞ , gðtÞ t


585

and integrating both sides over (1, z), we have log gðzÞ log gð1Þ pðxÞlog z: Then, gð z Þ zpðxÞ , gð1Þ so we conclude that Aðx, zÞ Aðx, ÞzpðxÞ : g


The proof is complete. LEMMA 2.2

Let p 2 Cþ .

(i) The functional is well-defined on E. (ii) The functional is of class C1(E, R) and

0 ðuÞ, ’ ¼

Z aðx, ruÞ r’ dx,

for all u,’ 2 E. (iii) The functional is weakly lower semi-continuous on E. (iv) For all u, 2 E

u þ 1 1 ðuÞ þ ðÞ kku kp : 2 2 2 (v) For all u, 2 E ðuÞ ðÞ 0 ðÞ, u : Proof (i) By (i) in Lemma 2.1 and Proposition 1.1 , we have Z Z Z ðuÞ ¼ Aðx, ruÞdx c1 h0 ðxÞjrujdx þ c1 jrujpðxÞ dx

pþ

c2 jh0 jp0 ðxÞ jrujpðxÞ þc1 jjujj þ

c3 jjujj þ c1 jjujjp 5 1: Hence, is well-defined on E. (ii) Let u, ’ 2 E, x 2 , and 0 < jrj < 1. Then, by the mean value theorem, there exists ! 2 [0, 1] such that Z Aðx, ruðxÞ þ rr’ðxÞÞ Aðx, ruðxÞÞ 1 ¼ aðx, ruðxÞ þ !rr’ðxÞÞ r’ðxÞd! r 0

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R.A. Mashiyev et al. Z

1

pðxÞ1 c0 ðh0 ðxÞ þ ruðxÞ þ !rr’ðxÞ Þr’ðxÞd!

0


pðxÞ1 Þr’ðxÞ c0 ðh0 ðxÞ þ ruðxÞ þ r’ðxÞ c0 h0 ðxÞr’ðxÞ þ c0 r’ðxÞðruðxÞ þ r’ðxÞÞpðxÞ1 pðxÞ1 pðxÞ1 þ c0 h0 ðxÞr’ðxÞ þ c0 2p 1 r’ðxÞðruðxÞ þr’ðxÞ Þ pðxÞ þ þ pðxÞ1 þc0 2p 1 r’ðxÞ : c0 h0 ðxÞr’ðxÞ þ c0 2p 1 r’ðxÞruðxÞ With the help of Proposition 1.1, we can see h0(x)jr’(x)j, jr’(x)j jru(x)jp(x)1 and jr’(x)jp(x)are integrable on , so the right-hand side is integrable on . Applying the Lebesgue dominated convergence theorem, we have Z 0 Aðx, ru þ rr’Þ Aðx, ruÞ dx ðuÞ, ’ ¼ lim r!0 r Z ¼ aðx, ruÞ r’ dx:

Next, we show the continuity of 0 on E. Suppose un ! u in E and define (x, u) ¼ a(x, ru). Using the hypothesis (A1) and Proposition 2.2 in Fan and 0 Zhang [3], we conclude that (x, un) ! (x, u) in ðLp ðxÞ ðÞÞN : Then, by Proposition 1.1, we have 0 ðun Þ 0 ðuÞ, ’ ðx, un Þ ðx, uÞ 0 r’ p ðx Þ p ðx Þ and so 0 ðun Þ 0 ðuÞ ðx, un Þ ðx, uÞ

p 0 ðx Þ

! 0,

as n ! 1. (iii) By corollary III.8 in Brezis [24], it is enough to show that is lower semicontinuous. For this purpose, we fix u 2 E and > 0. Since is convex (by condition (A2)), we deduce that for any 2 E, the following inequality holds: Z ðÞ ðuÞ þ aðx, ruÞ ðr ruÞdx:

Using condition (A1), Proposition 1.1 and Proposition 1.7, we have Z ðÞ ðuÞ aðx, ruÞjr rujdx Z Z ðuÞ c0 h0 ðxÞ rð uÞ dx c0 rð uÞjrujpðxÞ1 dx ðuÞ c4 jh0 jp0 ðxÞ rð uÞ c5 rð uÞ jrujpðxÞ1 0 p ðx Þ

ðuÞ c6 jj ujj c5 jj ujjkuk ðuÞ c6 jj ujj c7 jj ujj

p ðx Þ

p ðx Þ

pþ 1

ðuÞ ", " . So, we deduce that is weakly lower semifor all 2 E with jj ujj 5 ¼ c6 þc 7 continuous.

