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A remark on the existence of positive solutions for variable exponent elliptic systems
ARAB JOURNAL OF MATHEMATICAL SCIENCES
Arab J Math Sci 19(1) (2013), 85–94
A remark on the existence of positive solutions for variable exponent elliptic systems G.A. AFROUZI a,*, S. SHAKERI a, N.T. Chung
b
a
b
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam
Received 12 March 2012; revised 21 July 2012; accepted 26 August 2012
Available online 5 September 2012 Abstract.
In this article, we consider the system of differential equations
8 DpðxÞ u ¼ kpðxÞ ½aðxÞuaðxÞ vcðxÞ þ h1 ðxÞ in X; > > < DqðxÞ v ¼ kqðxÞ ½bðxÞudðxÞ vbðxÞ þ h2 ðxÞ in X; > > : u¼v¼0 on @X; where X RN is a bounded domain with C2-boundary @X; 1 < pðxÞ; qðxÞ 2 C1 ðXÞ are functions. The operator Dp(x) u = div(ŒuŒp(x)2u) is called the p(x)-Laplacian. When a, b, d, c satisfy some suitable conditions, we prove the existence of positive solution via sub-supersolution arguments without assuming sign conditions on the functions h1 and h2. Mathematics subject classifications: 35J25; 35J60 Keywords: Positive solutions; Variable exponent elliptic systems; Sub-supersolutions
1. INTRODUCTION
AND PRELIMINARIES
The study of differential equations and variational problems with nonstandard p(x)growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [1,2,14,20]). Many existence results have been obtained on this kind of problems, see for example [4,9,10,12,15–18]. In [6–8], Fan * Corresponding author. E-mail addresses: [email protected] (G.A. Afrouzi), [email protected] (S. Shakeri), ntchung82@yahoo. com (N.T. Chung). Peer review under responsibility of King Saud University.
Production and hosting by Elsevier 1319-5166 ª 2012 King Saud University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ajmsc.2012.08.002
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et al. studied the regularity of solutions for differential equations with nonstandard p(x)-growth conditions. In this paper, we mainly consider the existence of positive weak solutions for the system 8 DpðxÞ u ¼ kpðxÞ ½aðxÞuaðxÞ vcðxÞ þ h1 ðxÞ > > < DqðxÞ v ¼ kqðxÞ ½bðxÞudðxÞ vbðxÞ þ h2 ðxÞ > > : u¼v¼0
in X; in X;
ð1:1Þ
on @X;
where X RN is a bounded domain with C2-boundary @X; 1 < pðxÞ; qðxÞ 2 C1 ðXÞ are two functions. The operator Dp(x)u = div(ŒuŒp(x)2u) is called the p(x)Laplacian and the corresponding equation is called a variable exponent equation. Especially, if p(x) ” p (a constant), (1.1) is the well-known (p,q)-Laplacian system and the corresponding equation is called a constant exponent equation. We have known that the existence of solutions for p-Laplacian elliptic systems has been intensively studied in the last decades, we refer to [3,11,13]. In [11], Hai et al. considered the existence of positive weak solutions for the p-Laplacian problem 8 Dp u ¼ kfðvÞ in X; > > < ð1:2Þ Dp v ¼ kgðuÞ in X; > > : u¼v¼0 on @X; in which the first eigenfunction was used for constructing the subsolution of p-Laplacian problems. Under the condition f MðgðuÞÞ1=ðp1Þ lim ¼ 0 for all M > 0; u!þ1 up1 the authors showed that the problem (1.2) has at least one positive solution provided that k > 0 is large enough. In [3], the author studied the existence and nonexistence of positive weak solution to the following quasilinear elliptic system 8 a c in X; > < Dp u ¼ kfðu; vÞ ¼ ku v Dq v ¼ kgðu; vÞ ¼ kud vb in X; ð1:3Þ > : u¼v¼0 on @X: The first eigenfunction is used to construct the subsolution of problem (1.3), the main results are as follows: (i) If a,b P 0,c,d > 0, h = (p 1 a)(q 1 b) cd > 0, then problem (1.3) has a positive weak solution for each k > 0; (ii) If h = 0 and pc = q(p 1 a), then there exists k0 > 0 such that for 0 < k < k0, then problem (1.3) has no nontrivial nonnegative weak solution.
A remark on the existence of positive solutions for variable exponent elliptic systems
87
In recent papers [15–18], Zhang has developed problems (1.2) and (1.3) in the variable exponent Sobolev space. On the p(x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist. Even if the first eigenfunction of p(x)-Laplacian exists, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of p(x)-Laplacian problems. Motivated by the above papers, in this note, we are interested in the existence of positive solution for problem (1.1), where a, b are continuous functions in X and k is a positive parameter. Our main goal is to improve the result introduced in [18], in which a = b ” 1. To be more precise, we assume that X RN is an open bounded domain with 2 C -boundary oX and the following conditions hold: (H1) pðxÞ; qðxÞ 2 C 1 ðXÞ and 1 < p 6 p+ and 1 < q 6 q+; (H2) h1 ; h2 ; a; b; c; d 2 C 1 ðXÞ satisfy a,b P 0 on X, and c,d > 0 on X; (H3) 0 6 a+ < p 1,0 6 b+ < q 1 and - :¼(p 1 a+)(q 1 b+) d+c+ > 0; (H4) a; b : X ! ð0; 1Þ are continuous functions such that a1 ¼ minx2X aðxÞ; b1 ¼ minx2X bðxÞ; a2 ¼ maxx2X aðxÞ and b2 ¼ maxx2X bðxÞ. To study p(x)-Laplacian problems, we need some theories on the spaces Lp(x)(X), W (X) and properties of p(x)-Laplacian which we will use later (see [5]). If X RN is an open domain, write 1,p(x)
