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On the existence of positive solutions for a class of (p(x),q(x) )-Laplacian system
Applied Mathematics Letters 26 (2013) 367–372
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On the existence of positive solutions for a class of (p(x), q(x))-Laplacian system M.B. Ghaemi a , G.A. Afrouzi b , S.H. Rasouli c,∗ , M. Choubin d a
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
b
Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran
c
Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran
d
Department of Mathematics, Faculty of Basic Sciences, Payame Noor University, Tehran, Iran
article
abstract
info
Article history: Received 16 September 2012 Accepted 14 October 2012
We consider the system
−∆p(x) u = λ1 a(x)f (v) + µ1 α(x)h(u), −∆q(x) v = λ2 b(x)g (u) + µ2 β(x)γ (v), u = 0 = v,
Keywords: Positive radial solutions p(x)-Laplacian problems Boundary value problems
in Ω , in Ω , on ∂ Ω ,
where p(x) ∈ C 1 (RN ) is a radial symmetric function such that sup |∇ p(x)| < ∞, 1 < inf p(x) ≤ sup p(x) < ∞, a, b, α, β : [0, +∞) → (0, ∞) is a continuous function and Ω = B(0, R) ⊂ RN is a bounded radial symmetric domain, and where −∆p(x) u = −div(|∇ u|p(x)−2 ∇ u) which is called the p(x)-Laplacian. We discuss the existence of positive solution via sub-super-solutions without assuming sign conditions on f (0), h(0), g (0) and γ (0). © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The study of differential equations and variational problems with nonstandard p(x)-growth conditions has been a new and interesting topic. Many results have been obtained on this kind of problems; see for example [1–7]. In [3,4] Fan and Zhao give the regularity of weak solutions for differential equations with nonstandard p(x)-growth conditions. Zhang [8] investigated the existence of positive solutions of the system
−∆p(x) u = f (v), −∆p(x) v = g (u), u = v = 0,
in Ω , in Ω , on ∂ Ω .
(1.1)
where p(x) ∈ C 1 (RN ) is a function, Ω = B(0, R) ⊂ RN is a bounded domain. The operator −∆p(x) u = −div(|∇ u|p(x)−2 ∇ u) is called p(x)-Laplacian. Especially, if p(x) is a constant p, System (1.1) is the well-known p-Laplacian system. In [9], Afrouzi and Ghorbani extended the study of [8] to the problem
−∆p(x) u = λ[a(x)h(u) + f (v)], −∆p(x) v = λ[a(x)γ (v) + g (u)], u = 0 = v,
in Ω , in Ω , on ∂ Ω .
∗
Corresponding author. E-mail addresses: [email protected] (M.B. Ghaemi), [email protected] (G.A. Afrouzi), [email protected] (S.H. Rasouli), [email protected] (M. Choubin). 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.10.007
(1.2)
368
M.B. Ghaemi et al. / Applied Mathematics Letters 26 (2013) 367–372
In this paper, we consider the existence of positive solutions of the system
−∆p(x) u = λ1 a(x)f (v) + µ1 α(x)h(u), −∆q(x) v = λ2 b(x)g (u) + µ2 β(x)γ (v), u = 0 = v,
in Ω , in Ω , on ∂ Ω
(1.3)
where p(x) ∈ C 1 (RN ) is a function, λ1 , λ2 , µ1 and µ2 are positive parameters and Ω = B(0, R) ⊂ RN is a bounded domain. Here we focus on further extending the study in [9] to system (1.3) which features multiple parameters, weight functions and stronger coupling. Our approach is based on the method of sub-supersolutions. In [10], the authors consider the existence of positive weak solutions for the problem
−∆p u = λ1 f (v) + µ1 h(u), −∆q v = λ2 g (u) + µ2 γ (v), u = 0 = v,
in Ω , in Ω , on ∂ Ω .