587

Complex Variables and Elliptic Equations (iv) Using condition (A2) and Proposition 1.2, we have

Z

uþ ru þ r A x, ¼ dx 2 2 Z Z Z 1 1 Aðx, ruÞdx þ Aðx, rÞdx k jru rjpðxÞ dx 2 2 1 1 p ðuÞ þ ðÞ kku k : 2 2


(v) Since is convex (by condition (A2)), we can find t 2 (0, 1) such that ð þ tðu ÞÞ ðÞ ðð1 tÞ þ tuÞ ðÞ ¼ t t ð1 tÞðÞ þ tðuÞ ðÞ ¼ ðuÞ ðÞ: t Then we have ð þ tðu ÞÞ ðÞ 0 ¼ ðÞ, u : t!0 t

lim Thus, we obtain

0 ðÞ, u ðuÞ ðÞ: g

The proof of Lemma 2.2 is complete. 1

LEMMA 2.3 (Mountain Pass lemma) [25] Let E be a Banach space and I 2 C (E, R) satisfies the Palais–Smale condition. Assume that I(0) ¼ 0, and there exists a positive real number and u 2 E such that (i) kuk > , I(u) I(0). (ii) ¼ inf {I(u) : u 2 E, kuk ¼ } > 0. Put G ¼ {’ 2 C([0, 1], E) : ’(0) ¼ 0, ’(1) ¼ u}. Set ¼ inf{maxI(’([0, 1])) : ’ 2 G}. Then, and is a critical value of I. LEMMA 2.4 (Symmetric Mountain Pass lemma) [25] Let E be an infinite-dimensional real Banach space and I 2 C1(E, R) satisfies the Palais–Smale condition. Assume that I(0) ¼ 0 and (i) There exist two positive real numbers and such that inf IðuÞ 4 0,

u2@B

where B is an open ball in E of radius centred at the origin and @B is its boundary. (ii) For each finite-dimensional linear subspace E1 E, the set u 2 E1 : IðuÞ 0 , is bounded. Then, I has an unbounded sequence of critical values.

588


LEMMA2.5 [26] Assume that the boundary @ of possesses the cone property, and (x) p, s 2 C such that p(x) < s(x) < p(x). Suppose that n 2 L (), n(x) > 0, 0 þ 0 0 2 Cþ and ( ) (x) ( ) for all x 2 . If s 2 C , and 1 5 sðxÞ 5

ðxÞ 1 p ðxÞ 8x 2 ðxÞ

ð2:1Þ

or


1 5 ðxÞ 5

NpðxÞ NpðxÞ sðxÞðN pðxÞÞ

8x 2

the embedding from W1,p(x)() to LsnððxxÞÞ ðÞ is compact, and there is a constant c8 such that the inequality Z þ nðxÞjujsðxÞ dx c8 ðkuks þkuks Þ ð2:2Þ

holds. LEMMA @ of possesses the cone property, and r, 2.6 [26] Assume that the boundary , such that r(x) < p(x) < p (x). Suppose that m 2 L(x)(), m(x) > 0 and p2C ðxÞ 2 Cþ for all x 2 . If r 2 C , pðxÞ 5 ðxÞ1 qðxÞ and 1 5 rðxÞ 5

ðxÞ 1 p ðxÞ 8x 2 ðxÞ

or NpðxÞ pðxÞ 5 ðxÞ 5 NpðxÞ rðxÞðN pðxÞÞ pðxÞ rðxÞ

8x 2

the embedding from W1,p(x)() to LrmðxðxÞ Þ ðÞ is compact, and there is a constant c9 such that the inequality Z þ mðxÞjujrðxÞ dx c9 ðkukr þkukr Þ ð2:3Þ

holds. LEMMA 2.7

Let r, p, s 2 Cþ such that r rþ < p pþ < s sþ.