Cþ ðXÞ ¼ fh : h 2 CðXÞ; hðxÞ > 1 for x 2 Xg; h
+
= supx2Xh(x),h = infx2Xh(x), for any h 2 C(X), and Z pðxÞ LpðxÞ ðXÞ ¼ uju is a measurable real-valued function such that juj dx < 1 : X
Throughout the paper, we assume that p,q 2 C+(X) and 1 < infx2Xp(x) 6 supx2Xp(x) < N,1 < infx 2Xq(x) 6 supx2Xq(x) < N. We introduce the norm Lp(x)(X) by ( ) Z uðxÞpðxÞ jujpðxÞ ¼ inf k > 0 : k dx 6 1 ; X and (Lp(x)(X),Œ Æ Œp(x)) becomes a Banach space, we call it the generalized Lebesgue space. The space (Lp(x)(X), Œ Æ Œp(x)) is a separable, reflexive and uniform convex Banach space (see [5, Theorem 1.10, 1.14]). The space W1,p(x)(X) is defined by W1,p(x)(X) = {u 2 Lp(x)(X):ŒuŒ 2 Lp(x)(X)}, and it is equipped with the norm kuk ¼ jujpðxÞ þ jrujpðxÞ ; 1;pðxÞ
8u 2 W1;pðxÞ ðXÞ: 1;pðxÞ
1,p(x) We denote by W0 ðXÞ the closure of C1 (X). W1,p(x)(X) and W0 ðXÞ 0 ðXÞ in W are separable, reflexive and uniform convex Banach space (see [5, Theorem 2.1]). We define
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ðLðuÞ; vÞ ¼
Z
pðxÞ2
jruj
rurvdx;
8u; v 2 W1;pðxÞ ðXÞ;
RN
then L:W1,p(x)(X) fi (W1,p(x)(X))* is a continuous, bounded and is a strictly monotone operator, and it is a homeomorphism [9, Theorem 3.11]. 1;pðxÞ 1;qðxÞ ðXÞ; W0 ðXÞ ; ðu; vÞ is called a weak solution of Definition 1.1. If ðu; vÞ 2 W0 (1.1) if it satisfies (R X
R
X
jrujpðxÞ2 ru rudx ¼ jrvj
qðxÞ2
rv rwdx ¼
R R
1;pðxÞ
X X
kpðxÞ ½aðxÞuaðxÞ vcðxÞ þ h1 ðxÞudx; 8u 2 W0 k
qðxÞ
½bðxÞu
dðxÞ bðxÞ
v
þ h1 ðxÞwdx;
8w 2
ðXÞ;
1;qðxÞ W0 ðXÞ:
1;pðxÞ
Define A : W1;pðxÞ ðXÞ ! ðW0 ðXÞÞ as Z pðxÞ2 jruj ruru þ hðx; uÞu dx; hAu; ui ¼
8u; u 2 W1;pðxÞ ðXÞ;
X
where h(x, u) is continuous on X R, and h(x, Æ) is increasing and satisfies jhðx; tÞj 6 C1 þ C2 jtj where
(
p ðxÞ ¼
p ðxÞ1
;
NpðxÞ ; NpðxÞ
pðxÞ < N
1;
pðxÞ P N:
It is easy to check that A is a continuous bounded mapping. Copying the proof of [19], we have the following lemma. Lemma 1.2 (Comparison principle). Let u,v 2 W1,p(x)(X) satisfy Au Av P 0 in 1;pðxÞ 1;pðxÞ ðW0 ðXÞÞ ; uðxÞ ¼ minfuðxÞ vðxÞ; 0g. If uðxÞ 2 W0 ðXÞ (i.e. u P v on oX), then u P v a.e. in X. Here and hereafter, we will use the notation d(x, oX) to denote the distance of x 2 X to the boundary of X. Denote d(x) = d(x, oX) and oXe = {x 2 XŒd(x, oX) < e}. Since oX is C2 regularly, then there exists a constant l 2 (0, 1) such that dðxÞ 2 C2 ð@X3l Þ, and Œ d(x)Œ ” 1. Denote 8 fdðxÞ; dðxÞ < l; > > < R dðxÞ p21 2 ða1 þ 1Þp 1 dt; l 6 dðxÞ < 2l; v1 ðxÞ ¼ fl þ l f 2lt l > > R 2l p21 2 : fl þ l f 2lt ða1 þ 1Þp 1 dt; 2l 6 dðxÞ: l Obviously, 0 6 v1 ðxÞ 2 C1 ðXÞ. Consider the problem DpðxÞ wðxÞ ¼ g in X; w¼0
on @X;
ð1:4Þ
A remark on the existence of positive solutions for variable exponent elliptic systems
89
where g is a parameter. The following result plays an important role in our argument whose proof can be found in [18] or [4]. Lemma 1.3 (see [18]). If the positive parameter g is large enough and w is the unique solution of (1.4), then we have (i) For any h 2 (0, 1) there exists a positive constant C1 such that 1
C1 gpþ 1þh 6 maxwðxÞ; x2X
(ii) There exists a positive constant C2 such that 1
maxwðxÞ 6 C2 gp 1 : x2X
Proof (i) By computation, 8 pðxÞ1 ½ðrprdÞ ln f þ DdðxÞ; dðxÞ < r; f > > n io > > 2ldh 2ldp21 > < 1 2ðpðxÞ1Þ ln f rprd þ Dd l l l p 1 DpðxÞ v1 ðxÞ ¼ 2ðpðxÞ1Þ > 1 > pðxÞ1 2ld p 1 > ða1 þ 1Þ; l < dðxÞ < 2l; > 2lr > f : 0; 2l < dðxÞ: Then Œ Dp(x)v1(x)Œ 6 C*fp(x)1+h a.e. on X, for any h 2 (0,1), where C* = C*(l,h,p,X) is a positive constant depending on f. þ When C fp 1þh ¼ 12 g; we can see that v1(x) is a subsolution of (1.1). According to the comparison principle, it follows that v1(x) 6 X(x) on X. Obviously, fl 6 maxx2X v1 ðxÞ 6 2fl, there exists a positive constant C1 such that 1
maxwðxÞ P maxv1 ðxÞ P C1 gpþ 1þh : x2X
x2X
(ii) It is easy to see from Lemma 1.2 of [4]. This completes the proof.
2. EXISTENCE
h
OF SOLUTIONS
In the following, when there is no misunderstanding, we always use Ci to denote positive constants. Our main result of this paper is the following theorem. Theorem 2.1. On the conditions of (H1) (H4), then problem (1.1) has positive solution when k is large enough.