(1.4)
There the first eigenfunction is used for constructing the subsolution of p-Laplacian problems. Under the condition limx→+∞ f (M (g (x)))1/(q−1) /xp−1 = 0, for all M > 0, limx→0 h(x)/xp−1 = 0 and limx→0 γ (x)/xq−1 = 0, the authors show the existence of positive solutions for problem (1.4). But in the case p(x)-Laplacian, for example if Ω is bounded, then
λp =
inf
u∈W0 1,p(x) (Ω )\{0}
1
Ω p(x)
1
|∇ u|p(x) dx
Ω p(x)
|u|p(x) dx
is zero in general, and only under some special conditions λp (x) > 0 (see [11]), and maybe when the first eigenvalue and the first eigenfunction of p(x)-Laplacian are not existing. Even if the first eigenfunction of p(x)-Laplacian exists, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used for constructing the subsolution of p(x)-Laplacian problems. To study p(x)-Laplacian problems, we need some theory on the spaces Lp(x) (Ω ), W 1,p(x) (Ω ) and properties of p(x)Laplacian which we will use later (see [2]). If Ω ⊂ RN is an open domain, write C+ (Ω ) = {h : h ∈ C (Ω ), h(x) > 1 for x ∈ Ω } +
h
= supx∈Ω h(x), h− = infx∈Ω h(x), for any h ∈ C (Ω ), and Lp(x) (Ω ) = u | u is a measurable real-valued function, |u|p(x) dx < ∞ . Ω
Throughout the paper, we will assume that p ∈ C+ (Ω ) and 1 < infx∈RN p(x) ≤ supx∈RN p(x) < N. We introduce the norm on Lp(x) (Ω ) by
u(x) p(x) = inf λ > 0 : λ dx ≤ 1 , Ω
|u|p(x)
and (Lp(x) (Ω ), | · |p(x) ) becomes a Banach space, we call it generalized Lebesgue space. The space (Lp(x) (Ω ), | · |p(x) ) is a separable, reflexive and uniform convex Banach space (see [2, Theorems 1.10, 1.14]). The space W 1,p(x) (Ω ) is defined by W 1,p(x) (Ω ) = {u ∈ Lp(x) (Ω ) : |∇ u| ∈ Lp(x) (Ω )}, and it is equipped with the norm
∥u∥ = |u|p(x) + |∇ u|p(x) ,
∀u ∈ W 1,p(x) (Ω ). 1,p(x)
1,p(x)
We denote by W0 (Ω ) the closure of C0∞ (Ω ) in W 1,p(x) (Ω ). W 1,p(x) (Ω ) and W0 uniform convex Banach space (see [2, Theorem 2.1]). We define
(L(u), v) =
RN
|∇ u|p(x)−2 ∇ u∇v dx,
(Ω ) are separable, reflexive and
∀u, v ∈ W 1,p(x) (Ω ),
then L : W 1,p(x) (Ω ) → (W 1,p(x) (Ω ))∗ is a continuous, bounded and a strictly monotone operator, and it is a homeomorphism [5, Theorem 3.11]. 1,p(x) The pair (u, v) in W0 (Ω ) × W01,q(x) (Ω ), is called a weak solution of (1.3); it satisfies
p(x)−2
|∇ u| Ω
q(x)−2
|∇v| Ω
∇ u∇ξ1 dx = ∇v∇ξ2 dx =
Ω Ω
[λ1 a(x)f (v) + µ1 α(x)h(u)]ξ1 dx,
∀ξ1 ∈ W01,p(x) (Ω ),
[λ2 b(x)g (u) + µ2 β(x)γ (v)]ξ2 dx,
∀ξ2 ∈ W01,q(x) (Ω ).
We make the following assumptions: (H1) p(x) ∈ C 1 (RN ) is a radial symmetric function, i.e., p(x) = p(r ) where r = |x|, for all x ∈ RN . Further sup |∇ p(x)| < ∞ and 1 < inf p(x) ≤ sup p(x) < ∞.
M.B. Ghaemi et al. / Applied Mathematics Letters 26 (2013) 367–372
369
(H2) Ω = B(0, R) = {x ∈ RN : |x| < R} is a ball, where R > 0 is a sufficiently large constant. (H3) f , h, g , γ ∈ C 1 ([0, ∞)) are nondecreasing functions such that lim f (s) = lim h(s) = lim g (s) = lim γ (s) = +∞.
s→+∞
s→+∞
s→+∞
s→+∞
1
− f (M (g (s)) q −1 ) 0 M − sp −1 γ (s) h(s) lims→+∞ p− −1 0 lims→+∞ q− −1 s s
= ,∀
(H4) lims→+∞ (H5)
> 0.
= ,
= 0.