(i) I is weakly lower semi-continuous from E to R. (ii) I is well-defined on E and of class C1(E, R), and its derivative given by Z Z Z 0 aðx, ruÞ r dx mðxÞjujrðxÞ1 dx nðxÞjujsðxÞ1 dx, I ðuÞ, ¼

for all u, 2 E. (iii) I(0) ¼ 0. (iv) There exist two positive real numbers and such that inf IðuÞ : u 2 E, kuk ¼ 4 : (v) There exists ’ 2 E such that ’ 0, ’ 6¼ 0, and I(t’) < 0, for t > 0 small enough. (vi) There exists u 2 E such that kuk > , I(u) 0.


589

(vii) The set G ¼ ’ 2 Cð½0, 1 , EÞ : ’ð0Þ ¼ 0, ’ð1Þ ¼ u , is not empty. (viii) I satisfies the Palais–Smale condition on E, i.e. there exists a sequence un E which satisfies the properties; Ið Þ ! c and I 0 ðun Þ ! 0 in E as n ! 1 possesses a convergent subsequence in E. Proof (i) Let {un} E be a sequence such that


un * u in E: Since is weakly lower semi continuous, we have ðuÞ lim inf ðun Þ:

ð2:4Þ

Moreover, since we have the embeddings ðaÞ : E ,! LrmðxðxÞ Þ ðÞ and E ,! LsnððxxÞÞ ðÞ, it follows that ðbÞ : un ! u in LrmðxðxÞ Þ ðÞ and un ! u in LsnððxxÞÞ ðÞ: This fact together with (2.4) implies IðuÞ lim infIðun Þ: n!1

Hence, I is weakly lower semi-continuous. (ii) This comes from (i) and (ii) in Lemma 2.2 and the properties of J. (iii) This comes from the definition of I. (iv) From the definition of I we can write Z Z 1 1 IðuÞ ðuÞ mðxÞjujrðxÞ dx nðxÞjujsðxÞ dx: r s Let kuk < 1. Using the condition (A3), Lemma 2.5 and Lemma 2.6 we have 1 2c9 2c8 þ kukp kukr kuks þ p r s

1 2c9 2c þ þ 8 r p s pþ ku kp : ¼ þ ku k ku k p r s

IðuÞ

ð2:5Þ

Let us define the function : [0, 1] ! R by ðtÞ ¼

1 2c9 r pþ 2c8 s pþ t t : pþ r s

Since is positive 1 in a neighbourhood of the origin, for example, for a fixed t0 2 0, 2prþ c9 r pþ , the conclusion of the lemma follows at once.

590


(v) Let ’ 2 C1 0 ðÞ, ’ 0, ’ 6¼ 0, and t sufficiently small. Then, using (ii) in Lemma 2.1, we have Z Z rðxÞ sðxÞ tr ts pþ mðxÞ ’ dx þ nðxÞ’ dx: Iðt’Þ t ð’Þ þ r s Since r < pþ < s, we deduce that I(t’) < 0. (vi) If we choose t > 1 sufficiently large and use assumptions of Lemma 2.7, the desired result follows immediately from (v). (vii) If we consider the function ’ 2 C([0,1], E) defined by ’(t) ¼ tu, for every t 2 [0, 1], it is clear that ’ 2 G, and so G 6¼ 1. (viii) Assume that {un} E is a sequence which satisfies the properties I0 ðun Þ ! 0


Iðun Þ ! c,

in E as n ! 1,

ð2:6Þ

where E is dual space of E and c is a positive constant. We prove that {un} possesses a convergent subsequence. First, we show that {un} is bounded in E. Arguing by contradiction and passing to a subsequence, we have kunk ! 1 as n ! 1. Using (2.6), and considering kunk > 1, for n large enough, we can write 1 0 I ðun Þ, un s Z 1 ¼ ðun Þ Jðun Þ aðx, run Þ run dx s Z Z 1 1 r ðx Þ þ mðxÞjun j dx þ nðxÞjun jsðxÞ dx s s