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Proof. We shall establish Theorem 2.1 by constructing a positive subsolution (/1, /2) and supersolution (z1, z2) of (1.1), such that /1 6 z1 and /2 6 z2. That is (/1, /2) and (z1, z2) satisfies (R R aðxÞ cðxÞ pðxÞ2 jr/1 j r/1 rudx 6 X kpðxÞ ½aðxÞ/1 /2 þ h1 ðxÞudx X R R dðxÞ bðxÞ qðxÞ2 jr/2 j r/2 rwdx 6 X kqðxÞ ½bðxÞ/1 /2 þ h2 ðxÞwdx; X and
(R X
R
X
jrz1 j
pðxÞ2
jrz2 j
qðxÞ2
rz1 rudx P rz2 rwdx P
R R
aðxÞ cðxÞ
X X
kpðxÞ ½aðxÞz1 z2 k
pðxÞ
dðxÞ bðxÞ ½bðxÞz1 z2
þ h1 ðxÞudx þ h2 ðxÞwdx;
1;pðxÞ 1;qðxÞ ðW0 ðXÞ; W0 ðXÞÞ
for all ðu; wÞ 2 with u,w P 0. According to the sub-supersolution method for p(x)-Laplacian equations (see [4]), then (1.1) has a positive solution.
Step 1.
We construct a subsolution of (1.1). Let r 2 (0,l) 8 kdðxÞ e 1; dðxÞ < r; > > < R dðxÞ kr 2lt p21 2 ða1 þ 1Þp 1 dt; r 6 dðxÞ < 2l; /1 ðxÞ ¼ ekr 1 þ r ke 2lr > > 2lt p21 R 2l 2 : kr e 1 þ r kekr 2lr ða1 þ 1Þp 1 dt; 2l 6 dðxÞ: 8 kdðxÞ 1; dðxÞ < r; e > > < R dðxÞ kr 2lt p21 2 ðb1 þ 1Þp 1 dt; r 6 dðxÞ < 2l; /2 ðxÞ ¼ ekr 1 þ r ke 2lr > > 2lt p21 R 2l 2 : kr e 1 þ r kekr 2lr ðb1 þ 1Þp 1 dt; 2l 6 dðxÞ:
It is easy to see that /1 ; /2 2 C1 ðXÞ. Denote inf pðxÞ 1 inf qðxÞ 1 ; ;1 ; a ¼ min 4ðsup jrpðxÞj þ 1Þ 4ðsup jrqðxÞj þ 1Þ b ¼ min fa1 þ jh1 ð0Þj; b1 þ jh2 ð0Þj; 1g By computation
DpðxÞ /1 ¼
DpðxÞ /2 ¼
8 pðxÞ1 ðpðxÞ 1Þ þ dðxÞ þ lnkk rprd þ Dd kðkekdðxÞ Þ ; dðxÞ < r; > k > > n h io > 2 > < 1 2ðpðxÞ1Þ ln kekr 2ld p 1 rprd þ Dd 2ld 2lr
p 1
2lr
2lr
> 2ðpðxÞ1Þ > kr pðxÞ1 2ld p 1 1 > ðke Þ ða1 þ 1Þ; r < dðxÞ < 2l; > > 2lr : 0; 2l < dðxÞ:
8 pðxÞ1 ; dðxÞ < r; ðpðxÞ 1Þ þ dðxÞ þ lnkk rprd þ Dd kðkekdðxÞ Þ > k > > n h io > 2 > 2ðpðxÞ1Þ kr 1 2ld 2ld p 1 < ln ke rprd þ Dd 2lr
p 1
2lr
2lr
2ðpðxÞ1Þ > > pðxÞ1 2ld p 1 1 > ðkekr Þ ðb1 þ 1Þ; r < dðxÞ < 2l; > > 2lr : 0; 2l < dðxÞ:
A remark on the existence of positive solutions for variable exponent elliptic systems
91
From (H2) and (H3), there exists a positive constant M > 2 such that dðxÞ
bðxÞ
bðxÞ/1 /2
aðxÞ
cðxÞ
þ h2 ðxÞ P 1; aðxÞ/1 /2 2X
when
þ h1 ðxÞ P 1;
8x
/1 ; /2 P M 1:
Let r ¼ k1 ln M. Then rk ¼ ln M:
ð2:1Þ
If k is sufficiently large, from (2.1), we have DpðxÞ / 6 kpðxÞ a;
ð2:2Þ
dðxÞ < r:
Let kb = ka, then kpðxÞ a P kpðxÞ b; from (2.2) and the definition of b, we have aðxÞ
cðxÞ
DpðxÞ /1 6 kpðxÞ ða1 þ 1Þ 6 kpðxÞ ðaðxÞ/1 /2
þ h1 ðxÞÞ;
dðxÞ < r:
ð2:3Þ
Since dðxÞ 2 C2 ð@X3l Þ, then there exists a positive constant C3 such that 2ðpðxÞ1Þ p 1 1 kr pðxÞ1 2l d DpðxÞ /1 6 ðke Þ ða1 þ 1Þ 2l r # " p21 ! 2ðpðxÞ 1Þ 2l d 2l d kr rprd þ Dd lnke ð2l rÞðp 1Þ 2l r 2l r 6 C3 ðkekr Þ
pðxÞ1
ða1 þ 1Þlnk; r < dðxÞ < 2l:
If k is sufficiently large, let kf = ka, we have ða1 þ 1ÞC3 ðkekr Þ
pðxÞ1
ln k ¼ ða1 þ 1ÞC3 ðkMÞ
pðxÞ1
ln k 6 kpðxÞ ða1 þ 1Þ;
ð2:4Þ
then DpðxÞ /1 6 kpðxÞ ða1 þ 1Þ;
ð2:5Þ
r < dðxÞ < 2l:
Since /1(x),/2(x) P 0 and combining (2.4) and (2.5) when k is large enough, then we have aðxÞ
cðxÞ
DpðxÞ /1 6 kpðxÞ ðaðxÞ/1 /2
þ h1 ðxÞÞ;
ð2:6Þ
r < dðxÞ < 2l:
Obviously, aðxÞ
cðxÞ
DpðxÞ /1 ¼ 0 6 kpðxÞ ða1 þ 1Þ 6 kpðxÞ ðaðxÞ/1 /2
þ h1 ðxÞÞ;
2l ð2:7Þ
< dðxÞ: Combining (2.5)–(2.7), we can conclude that aðxÞ
cðxÞ
DpðxÞ /1 6 kpðxÞ ðk1 /1 /2
þ h1 ðxÞÞ;
a:e: in X:
ð2:8Þ
Similarly, dðxÞ
bðxÞ
DqðxÞ /2 6 kqðxÞ ðbðxÞ/1 /2
þ h2 ðxÞÞ;
a:e: in X:
From (2.8) and (2.9), we can see that (/1, /2) is a subsolution of (1.1).