(H6) a, b, α, β ∈ C ([0, +∞)) are positive functions. We shall establish the following result. Theorem 1.1. If (H1)–(H6) hold, then (1.3) has a positive solution for λ1 + µ1 ≫ 1 and λ2 + µ2 ≫ 1. Remark 1.2. Theorem 1.1 was established in [10, Theorem A] for the case p(x) ≡ p, q(x) ≡ q (p and q are constant) and weight functions equivalent to 1. Remark 1.3. Theorem 1.1 was established in [8, Theorem 2.3] for the case p(x) ≡ q(x), λ1 = λ2 , µ1 = µ2 = 0 and a(x) ≡ b(x) ≡ 1. Remark 1.4. Theorem 1.1 was established in [9, Theorem 1.2] for the case p(x) ≡ q(x), λ1 = λ2 = µ1 = µ2 , a(x) ≡ b(x) ≡ 1 and α(x) ≡ β(x). Example 1.5. Let
−∆p(x) u = λ1 a(x)(v − c1 )s1 + µ1 α(x)(u − c2 )s2 , −∆q(x) v = λ2 b(x)(u − c3 )s3 + µ2 β(x)(v − c4 )s4 , u = 0 = v,
in Ω , in Ω , on ∂ Ω ,
(1.5)
where s1 , s2 , s3 , s4 , c1 , c2 , c3 , c4 ≥ 0, s1 s3 < (p− − 1) × (q− − 1), s2 < (p− − 1) and s4 < (q− − 1). Then by Theorem 1.1, (1.5) has a positive solution provided λ1 + µ1 and λ2 + µ2 are large. 2. Proof of Theorem 1.1 Proof. We establish this theorem by constructing a positive subsolution (ψ1 , ψ2 ) and supersolution (z1 , z2 ) of (1.3), such that ψ1 ≤ z1 and ψ2 ≤ z2 . That is (ψ1 , ψ2 ) and (z1 , z2 ) satisfy
Ω
|∇ψ1 |p(x)−2 ∇ψ1 .∇ξ1 dx ≤
Ω
[λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 )]ξ1 dx,
|∇ψ2 |q(x)−2 ∇ψ2 .∇ξ2 dx ≤ [λ2 b(x)g (ψ1 ) + µ2 β(x)γ (ψ2 )]ξ2 dx, Ω |∇ z1 |p(x)−2 ∇ z1 .∇ξ1 dx ≥ [λ1 a(x)f (z2 ) + µ1 α(x)h(z1 )]ξ1 dx, Ω Ω q(x)−2 |∇ z2 | ∇ z2 .∇ξ2 dx ≥ [λ2 b(x)g (z1 ) + µ2 β(x)γ (z2 )]ξ2 dx Ω
Ω
Ω
1,p(x) W0
1,q(x) W0
for all ξ1 ∈ (Ω ) and ξ2 ∈ (Ω ) with ξ1 , ξ2 ≥ 0. Then (1.3) has a positive solution. We construct a subsolution of (1.3). Denote
η=
inf p(x) − 1 4(sup |∇ p(x)| + 1)
,
R0 =
R−η 2
,
and k0 > 0 be such that f (s), h(s), g (s), γ (s) > −k0 for all s ≥ 0. Let k1 := k0 l1 (λ1 + µ1 )/η and k2 := k0 l2 (λ2 + µ2 )/η, where l1 = min{minx∈Ω a(x), minx∈Ω α(x)} and l2 = min{minx∈Ω b(x), minx∈Ω β(x)}. Let
ψ1 (r ) =
−k1 (r −R) e − 1, 2R0 < r ≤ R, 1 2R0 p(2R )−1 (2R0 )N −1 π p(r )−1 ηk1 p(r 0)−1 eηk1 − 1 + (k1 e ) sin ε(r − 2R0 ) + dr , N −1 r
r
η k1 e − 1 +
2R0 2R0 − 2πε
(k1 eηk1 )
p(2R0 )−1 p(r )−1
2
(2R0 )
N −1
r N −1
2R0 −
1 π p(r )−1 sin ε0 (r − 2R0 ) + dr , 2
π < r ≤ 2R0 , 2ε
r ≤ 2R0 −
π 2ε
,
370
M.B. Ghaemi et al. / Applied Mathematics Letters 26 (2013) 367–372
and
ψ2 (r ) =
−k2 (r −R) e − 1, 2R0 < r ≤ R, 1 2R0 q(2R )−1 (2R0 )N −1 π q(r )−1 ηk2 q(r 0)−1 eηk2 − 1 + (k2 e ) sin ε(r − 2R0 ) + dr , N −1 r
r
ηk2 − 1 + e
2R0
2R0 − 2πε
(k2 eηk2 )
q(2R0 )−1 q(r )−1
2R0 −
2
(2R0 )N −1 r N −1
sin ε0 (r − 2R0 ) +
1 π q(r )−1 2
dr ,
π 2ε
< r ≤ 2R0 ,
r ≤ 2R0 −
π 2ε
,
where R0 is sufficiently large, ε is a small positive constant which satisfies R0 ≤ 2R0 − 2πε , In the following, we will prove that (ψ1 , ψ2 ) is a subsolution of (1.3). Since
−k1 e−k1 (r −R) , 1 p(2R )−1 (2R0 )N −1 π p(r )−1 ηk1 p(r 0)−1 ′ sin ε(r − 2R0 ) + dr , ψ1 (r ) = −(k1 e ) r N −1 2 0,
2R0 < r ≤ R, 2R0 −
π 2ε
< r ≤ 2R0 ,
0 ≤ r ≤ 2R0 −
π 2ε
,
¯ ). Let r = |x|. By computation, it is easy to see that ψ1 ≥ 0 is decreasing and ψ1 ∈ C 1 ([0, R]), ψ1 (x) = ψ1 (|x|) ∈ C 1 (Ω −∆p(x) ψ1 = −div(|∇ψ1 (x)|p(x)−2 ∇ψ1 (x)) = −(r N −1 |ψ1 ′ (r )|p(r )−2 ψ1 ′ (r ))′ /r N −1 . Then
N − 1 − k ( r − R ) p ( r )− 1 ′ ′ , 2R0 < r ≤ R, (k1 e 1 ) −k1 (p(r ) − 1) + p (r ) ln k1 − k1 p (r )(r − R) + r N −1 −∆p(x) ψ1 = π π 2R0 (k1 eηk1 )(p(2R0 )−1) cos ε(r − 2R0 ) + , 2R0 − < r ≤ 2R0 , ε r 2 2ε 0, 0 ≤ r ≤ 2R − π , 0 2ε If λ1 + µ1 ≫ 1 (and so k1 is sufficiently large), when 2R0 < r ≤ R, then
ln k1 N −1 −∆p(x) ψ1 ≤ −k1 inf p(x) − 1 − sup |∇ p(x)| +R−r + k1
k1 r
≤ −k1 η = −k0 l1 (λ1 + µ1 ). Since f (ψ2 ), h(ψ1 ) ≥ −k0 , this implies
− ∆p(x) ψ1 ≤ λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 ),
2R0 < |x| ≤ R.