Z Z 1 1 1 mðxÞjun jrðxÞ dx: ðun Þ aðx, run Þ run dx þ s s r

1 þ c10 þ jjun jj Iðun Þ

By the condition (A3), we have Z Z jrun jpðxÞ dx aðx, run Þ run dx pþ ðun Þ:

Thus, we can write

Z pþ 1 1 1 þ c10 þ jjun jj 1 ðun Þ þ mðxÞjun jrðxÞ dx s r s

Z

Z 1 pþ 1 1 jrun jpðxÞ dx þ mðxÞjun jrðxÞ dx: þ 1 p s r s Moreover, by (2.2), we can write 1 þ c10 þ jjun jj þ

1 1 1 1 rþ j j j j jjun jjp : u n r s pþ s

Now dividing the above inequality by jjun jjp , and passing to the limit as n ! 1 we obtain s pþ. But this result contradicts with the fact that s > pþ. So, {un} is bounded in E. Therefore, there exists a subsequence, again denoted by {un},


591

and u 2 E such that un * u in E: Using the embeddings (a) and (b) (see proof of Lemma 2.7) and relation (2.6), we have 0 I ðun Þ, un u ! 0 as n ! 1: On the other hand, we have Z Z 0 aðx, run Þ ðrun ruÞ dx mðxÞjun jrðxÞ2 un ðun uÞ dx I ðun Þ, un u ¼ Z nðxÞjun jsðxÞ2 un ðun uÞdx


or Z

aðx, run Þ ðrun ruÞdx Z Z 0 rðxÞ2 ¼ I ðun Þ, un u þ mðxÞjun j un ðun uÞdx þ nðxÞjun jsðxÞ2 un ðun uÞdx:

Using the fact that {un} strongly converges to u in LrmðxðxÞ Þ ðÞ, Proposition 1.1, Proposition 1.6 and Proposition 1.7, we have Z mðxÞjun jrðxÞ2 un ðun uÞdx c11 jmjðxÞ jun jrðxÞ1 jun ujmðxÞ,rðxÞ ðx Þ rðxÞ1 0 ju ujmðxÞ,rðxÞ c12 jun j ðxÞrðxÞ n þ

c13 jun jrðrþ1 1Þ0 ðxÞrðxÞ jun ujmðxÞ,rðxÞ c14 kun kjun ujmðxÞ,rðxÞ , where 2 Cþ such that ð1xÞ þ ð1xÞ þ rð1xÞ ¼ 1. Since jun ujm(x),r(x) ! 0 as n ! 1, by Proposition 1.4, it follows that Z mðxÞjun jrðxÞ2 un ðun uÞdx ¼ 0: lim n!1

With similar arguments we can obtain that Z nðxÞjun jsðxÞ2 un ðun uÞdx ¼ 0: lim n!1

All the above pieces of information imply Z aðx, run Þ ðrun ruÞdx ¼ 0,

that is, lim 0 ðun Þ, un u ¼ 0:

n!1

592


By using (v) in Lemma 2.2, we have 0 ¼ lim 0 ðun Þ, u un lim ððuÞ ðun ÞÞ ¼ ðuÞ lim ðun Þ n!1

n!1

n!1

or lim ðun Þ ðuÞ:

n!1

This fact and relation (iii) in Lemma 2.2 imply lim ðun Þ ¼ ðuÞ:


n!1

We assume by contradiction that un does not strongly converge tou in E. Then, there exists " > 0 and a subsequence funm g of {un} such that unm u ". On the other hand, by (iv) in Lemma 2.2, we have

u þ u p 1 1 n ðuÞ þ unm m kunm u k"p : 2 2 2 Letting m ! 1 in the above inequality, we obtain

u þ u n lim sup m ðuÞ k"p : 2 n!1 Moreover, we have funm2þug converges weakly to u in E. Using (iii) in Lemma 2.2, we obtain

u þ u n , ðuÞ lim inf m n!1 2 and that is a contradiction. It follows that {un} strongly converges to u in E. The proof of Lemma 2.7 is complete. g