ð2:9Þ
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Step 2.
We construct a supersolution of (1.1). We consider 8 pþ > < DpðxÞ z1 ¼ k ða2 þ 1Þl1 in X; þ DqðxÞ z2 ¼ kq ðb2 þ 1Þl2 in X; > : z1 ¼ z 2 ¼ 0 on @X;
when l1, l2 satisfy some conditions. If we could prove that
aþ
ða2 þ 1Þl1 P aðxÞ maxz1
cþ maxz2
x2X
þ maxjh1 ðxÞj;
x2X
ð2:10Þ
x2X
and
dþ bþ ðb2 þ 1Þl2 P bðxÞ maxz1 maxz2 þ maxjh2 ðxÞj; x2X
x2X
ð2:11Þ
x2X
we would see that (z1, z2) is a supersolution for (1.1). From Lemma 1.3, we have þ p11 maxz1 ðxÞ 6 C2 kp ða2 þ 1Þl1 x2X
pþ
6 C2 k ða2 þ 1Þl1 Let
p11
and
maxz2 ðxÞ x2X
:
h iaþ h icþ 1 1 þ þ l1 ¼ 2 C2 ðkp ða2 þ 1Þl1 Þp 1 C2 ðkp ðb2 þ 1Þl2 Þq 1 :
We only need
p11 dþ h ibþ 1 þ þ l2 P 2 C2 kp ða2 þ 1Þl1 C2 ðkp ðb2 þ 1Þl2 Þq 1 ;
ð2:12Þ
when l1, l2 are large enough. Indeed, since 0 6 a+ < p 1 and 0 6 b+ < q 1, from (2.11), we can see that l2 is large enough when l1 is large enough. From (H2) and (H3), relation (2.12) is satisfied. According to (2.10) and (2.11), we can conclude that (z1, z2) is a supersolution for (1.1). It only remains to prove that /1 6 z1 and /2 6 z2. In the definition of v1(x), let 2 max/ ðxÞ þ maxjr/1 ðxÞj : f¼ l x2X 1 x2X We will claim that /1 ðxÞ 6 v1 ðxÞ;
8x 2 X:
ð2:13Þ
A remark on the existence of positive solutions for variable exponent elliptic systems
93
From the definition of v1, it is easy to see that /1 ðxÞ 6 2max/1 ðxÞ 6 v1 ðxÞ;
when dðxÞ ¼ l;
x2X
and /1 ðxÞ 6 2max/1 ðxÞ 6 v1 ðxÞ;
when dðxÞ P l:
x2X
It only remains to prove that /1 ðxÞ 6 v1 ðxÞ;
when dðxÞ < l:
Since v1 /1 2 C1 ð@Xl Þ; then there exists a point x0 2 @Xl such that v1 ðx0 Þ /1 ðx0 Þ ¼ min ½v1 ðxÞ /1 ðxÞ: x0 2@Xl
If v1(x0) /1(x0) < 0, it is easy to see that 0 < d(x) < l, and then rv1 ðx0 Þ r/1 ðx0 Þ ¼ 0: From the definition of v1, we have 2 max/ ðxÞ þ maxjr/1 ðxÞj > jr/1 ðx0 Þj: jrv1 ðx0 Þj ¼ f ¼ l x2X 1 x2X It is a contradiction to v1(x0) /1(x0) = 0. Thus (2.13) is valid. Obviously, there exists a positive constant C3 such that f 6 C3 k: Since dðxÞ 2 C2 ð@X3l Þ, according to the proof of Lemma 1.3, then there exists a positive constant C4 such that DpðxÞ v1 ðxÞ 6 C fpðxÞ1þh 6 C4 kpðxÞ1þh ; When g P k
pþ
a:e: in X; where h 2 ð0; 1Þ:
is large enough, we have
DpðxÞ v1 ðxÞ 6 g: According to the comparison principle, we have v1 ðxÞ 6 wðxÞ;
8x 2 X:
ð2:14Þ
From (2.13) and (2.14), when g P k have /1 ðxÞ 6 v1 ðxÞ 6 wðxÞ;
pþ
and the parameter k P 1 is sufficiently large, we
8x 2 X:
ð2:15Þ
According to the comparison principle, when l is large enough, we have v1 ðxÞ 6 wðxÞ 6 z1 ðxÞ;
8x 2 X:
Combining the definition of v1(x) and (2.15), it is easy to see that /1 ðxÞ 6 v1 ðxÞ 6 wðxÞ 6 z1 ðxÞ;
8x 2 X:
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G.A. Afrouzi et al.
When from 1.3, we can see that þ l P 1 and the parameter k is large enough, Lemma p pþ pþ b k ðk1 þ l1 Þl is large enough, then k ðk2 þ l2 Þh bðk ðk1 þ l1 ÞlÞ is large enough. Similarly, we have /2 6 z2. This completes the proof.
h
ACKNOWLEDGMENT The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript. REFERENCES [1] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002) 213–259. [2] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (4) (2006) 1383–1406. [3] M. Chen, On positive weak solutions for a class of quasilinear elliptic systems, Nonlinear Anal. 62 (2005) 751–756. [4] X.L. Fan, On the sub-supersolution method for p(x)-Laplacian equations, J. Math. Anal. Appl. 330 (2007) 665–682. [5] X.L. Fan, D. Zhao, On the spaces Lp(x)(X) and Wm,p(x)(X), J. Math. Anal. Appl. 263 (2001) 424–446. [6] X.L. Fan, D. Zhao, A class of De Giorgi type and Ho¨lder continuity, Nonlinear Anal. 36 (1999) 295–318. [7] X.L. Fan, D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear Anal. 39 (2000) 807–816. [8] X.L. Fan, D. Zhao, Regularity of minimizers of variational integrals with continuous p(x)-growth conditions, Chin. Ann. Math. 17A (5) (1996) 557–564. [9] X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843–1852. [10] X.L. Fan, Q.H. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005) 306–317. [11] D.D. Hai, R. Shivaji, An existence result on positive solutions of p-Laplacian systems, Nonlinear Anal. 56 (2004) 1007–1010. [12] A. El Hamidi, Existence results to elliptic systems with nonstandard growth conditions, J. Math. Anal. Appl. 300 (2004) 30–42. [13] P. Marcellini, Regularity and existence of solutions of elliptic equations with (p, q)-growth conditions, J. Differ. Equat. 90 (1991) 1–30. [14] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002. [15] Q.H. Zhang, Existence of positive solutions for a class of p(x)-Laplacian systems, J. Math. Anal. Appl. 333 (2007) 591–603. [16] Q.H. Zhang, Existence of positive solutions for elliptic systems with nonstandard p(x)-growth conditions via sub-supersolution method, Nonlinear Anal. 67 (2007) 1055–1067. [17] Q.H. Zhang, Existence of radial solutions for p(x)-Laplacian equations in RN , J. Math. Anal. Appl. 315 (2) (2006) 506–516. [18] Q.H. Zhang, Existence and asymptotic behavior of positive solutions for variable exponent elliptic systems without variational structure, Nonlinear Anal. 72 (2010) 354–363. [19] Q.H. Zhang, A strong maximum principle for differential equations with nonstandard p(x)-growth conditions, J. Math. Anal. Appl. 312 (2005) 24–32. [20] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv29 (1987) 33–36.