(2.1)
For λ1 + µ1 (and so k1 ) large enough we have f (eηk1 − 1) ≥ 1,
h(eηk1 − 1) ≥ 1,
g (eηk1 − 1) ≥ 1,
γ (eηk1 − 1) ≥ 1.
When 2R0 − 2πε < |x| < 2R0 , since
−∆p(x) ψ1 = ε
2R0
N −1
r
2
p+ ηk1 p+
≤ ε 2 k1 e N
η
for ε = ( k )2−N k1 0
π (k1 eηk1 )(p(2R0 )−1) cos ε(r − 2R0 ) +
−p+ +1 −ηk1 p+
e
,
, we have
− ∆p(x) ψ1 ≤ l1 (λ1 + µ1 ) ≤ λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 ).
(2.2)
Obviously, when |x| < 2R0 − 2πε , then
− ∆p(x) ψ1 = 0 ≤ l1 (λ1 + µ1 ) ≤ λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 ). Since ψ1 (x) ∈ C (Ω ), combining (2.1), (2.2), (2.3), we have 1
−∆p(x) ψ1 ≤ λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 ),
(2.3)
M.B. Ghaemi et al. / Applied Mathematics Letters 26 (2013) 367–372
371
for a.e. x ∈ Ω . Similarly for a.e. x ∈ Ω and λ2 + µ2 ≫ 1 we have
−∆q(x) ψ2 ≤ λ2 b(x)g (ψ1 ) + µ2 β(x)γ (ψ2 ). Thus (ψ1 , ψ2 ) is a subsolution of (1.3). Next we construct a supersolution of (1.3). Let e1 and e2 be radial solutions of
−∆p(x) e1 (x) = 1, in Ω , −∆q(x) e2 (x) = 1, in Ω , e1 = 0 = e2 on ∂ Ω . We denote ei = ei (r ) = ei (|x|) for i = 1, 2, then
−(r N −1 |e′1 |p(r )−2 e′1 )′ = r N −1 ,
e1 (R) = 0,
e′1 (0) = 0,
−(r N −1 |e′2 |q(r )−2 e′2 )′ = r N −1 ,
e2 (R) = 0,
e′2 (0) = 0.
Then
r 1 p(r )−1 ,
e′1 = −
r 1 q(r )−1 ,
e′2 = −
N
N
and R
e1 =
r 1 p(r )−1 dr ,
e2 =
N
r
R
r 1 q(r )−1 dr . N
r
We denote e∗i = max0≤r ≤R ei (r ) for i = 1, 2, and let
1 1 (z1 , z2 ) = Ce1 , (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 g (Ce∗1 ) q− −1 e2 . By (H4)–(H5) we can choose C large enough so that 1
1
− C p −1 ≥ λ1 ∥a∥∞ f (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce∗1 )] q− −1 e∗2 + µ1 ∥α∥∞ h(Ce∗1 ).
(2.4)
1 1 g (Ce∗1 ) ≥ γ (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce∗1 )] q− −1 e∗2 .
(2.5)
Hence from (2.4) we have
p−2
Ω
|∇ z1 |
∇ z1 .∇ξ1 dx =
Ω
C p(r )−1 ξ1 dx
− −1
≥ Cp
Ω
≥ Ω
≥ Ω
ξ1 dx 1
1
λ1 ∥a∥∞ f (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce1 )] q− −1 e2 + µ1 ∥α∥∞ h(Ce1 ) ξ1 dx ∗
∗
∗
[λ1 a(x)f (z2 ) + µ1 α(x)h(z1 )]ξ1 dx,
for ξ1 ∈ W 1,p(x) (Ω ) with ξ1 ≥ 0. Also for ξ2 ∈ W 1,q(x) (Ω ) with ξ2 ≥ 0 from (2.5) we have
q(x)−2
Ω
|∇ z2 |
q(r )−1
∇ z2 · ∇ξ2 dx =
Ω
q(r )−1
(λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce∗1 )] q− −1 ξ2 dx
(λ2 ∥b∥∞ + µ2 ∥β∥∞ )g (Ce∗1 )ξ2 dx 1 1 ≥ λ2 b(x)g (Ce∗1 ) + µ2 β(x)γ (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce∗1 )] q− −1 e∗2 ξ2 dx
≥
Ω
Ω
≥ Ω
[λ2 b(x)g (z1 ) + µ2 β(x)γ (z2 )]ξ2 dx.