3. Proofs Proof of Theorem 1.8 From Lemma 2.3 and Lemma 2.7 we deduce the existence of u1 2 E as a nontrivial weak solution of (P). We will prove now that there exists a second weak solution u2 2 E such that u1 6¼ u2. By Lemma 2.7 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 I 4 0:

@B ð0Þ

On the other hand, from (v) in Lemma 2.7, there exists ’ 2 E such that I(t’) < 0 for all t > 0 small enough. Moreover, since relations (2.5) holds for all u 2 E, i.e. IðuÞ

1 2c9 2c8 jjujjp jjujjr jjujjs , þ p r s

it follows that 1 5 c :¼ inf I 5 0: B ð0 Þ


593

So, we have 0 5 " 5 inf I inf I: @B ð0Þ

B ð0Þ

Applying Ekeland’s variational principle [27] to the functional I : B ð0Þ ! R, we can find u" 2 B ð0Þ such that u" 2 B(0). Now, let define : B ð0Þ ! R by (u) ¼ I(u) þ " ku u k. It is clear that u is a minimum point of , and this implies that kI0 (u")k ". So, we deduce that there exists a sequence {un} B (0) such that I0 ðun Þ ! 0:


Iðun Þ ! c and

Since I satisfies the Palais–Smale condition on E we conclude that {un} strongly converges to u2 in E. Hence, u2 is a weak solution for (P). From the relation 0 > c ¼ I(u2) it follows that u2 is nontrivial. Moreover, since Iðu1 Þ ¼ c 4 0 4 c ¼ Iðu2 Þ, g

it is clear that u1 6¼ u2. The proof is complete.

Proof of Theorem 1.9 By the condition (A5), I is even. So, in order to apply Lemma 2.4, it is enough to show that (ii) in Lemma 2.4 holds. By proof of (i) in Lemma 2.2, we know þ

ðuÞ c3 jjujj þ c1 jjujjp , for all u 2 E. Thus, we can write 1 Iðun Þ c3 jjun jj þ c1 jjun jj þ s pþ

Z

nðxÞjun jsðxÞ dx:

Let u 2 E be arbitrary but fixed. Set ¼ < [ where < :¼ {x 2 : j u(x) j < 1} and :¼ /{x 2 : ju(x)j < 1}. Then, we have Z 1 pþ Iðun Þ c3 jjun jj þ c1 jjun jj þ nðxÞjun jsðxÞ dx s Z 1 pþ c3 jjun jj þ c1 jjun jj þ nðxÞjun js dx s Z Z 1 1 pþ s c3 jjun jj þ c1 jjun jj þ nðxÞjun j dx þ þ nðxÞjun js dx: s s 5 On the other hand, we can find a constant c15 such that Z nðxÞjun js dx c15 : 5

Thus þ

Iðun Þ c3 jjun jj þ c1 jjun jjp for all u 2 E.

1 sþ

Z

nðxÞjun js dx þ c16 ,

594


The functional jjnðxÞ,s : E ! R defined by Z

1=s s jujnðxÞ,s ¼ nðxÞjuj dx

is a norm in E. Since in the finite-dimensional subspace E1, the norms jjnðxÞ,s and kk are equivalent then, there exists a constant C ¼ C(E1) such that kuk CjujnðxÞ,s

8u 2 E1 :

Therefore, there exists a constant c17 such that þ

Iðun Þ c3 jjun jj þ c1 jjun jjp c17 kun ks þc16 ,


for all u 2 E1. Since pþ < s then {u 2 E1 : I(u) 0} is bounded. Hence, I has an unbounded sequence of critical values in E. The proof is complete. g

Acknowledgements The authors thank the referees for their valuable suggestions and helpful corrections, which have improved the presentation of this article. This research project was supported by DUBAP -10-FF-15, Dicle University, Turkey.

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