Arab J Math Sci 19(1) (2013), 85–94
A remark on the existence of positive solutions for variable exponent elliptic systems G.A. AFROUZI a,*, S. SHAKERI a, N.T. Chung
b
a
b
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam
Received 12 March 2012; revised 21 July 2012; accepted 26 August 2012
Available online 5 September 2012 Abstract.
In this article, we consider the system of differential equations
8 DpðxÞ u ¼ kpðxÞ ½aðxÞuaðxÞ vcðxÞ þ h1 ðxÞ in X; > > < DqðxÞ v ¼ kqðxÞ ½bðxÞudðxÞ vbðxÞ þ h2 ðxÞ in X; > > : u¼v¼0 on @X; where X RN is a bounded domain with C2-boundary @X; 1 < pðxÞ; qðxÞ 2 C1 ðXÞ are functions. The operator Dp(x) u = div(ŒuŒp(x)2u) is called the p(x)-Laplacian. When a, b, d, c satisfy some suitable conditions, we prove the existence of positive solution via sub-supersolution arguments without assuming sign conditions on the functions h1 and h2. Mathematics subject classifications: 35J25; 35J60 Keywords: Positive solutions; Variable exponent elliptic systems; Sub-supersolutions
1. INTRODUCTION
AND PRELIMINARIES
The study of differential equations and variational problems with nonstandard p(x)growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [1,2,14,20]). Many existence results have been obtained on this kind of problems, see for example [4,9,10,12,15–18]. In [6–8], Fan * Corresponding author. E-mail addresses: [email protected] (G.A. Afrouzi), [email protected] (S. Shakeri), ntchung82@yahoo. com (N.T. Chung). Peer review under responsibility of King Saud University.
Production and hosting by Elsevier 1319-5166 ª 2012 King Saud University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ajmsc.2012.08.002
86
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et al. studied the regularity of solutions for differential equations with nonstandard p(x)-growth conditions. In this paper, we mainly consider the existence of positive weak solutions for the system 8 DpðxÞ u ¼ kpðxÞ ½aðxÞuaðxÞ vcðxÞ þ h1 ðxÞ > > < DqðxÞ v ¼ kqðxÞ ½bðxÞudðxÞ vbðxÞ þ h2 ðxÞ > > : u¼v¼0
in X; in X;
ð1:1Þ
on @X;
where X RN is a bounded domain with C2-boundary @X; 1 < pðxÞ; qðxÞ 2 C1 ðXÞ are two functions. The operator Dp(x)u = div(ŒuŒp(x)2u) is called the p(x)Laplacian and the corresponding equation is called a variable exponent equation. Especially, if p(x) ” p (a constant), (1.1) is the well-known (p,q)-Laplacian system and the corresponding equation is called a constant exponent equation. We have known that the existence of solutions for p-Laplacian elliptic systems has been intensively studied in the last decades, we refer to [3,11,13]. In [11], Hai et al. considered the existence of positive weak solutions for the p-Laplacian problem 8 Dp u ¼ kfðvÞ in X; > > < ð1:2Þ Dp v ¼ kgðuÞ in X; > > : u¼v¼0 on @X; in which the first eigenfunction was used for constructing the subsolution of p-Laplacian problems. Under the condition f MðgðuÞÞ1=ðp1Þ lim ¼ 0 for all M > 0; u!þ1 up1 the authors showed that the problem (1.2) has at least one positive solution provided that k > 0 is large enough. In [3], the author studied the existence and nonexistence of positive weak solution to the following quasilinear elliptic system 8 a c in X; > < Dp u ¼ kfðu; vÞ ¼ ku v Dq v ¼ kgðu; vÞ ¼ kud vb in X; ð1:3Þ > : u¼v¼0 on @X: The first eigenfunction is used to construct the subsolution of problem (1.3), the main results are as follows: (i) If a,b P 0,c,d > 0, h = (p 1 a)(q 1 b) cd > 0, then problem (1.3) has a positive weak solution for each k > 0; (ii) If h = 0 and pc = q(p 1 a), then there exists k0 > 0 such that for 0 < k < k0, then problem (1.3) has no nontrivial nonnegative weak solution.