Hence (z1 , z2 ) is a supersolution of (1.3) such that zi ≥ ψi for C large, i = 1, 2. This completes the proof.
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M.B. Ghaemi et al. / Applied Mathematics Letters 26 (2013) 367–372
References [1] [2] [3] [4] [5] [6] [7] [8]
X.L. Fan, H.Q. Wu, F.Z. Wang, Hartman-type results for p(t)-Laplacian systems, Nonlinear Anal. 52 (2003) 585–594. X.L. Fan, D. Zhao, On the spaces Lp(x) (Ω ) and W m,p(x) (Ω ), J. Math. Anal. Appl. 263 (2001) 424–446. X.L. Fan, D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal. TMA 36 (1999) 295–318. X.L. Fan, D. Zhao, The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear Anal. TMA 39 (2000) 807–816. X.L. Fan, Q.H. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003) 1843–1852. X.L. Fan, Q.H. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005) 306–317. V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987) 33–36. 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. [9] G.A. Afrouzi, H. Ghorbani, Existence of positive solutions for p(x)-Laplacian problems, Electron. J. Differential Equations 177 (2007) 1–9. [10] Jaffar Ali, R. Shivaji, Positive solutions for a class of p-Laplacian systems with multiple parameters, J. Math. Anal. Appl. 335 (2007) 1013–1019. [11] X.L. Fan, Q.H. Zhang, Existence of positive solutions for a class of p(x)-Laplacian systems, J. Math. Anal. Appl. 302 (2005) 306–317.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
On the existence of positive solutions for a class of (p(x), q(x))-Laplacian system M.B. Ghaemi a , G.A. Afrouzi b , S.H. Rasouli c,∗ , M. Choubin d a
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
b
Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran
c
Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran
d
Department of Mathematics, Faculty of Basic Sciences, Payame Noor University, Tehran, Iran
article
abstract
info
Article history: Received 16 September 2012 Accepted 14 October 2012
We consider the system
−∆p(x) u = λ1 a(x)f (v) + µ1 α(x)h(u), −∆q(x) v = λ2 b(x)g (u) + µ2 β(x)γ (v), u = 0 = v,
Keywords: Positive radial solutions p(x)-Laplacian problems Boundary value problems
in Ω , in Ω , on ∂ Ω ,
where p(x) ∈ C 1 (RN ) is a radial symmetric function such that sup |∇ p(x)| < ∞, 1 < inf p(x) ≤ sup p(x) < ∞, a, b, α, β : [0, +∞) → (0, ∞) is a continuous function and Ω = B(0, R) ⊂ RN is a bounded radial symmetric domain, and where −∆p(x) u = −div(|∇ u|p(x)−2 ∇ u) which is called the p(x)-Laplacian. We discuss the existence of positive solution via sub-super-solutions without assuming sign conditions on f (0), h(0), g (0) and γ (0). © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The study of differential equations and variational problems with nonstandard p(x)-growth conditions has been a new and interesting topic. Many results have been obtained on this kind of problems; see for example [1–7]. In [3,4] Fan and Zhao give the regularity of weak solutions for differential equations with nonstandard p(x)-growth conditions. Zhang [8] investigated the existence of positive solutions of the system
−∆p(x) u = f (v), −∆p(x) v = g (u), u = v = 0,
in Ω , in Ω , on ∂ Ω .
(1.1)
where p(x) ∈ C 1 (RN ) is a function, Ω = B(0, R) ⊂ RN is a bounded domain. The operator −∆p(x) u = −div(|∇ u|p(x)−2 ∇ u) is called p(x)-Laplacian. Especially, if p(x) is a constant p, System (1.1) is the well-known p-Laplacian system. In [9], Afrouzi and Ghorbani extended the study of [8] to the problem
−∆p(x) u = λ[a(x)h(u) + f (v)], −∆p(x) v = λ[a(x)γ (v) + g (u)], u = 0 = v,
in Ω , in Ω , on ∂ Ω .