A remark on the existence of positive solutions for variable exponent elliptic systems
87
In recent papers [15–18], Zhang has developed problems (1.2) and (1.3) in the variable exponent Sobolev space. On the p(x)-Laplacian problems, maybe the first eigenvalue and the first eigenfunction of p(x)-Laplacian do not exist. Even if the first eigenfunction of p(x)-Laplacian exists, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used to construct the subsolution of p(x)-Laplacian problems. Motivated by the above papers, in this note, we are interested in the existence of positive solution for problem (1.1), where a, b are continuous functions in X and k is a positive parameter. Our main goal is to improve the result introduced in [18], in which a = b ” 1. To be more precise, we assume that X RN is an open bounded domain with 2 C -boundary oX and the following conditions hold: (H1) pðxÞ; qðxÞ 2 C 1 ðXÞ and 1 < p 6 p+ and 1 < q 6 q+; (H2) h1 ; h2 ; a; b; c; d 2 C 1 ðXÞ satisfy a,b P 0 on X, and c,d > 0 on X; (H3) 0 6 a+ < p 1,0 6 b+ < q 1 and - :¼(p 1 a+)(q 1 b+) d+c+ > 0; (H4) a; b : X ! ð0; 1Þ are continuous functions such that a1 ¼ minx2X aðxÞ; b1 ¼ minx2X bðxÞ; a2 ¼ maxx2X aðxÞ and b2 ¼ maxx2X bðxÞ. To study p(x)-Laplacian problems, we need some theories on the spaces Lp(x)(X), W (X) and properties of p(x)-Laplacian which we will use later (see [5]). If X RN is an open domain, write 1,p(x)
Cþ ðXÞ ¼ fh : h 2 CðXÞ; hðxÞ > 1 for x 2 Xg; h
+
= supx2Xh(x),h = infx2Xh(x), for any h 2 C(X), and Z pðxÞ LpðxÞ ðXÞ ¼ uju is a measurable real-valued function such that juj dx < 1 : X
Throughout the paper, we assume that p,q 2 C+(X) and 1 < infx2Xp(x) 6 supx2Xp(x) < N,1 < infx 2Xq(x) 6 supx2Xq(x) < N. We introduce the norm Lp(x)(X) by ( ) Z uðxÞpðxÞ jujpðxÞ ¼ inf k > 0 : k dx 6 1 ; X and (Lp(x)(X),Œ Æ Œp(x)) becomes a Banach space, we call it the generalized Lebesgue space. The space (Lp(x)(X), Œ Æ Œp(x)) is a separable, reflexive and uniform convex Banach space (see [5, Theorem 1.10, 1.14]). The space W1,p(x)(X) is defined by W1,p(x)(X) = {u 2 Lp(x)(X):ŒuŒ 2 Lp(x)(X)}, and it is equipped with the norm kuk ¼ jujpðxÞ þ jrujpðxÞ ; 1;pðxÞ
8u 2 W1;pðxÞ ðXÞ: 1;pðxÞ
1,p(x) We denote by W0 ðXÞ the closure of C1 (X). W1,p(x)(X) and W0 ðXÞ 0 ðXÞ in W are separable, reflexive and uniform convex Banach space (see [5, Theorem 2.1]). We define
88
G.A. Afrouzi et al.
ðLðuÞ; vÞ ¼
Z
pðxÞ2
jruj
rurvdx;
8u; v 2 W1;pðxÞ ðXÞ;
RN
then L:W1,p(x)(X) fi (W1,p(x)(X))* is a continuous, bounded and is a strictly monotone operator, and it is a homeomorphism [9, Theorem 3.11]. 1;pðxÞ 1;qðxÞ ðXÞ; W0 ðXÞ ; ðu; vÞ is called a weak solution of Definition 1.1. If ðu; vÞ 2 W0 (1.1) if it satisfies (R X
R
X
jrujpðxÞ2 ru rudx ¼ jrvj
qðxÞ2
rv rwdx ¼
R R
1;pðxÞ
X X
kpðxÞ ½aðxÞuaðxÞ vcðxÞ þ h1 ðxÞudx; 8u 2 W0 k
qðxÞ
½bðxÞu
dðxÞ bðxÞ
v
þ h1 ðxÞwdx;
8w 2
ðXÞ;
1;qðxÞ W0 ðXÞ:
1;pðxÞ
Define A : W1;pðxÞ ðXÞ ! ðW0 ðXÞÞ as Z pðxÞ2 jruj ruru þ hðx; uÞu dx; hAu; ui ¼
8u; u 2 W1;pðxÞ ðXÞ;
X
where h(x, u) is continuous on X R, and h(x, Æ) is increasing and satisfies jhðx; tÞj 6 C1 þ C2 jtj where
(
p ðxÞ ¼
p ðxÞ1
;
NpðxÞ ; NpðxÞ
pðxÞ < N
1;
pðxÞ P N:
It is easy to check that A is a continuous bounded mapping. Copying the proof of [19], we have the following lemma. Lemma 1.2 (Comparison principle). Let u,v 2 W1,p(x)(X) satisfy Au Av P 0 in 1;pðxÞ 1;pðxÞ ðW0 ðXÞÞ ; uðxÞ ¼ minfuðxÞ vðxÞ; 0g. If uðxÞ 2 W0 ðXÞ (i.e. u P v on oX), then u P v a.e. in X. Here and hereafter, we will use the notation d(x, oX) to denote the distance of x 2 X to the boundary of X. Denote d(x) = d(x, oX) and oXe = {x 2 XŒd(x, oX) < e}. Since oX is C2 regularly, then there exists a constant l 2 (0, 1) such that dðxÞ 2 C2 ð@X3l Þ, and Œ d(x)Œ ” 1. Denote 8 fdðxÞ; dðxÞ < l; > > < R dðxÞ p21 2 ða1 þ 1Þp 1 dt; l 6 dðxÞ < 2l; v1 ðxÞ ¼ fl þ l f 2lt l > > R 2l p21 2 : fl þ l f 2lt ða1 þ 1Þp 1 dt; 2l 6 dðxÞ: l Obviously, 0 6 v1 ðxÞ 2 C1 ðXÞ. Consider the problem DpðxÞ wðxÞ ¼ g in X; w¼0
on @X;
ð1:4Þ
A remark on the existence of positive solutions for variable exponent elliptic systems
89
where g is a parameter. The following result plays an important role in our argument whose proof can be found in [18] or [4]. Lemma 1.3 (see [18]). If the positive parameter g is large enough and w is the unique solution of (1.4), then we have (i) For any h 2 (0, 1) there exists a positive constant C1 such that 1
C1 gpþ 1þh 6 maxwðxÞ; x2X
(ii) There exists a positive constant C2 such that 1
maxwðxÞ 6 C2 gp 1 : x2X
Proof (i) By computation, 8 pðxÞ1 ½ðrprdÞ ln f þ DdðxÞ; dðxÞ < r; f > > n io > > 2ldh 2ldp21 > < 1 2ðpðxÞ1Þ ln f rprd þ Dd l l l p 1 DpðxÞ v1 ðxÞ ¼ 2ðpðxÞ1Þ > 1 > pðxÞ1 2ld p 1 > ða1 þ 1Þ; l < dðxÞ < 2l; > 2lr > f : 0; 2l < dðxÞ: Then Œ Dp(x)v1(x)Œ 6 C*fp(x)1+h a.e. on X, for any h 2 (0,1), where C* = C*(l,h,p,X) is a positive constant depending on f. þ When C fp 1þh ¼ 12 g; we can see that v1(x) is a subsolution of (1.1). According to the comparison principle, it follows that v1(x) 6 X(x) on X. Obviously, fl 6 maxx2X v1 ðxÞ 6 2fl, there exists a positive constant C1 such that 1
maxwðxÞ P maxv1 ðxÞ P C1 gpþ 1þh : x2X
x2X
(ii) It is easy to see from Lemma 1.2 of [4]. This completes the proof.
2. EXISTENCE
h
OF SOLUTIONS
In the following, when there is no misunderstanding, we always use Ci to denote positive constants. Our main result of this paper is the following theorem. Theorem 2.1. On the conditions of (H1) (H4), then problem (1.1) has positive solution when k is large enough.
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Proof. We shall establish Theorem 2.1 by constructing a positive subsolution (/1, /2) and supersolution (z1, z2) of (1.1), such that /1 6 z1 and /2 6 z2. That is (/1, /2) and (z1, z2) satisfies (R R aðxÞ cðxÞ pðxÞ2 jr/1 j r/1 rudx 6 X kpðxÞ ½aðxÞ/1 /2 þ h1 ðxÞudx X R R dðxÞ bðxÞ qðxÞ2 jr/2 j r/2 rwdx 6 X kqðxÞ ½bðxÞ/1 /2 þ h2 ðxÞwdx; X and
(R X
R
X
jrz1 j
pðxÞ2
jrz2 j
qðxÞ2
rz1 rudx P rz2 rwdx P
R R
aðxÞ cðxÞ
X X
kpðxÞ ½aðxÞz1 z2 k
pðxÞ
dðxÞ bðxÞ ½bðxÞz1 z2
þ h1 ðxÞudx þ h2 ðxÞwdx;
1;pðxÞ 1;qðxÞ ðW0 ðXÞ; W0 ðXÞÞ
for all ðu; wÞ 2 with u,w P 0. According to the sub-supersolution method for p(x)-Laplacian equations (see [4]), then (1.1) has a positive solution.
Step 1.
We construct a subsolution of (1.1). Let r 2 (0,l) 8 kdðxÞ e 1; dðxÞ < r; > > < R dðxÞ kr 2lt p21 2 ða1 þ 1Þp 1 dt; r 6 dðxÞ < 2l; /1 ðxÞ ¼ ekr 1 þ r ke 2lr > > 2lt p21 R 2l 2 : kr e 1 þ r kekr 2lr ða1 þ 1Þp 1 dt; 2l 6 dðxÞ: 8 kdðxÞ 1; dðxÞ < r; e > > < R dðxÞ kr 2lt p21 2 ðb1 þ 1Þp 1 dt; r 6 dðxÞ < 2l; /2 ðxÞ ¼ ekr 1 þ r ke 2lr > > 2lt p21 R 2l 2 : kr e 1 þ r kekr 2lr ðb1 þ 1Þp 1 dt; 2l 6 dðxÞ:
It is easy to see that /1 ; /2 2 C1 ðXÞ. Denote inf pðxÞ 1 inf qðxÞ 1 ; ;1 ; a ¼ min 4ðsup jrpðxÞj þ 1Þ 4ðsup jrqðxÞj þ 1Þ b ¼ min fa1 þ jh1 ð0Þj; b1 þ jh2 ð0Þj; 1g By computation
DpðxÞ /1 ¼
DpðxÞ /2 ¼
8 pðxÞ1 ðpðxÞ 1Þ þ dðxÞ þ lnkk rprd þ Dd kðkekdðxÞ Þ ; dðxÞ < r; > k > > n h io > 2 > < 1 2ðpðxÞ1Þ ln kekr 2ld p 1 rprd þ Dd 2ld 2lr
p 1
2lr
2lr
> 2ðpðxÞ1Þ > kr pðxÞ1 2ld p 1 1 > ðke Þ ða1 þ 1Þ; r < dðxÞ < 2l; > > 2lr : 0; 2l < dðxÞ:
8 pðxÞ1 ; dðxÞ < r; ðpðxÞ 1Þ þ dðxÞ þ lnkk rprd þ Dd kðkekdðxÞ Þ > k > > n h io > 2 > 2ðpðxÞ1Þ kr 1 2ld 2ld p 1 < ln ke rprd þ Dd 2lr
p 1
2lr
2lr
2ðpðxÞ1Þ > > pðxÞ1 2ld p 1 1 > ðkekr Þ ðb1 þ 1Þ; r < dðxÞ < 2l; > > 2lr : 0; 2l < dðxÞ:
A remark on the existence of positive solutions for variable exponent elliptic systems
91
From (H2) and (H3), there exists a positive constant M > 2 such that dðxÞ
bðxÞ
bðxÞ/1 /2
aðxÞ
cðxÞ
þ h2 ðxÞ P 1; aðxÞ/1 /2 2X
when
þ h1 ðxÞ P 1;
8x
/1 ; /2 P M 1:
Let r ¼ k1 ln M. Then rk ¼ ln M:
ð2:1Þ
If k is sufficiently large, from (2.1), we have DpðxÞ / 6 kpðxÞ a;
ð2:2Þ
dðxÞ < r:
Let kb = ka, then kpðxÞ a P kpðxÞ b; from (2.2) and the definition of b, we have aðxÞ
cðxÞ
DpðxÞ /1 6 kpðxÞ ða1 þ 1Þ 6 kpðxÞ ðaðxÞ/1 /2
þ h1 ðxÞÞ;
dðxÞ < r:
ð2:3Þ
Since dðxÞ 2 C2 ð@X3l Þ, then there exists a positive constant C3 such that 2ðpðxÞ1Þ p 1 1 kr pðxÞ1 2l d DpðxÞ /1 6 ðke Þ ða1 þ 1Þ 2l r # " p21 ! 2ðpðxÞ 1Þ 2l d 2l d kr rprd þ Dd lnke ð2l rÞðp 1Þ 2l r 2l r 6 C3 ðkekr Þ
pðxÞ1
ða1 þ 1Þlnk; r < dðxÞ < 2l:
If k is sufficiently large, let kf = ka, we have ða1 þ 1ÞC3 ðkekr Þ
pðxÞ1
ln k ¼ ða1 þ 1ÞC3 ðkMÞ
pðxÞ1
ln k 6 kpðxÞ ða1 þ 1Þ;
ð2:4Þ
then DpðxÞ /1 6 kpðxÞ ða1 þ 1Þ;
ð2:5Þ
r < dðxÞ < 2l:
Since /1(x),/2(x) P 0 and combining (2.4) and (2.5) when k is large enough, then we have aðxÞ
cðxÞ
DpðxÞ /1 6 kpðxÞ ðaðxÞ/1 /2
þ h1 ðxÞÞ;
ð2:6Þ
r < dðxÞ < 2l:
Obviously, aðxÞ
cðxÞ
DpðxÞ /1 ¼ 0 6 kpðxÞ ða1 þ 1Þ 6 kpðxÞ ðaðxÞ/1 /2
þ h1 ðxÞÞ;
2l ð2:7Þ
< dðxÞ: Combining (2.5)–(2.7), we can conclude that aðxÞ
cðxÞ
DpðxÞ /1 6 kpðxÞ ðk1 /1 /2
þ h1 ðxÞÞ;
a:e: in X:
ð2:8Þ
Similarly, dðxÞ
bðxÞ
DqðxÞ /2 6 kqðxÞ ðbðxÞ/1 /2
þ h2 ðxÞÞ;
a:e: in X:
From (2.8) and (2.9), we can see that (/1, /2) is a subsolution of (1.1).