∗
Corresponding author. E-mail addresses: [email protected] (M.B. Ghaemi), [email protected] (G.A. Afrouzi), [email protected] (S.H. Rasouli), [email protected] (M. Choubin). 0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.10.007
(1.2)
368
M.B. Ghaemi et al. / Applied Mathematics Letters 26 (2013) 367–372
In this paper, we consider the existence of positive solutions of the system
−∆p(x) u = λ1 a(x)f (v) + µ1 α(x)h(u), −∆q(x) v = λ2 b(x)g (u) + µ2 β(x)γ (v), u = 0 = v,
in Ω , in Ω , on ∂ Ω
(1.3)
where p(x) ∈ C 1 (RN ) is a function, λ1 , λ2 , µ1 and µ2 are positive parameters and Ω = B(0, R) ⊂ RN is a bounded domain. Here we focus on further extending the study in [9] to system (1.3) which features multiple parameters, weight functions and stronger coupling. Our approach is based on the method of sub-supersolutions. In [10], the authors consider the existence of positive weak solutions for the problem
−∆p u = λ1 f (v) + µ1 h(u), −∆q v = λ2 g (u) + µ2 γ (v), u = 0 = v,
in Ω , in Ω , on ∂ Ω .
(1.4)
There the first eigenfunction is used for constructing the subsolution of p-Laplacian problems. Under the condition limx→+∞ f (M (g (x)))1/(q−1) /xp−1 = 0, for all M > 0, limx→0 h(x)/xp−1 = 0 and limx→0 γ (x)/xq−1 = 0, the authors show the existence of positive solutions for problem (1.4). But in the case p(x)-Laplacian, for example if Ω is bounded, then
λp =
inf
u∈W0 1,p(x) (Ω )\{0}
1
Ω p(x)
1
|∇ u|p(x) dx
Ω p(x)
|u|p(x) dx
is zero in general, and only under some special conditions λp (x) > 0 (see [11]), and maybe when the first eigenvalue and the first eigenfunction of p(x)-Laplacian are not existing. Even if the first eigenfunction of p(x)-Laplacian exists, because of the nonhomogeneity of p(x)-Laplacian, the first eigenfunction cannot be used for constructing the subsolution of p(x)-Laplacian problems. To study p(x)-Laplacian problems, we need some theory on the spaces Lp(x) (Ω ), W 1,p(x) (Ω ) and properties of p(x)Laplacian which we will use later (see [2]). If Ω ⊂ RN is an open domain, write C+ (Ω ) = {h : h ∈ C (Ω ), h(x) > 1 for x ∈ Ω } +
h
= supx∈Ω h(x), h− = infx∈Ω h(x), for any h ∈ C (Ω ), and Lp(x) (Ω ) = u | u is a measurable real-valued function, |u|p(x) dx < ∞ . Ω
Throughout the paper, we will assume that p ∈ C+ (Ω ) and 1 < infx∈RN p(x) ≤ supx∈RN p(x) < N. We introduce the norm on Lp(x) (Ω ) by
u(x) p(x) = inf λ > 0 : λ dx ≤ 1 , Ω
|u|p(x)
and (Lp(x) (Ω ), | · |p(x) ) becomes a Banach space, we call it generalized Lebesgue space. The space (Lp(x) (Ω ), | · |p(x) ) is a separable, reflexive and uniform convex Banach space (see [2, Theorems 1.10, 1.14]). The space W 1,p(x) (Ω ) is defined by W 1,p(x) (Ω ) = {u ∈ Lp(x) (Ω ) : |∇ u| ∈ Lp(x) (Ω )}, and it is equipped with the norm
∥u∥ = |u|p(x) + |∇ u|p(x) ,
∀u ∈ W 1,p(x) (Ω ). 1,p(x)
1,p(x)
We denote by W0 (Ω ) the closure of C0∞ (Ω ) in W 1,p(x) (Ω ). W 1,p(x) (Ω ) and W0 uniform convex Banach space (see [2, Theorem 2.1]). We define
(L(u), v) =
RN
|∇ u|p(x)−2 ∇ u∇v dx,
(Ω ) are separable, reflexive and
∀u, v ∈ W 1,p(x) (Ω ),
then L : W 1,p(x) (Ω ) → (W 1,p(x) (Ω ))∗ is a continuous, bounded and a strictly monotone operator, and it is a homeomorphism [5, Theorem 3.11]. 1,p(x) The pair (u, v) in W0 (Ω ) × W01,q(x) (Ω ), is called a weak solution of (1.3); it satisfies
p(x)−2
|∇ u| Ω
q(x)−2
|∇v| Ω
∇ u∇ξ1 dx = ∇v∇ξ2 dx =
Ω Ω
[λ1 a(x)f (v) + µ1 α(x)h(u)]ξ1 dx,
∀ξ1 ∈ W01,p(x) (Ω ),
[λ2 b(x)g (u) + µ2 β(x)γ (v)]ξ2 dx,
∀ξ2 ∈ W01,q(x) (Ω ).
We make the following assumptions: (H1) p(x) ∈ C 1 (RN ) is a radial symmetric function, i.e., p(x) = p(r ) where r = |x|, for all x ∈ RN . Further sup |∇ p(x)| < ∞ and 1 < inf p(x) ≤ sup p(x) < ∞.