ð2:9Þ
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G.A. Afrouzi et al.
Step 2.
We construct a supersolution of (1.1). We consider 8 pþ > < DpðxÞ z1 ¼ k ða2 þ 1Þl1 in X; þ DqðxÞ z2 ¼ kq ðb2 þ 1Þl2 in X; > : z1 ¼ z 2 ¼ 0 on @X;
when l1, l2 satisfy some conditions. If we could prove that
aþ
ða2 þ 1Þl1 P aðxÞ maxz1
cþ maxz2
x2X
þ maxjh1 ðxÞj;
x2X
ð2:10Þ
x2X
and
dþ bþ ðb2 þ 1Þl2 P bðxÞ maxz1 maxz2 þ maxjh2 ðxÞj; x2X
x2X
ð2:11Þ
x2X
we would see that (z1, z2) is a supersolution for (1.1). From Lemma 1.3, we have þ p11 maxz1 ðxÞ 6 C2 kp ða2 þ 1Þl1 x2X
pþ
6 C2 k ða2 þ 1Þl1 Let
p11
and
maxz2 ðxÞ x2X
:
h iaþ h icþ 1 1 þ þ l1 ¼ 2 C2 ðkp ða2 þ 1Þl1 Þp 1 C2 ðkp ðb2 þ 1Þl2 Þq 1 :
We only need
p11 dþ h ibþ 1 þ þ l2 P 2 C2 kp ða2 þ 1Þl1 C2 ðkp ðb2 þ 1Þl2 Þq 1 ;
ð2:12Þ
when l1, l2 are large enough. Indeed, since 0 6 a+ < p 1 and 0 6 b+ < q 1, from (2.11), we can see that l2 is large enough when l1 is large enough. From (H2) and (H3), relation (2.12) is satisfied. According to (2.10) and (2.11), we can conclude that (z1, z2) is a supersolution for (1.1). It only remains to prove that /1 6 z1 and /2 6 z2. In the definition of v1(x), let 2 max/ ðxÞ þ maxjr/1 ðxÞj : f¼ l x2X 1 x2X We will claim that /1 ðxÞ 6 v1 ðxÞ;
8x 2 X:
ð2:13Þ
A remark on the existence of positive solutions for variable exponent elliptic systems
93
From the definition of v1, it is easy to see that /1 ðxÞ 6 2max/1 ðxÞ 6 v1 ðxÞ;
when dðxÞ ¼ l;
x2X
and /1 ðxÞ 6 2max/1 ðxÞ 6 v1 ðxÞ;
when dðxÞ P l:
x2X
It only remains to prove that /1 ðxÞ 6 v1 ðxÞ;
when dðxÞ < l:
Since v1 /1 2 C1 ð@Xl Þ; then there exists a point x0 2 @Xl such that v1 ðx0 Þ /1 ðx0 Þ ¼ min ½v1 ðxÞ /1 ðxÞ: x0 2@Xl
If v1(x0) /1(x0) < 0, it is easy to see that 0 < d(x) < l, and then rv1 ðx0 Þ r/1 ðx0 Þ ¼ 0: From the definition of v1, we have 2 max/ ðxÞ þ maxjr/1 ðxÞj > jr/1 ðx0 Þj: jrv1 ðx0 Þj ¼ f ¼ l x2X 1 x2X It is a contradiction to v1(x0) /1(x0) = 0. Thus (2.13) is valid. Obviously, there exists a positive constant C3 such that f 6 C3 k: Since dðxÞ 2 C2 ð@X3l Þ, according to the proof of Lemma 1.3, then there exists a positive constant C4 such that DpðxÞ v1 ðxÞ 6 C fpðxÞ1þh 6 C4 kpðxÞ1þh ; When g P k
pþ
a:e: in X; where h 2 ð0; 1Þ:
is large enough, we have
DpðxÞ v1 ðxÞ 6 g: According to the comparison principle, we have v1 ðxÞ 6 wðxÞ;
8x 2 X:
ð2:14Þ
From (2.13) and (2.14), when g P k have /1 ðxÞ 6 v1 ðxÞ 6 wðxÞ;
pþ
and the parameter k P 1 is sufficiently large, we
8x 2 X:
ð2:15Þ
According to the comparison principle, when l is large enough, we have v1 ðxÞ 6 wðxÞ 6 z1 ðxÞ;
8x 2 X:
Combining the definition of v1(x) and (2.15), it is easy to see that /1 ðxÞ 6 v1 ðxÞ 6 wðxÞ 6 z1 ðxÞ;
8x 2 X:
94
G.A. Afrouzi et al.
When from 1.3, we can see that þ l P 1 and the parameter k is large enough, Lemma p pþ pþ b k ðk1 þ l1 Þl is large enough, then k ðk2 þ l2 Þh bðk ðk1 þ l1 ÞlÞ is large enough. Similarly, we have /2 6 z2. This completes the proof.
h
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