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369
(H2) Ω = B(0, R) = {x ∈ RN : |x| < R} is a ball, where R > 0 is a sufficiently large constant. (H3) f , h, g , γ ∈ C 1 ([0, ∞)) are nondecreasing functions such that lim f (s) = lim h(s) = lim g (s) = lim γ (s) = +∞.
s→+∞
s→+∞
s→+∞
s→+∞
1
− f (M (g (s)) q −1 ) 0 M − sp −1 γ (s) h(s) lims→+∞ p− −1 0 lims→+∞ q− −1 s s
= ,∀
(H4) lims→+∞ (H5)
> 0.
= ,
= 0.
(H6) a, b, α, β ∈ C ([0, +∞)) are positive functions. We shall establish the following result. Theorem 1.1. If (H1)–(H6) hold, then (1.3) has a positive solution for λ1 + µ1 ≫ 1 and λ2 + µ2 ≫ 1. Remark 1.2. Theorem 1.1 was established in [10, Theorem A] for the case p(x) ≡ p, q(x) ≡ q (p and q are constant) and weight functions equivalent to 1. Remark 1.3. Theorem 1.1 was established in [8, Theorem 2.3] for the case p(x) ≡ q(x), λ1 = λ2 , µ1 = µ2 = 0 and a(x) ≡ b(x) ≡ 1. Remark 1.4. Theorem 1.1 was established in [9, Theorem 1.2] for the case p(x) ≡ q(x), λ1 = λ2 = µ1 = µ2 , a(x) ≡ b(x) ≡ 1 and α(x) ≡ β(x). Example 1.5. Let
−∆p(x) u = λ1 a(x)(v − c1 )s1 + µ1 α(x)(u − c2 )s2 , −∆q(x) v = λ2 b(x)(u − c3 )s3 + µ2 β(x)(v − c4 )s4 , u = 0 = v,
in Ω , in Ω , on ∂ Ω ,
(1.5)
where s1 , s2 , s3 , s4 , c1 , c2 , c3 , c4 ≥ 0, s1 s3 < (p− − 1) × (q− − 1), s2 < (p− − 1) and s4 < (q− − 1). Then by Theorem 1.1, (1.5) has a positive solution provided λ1 + µ1 and λ2 + µ2 are large. 2. Proof of Theorem 1.1 Proof. We establish this theorem by constructing a positive subsolution (ψ1 , ψ2 ) and supersolution (z1 , z2 ) of (1.3), such that ψ1 ≤ z1 and ψ2 ≤ z2 . That is (ψ1 , ψ2 ) and (z1 , z2 ) satisfy
Ω
|∇ψ1 |p(x)−2 ∇ψ1 .∇ξ1 dx ≤
Ω
[λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 )]ξ1 dx,
|∇ψ2 |q(x)−2 ∇ψ2 .∇ξ2 dx ≤ [λ2 b(x)g (ψ1 ) + µ2 β(x)γ (ψ2 )]ξ2 dx, Ω |∇ z1 |p(x)−2 ∇ z1 .∇ξ1 dx ≥ [λ1 a(x)f (z2 ) + µ1 α(x)h(z1 )]ξ1 dx, Ω Ω q(x)−2 |∇ z2 | ∇ z2 .∇ξ2 dx ≥ [λ2 b(x)g (z1 ) + µ2 β(x)γ (z2 )]ξ2 dx Ω
Ω
Ω
1,p(x) W0
1,q(x) W0
for all ξ1 ∈ (Ω ) and ξ2 ∈ (Ω ) with ξ1 , ξ2 ≥ 0. Then (1.3) has a positive solution. We construct a subsolution of (1.3). Denote
η=
inf p(x) − 1 4(sup |∇ p(x)| + 1)
,
R0 =
R−η 2
,
and k0 > 0 be such that f (s), h(s), g (s), γ (s) > −k0 for all s ≥ 0. Let k1 := k0 l1 (λ1 + µ1 )/η and k2 := k0 l2 (λ2 + µ2 )/η, where l1 = min{minx∈Ω a(x), minx∈Ω α(x)} and l2 = min{minx∈Ω b(x), minx∈Ω β(x)}. Let
ψ1 (r ) =
−k1 (r −R) e − 1, 2R0 < r ≤ R, 1 2R0 p(2R )−1 (2R0 )N −1 π p(r )−1 ηk1 p(r 0)−1 eηk1 − 1 + (k1 e ) sin ε(r − 2R0 ) + dr , N −1 r
r
η k1 e − 1 +
2R0 2R0 − 2πε
(k1 eηk1 )
p(2R0 )−1 p(r )−1
2
(2R0 )
N −1
r N −1
2R0 −
1 π p(r )−1 sin ε0 (r − 2R0 ) + dr , 2
π < r ≤ 2R0 , 2ε
r ≤ 2R0 −
π 2ε
,
370
M.B. Ghaemi et al. / Applied Mathematics Letters 26 (2013) 367–372
and
ψ2 (r ) =
−k2 (r −R) e − 1, 2R0 < r ≤ R, 1 2R0 q(2R )−1 (2R0 )N −1 π q(r )−1 ηk2 q(r 0)−1 eηk2 − 1 + (k2 e ) sin ε(r − 2R0 ) + dr , N −1 r
r
ηk2 − 1 + e
2R0
2R0 − 2πε
(k2 eηk2 )
q(2R0 )−1 q(r )−1
2R0 −
2
(2R0 )N −1 r N −1
sin ε0 (r − 2R0 ) +
1 π q(r )−1 2
dr ,
π 2ε
< r ≤ 2R0 ,
r ≤ 2R0 −
π 2ε
,
where R0 is sufficiently large, ε is a small positive constant which satisfies R0 ≤ 2R0 − 2πε , In the following, we will prove that (ψ1 , ψ2 ) is a subsolution of (1.3). Since
−k1 e−k1 (r −R) , 1 p(2R )−1 (2R0 )N −1 π p(r )−1 ηk1 p(r 0)−1 ′ sin ε(r − 2R0 ) + dr , ψ1 (r ) = −(k1 e ) r N −1 2 0,
2R0 < r ≤ R, 2R0 −
π 2ε
< r ≤ 2R0 ,
0 ≤ r ≤ 2R0 −
π 2ε
,
¯ ). Let r = |x|. By computation, it is easy to see that ψ1 ≥ 0 is decreasing and ψ1 ∈ C 1 ([0, R]), ψ1 (x) = ψ1 (|x|) ∈ C 1 (Ω −∆p(x) ψ1 = −div(|∇ψ1 (x)|p(x)−2 ∇ψ1 (x)) = −(r N −1 |ψ1 ′ (r )|p(r )−2 ψ1 ′ (r ))′ /r N −1 . Then
N − 1 − k ( r − R ) p ( r )− 1 ′ ′ , 2R0 < r ≤ R, (k1 e 1 ) −k1 (p(r ) − 1) + p (r ) ln k1 − k1 p (r )(r − R) + r N −1 −∆p(x) ψ1 = π π 2R0 (k1 eηk1 )(p(2R0 )−1) cos ε(r − 2R0 ) + , 2R0 − < r ≤ 2R0 , ε r 2 2ε 0, 0 ≤ r ≤ 2R − π , 0 2ε If λ1 + µ1 ≫ 1 (and so k1 is sufficiently large), when 2R0 < r ≤ R, then
ln k1 N −1 −∆p(x) ψ1 ≤ −k1 inf p(x) − 1 − sup |∇ p(x)| +R−r + k1
k1 r
≤ −k1 η = −k0 l1 (λ1 + µ1 ). Since f (ψ2 ), h(ψ1 ) ≥ −k0 , this implies
− ∆p(x) ψ1 ≤ λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 ),
2R0 < |x| ≤ R.
(2.1)
For λ1 + µ1 (and so k1 ) large enough we have f (eηk1 − 1) ≥ 1,
h(eηk1 − 1) ≥ 1,
g (eηk1 − 1) ≥ 1,
γ (eηk1 − 1) ≥ 1.
When 2R0 − 2πε < |x| < 2R0 , since
−∆p(x) ψ1 = ε
2R0
N −1
r
2
p+ ηk1 p+
≤ ε 2 k1 e N
η
for ε = ( k )2−N k1 0
π (k1 eηk1 )(p(2R0 )−1) cos ε(r − 2R0 ) +
−p+ +1 −ηk1 p+
e
,
, we have
− ∆p(x) ψ1 ≤ l1 (λ1 + µ1 ) ≤ λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 ).
(2.2)
Obviously, when |x| < 2R0 − 2πε , then
− ∆p(x) ψ1 = 0 ≤ l1 (λ1 + µ1 ) ≤ λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 ). Since ψ1 (x) ∈ C (Ω ), combining (2.1), (2.2), (2.3), we have 1
−∆p(x) ψ1 ≤ λ1 a(x)f (ψ2 ) + µ1 α(x)h(ψ1 ),
(2.3)
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371
for a.e. x ∈ Ω . Similarly for a.e. x ∈ Ω and λ2 + µ2 ≫ 1 we have
−∆q(x) ψ2 ≤ λ2 b(x)g (ψ1 ) + µ2 β(x)γ (ψ2 ). Thus (ψ1 , ψ2 ) is a subsolution of (1.3). Next we construct a supersolution of (1.3). Let e1 and e2 be radial solutions of
−∆p(x) e1 (x) = 1, in Ω , −∆q(x) e2 (x) = 1, in Ω , e1 = 0 = e2 on ∂ Ω . We denote ei = ei (r ) = ei (|x|) for i = 1, 2, then
−(r N −1 |e′1 |p(r )−2 e′1 )′ = r N −1 ,
e1 (R) = 0,
e′1 (0) = 0,
−(r N −1 |e′2 |q(r )−2 e′2 )′ = r N −1 ,
e2 (R) = 0,
e′2 (0) = 0.
Then
r 1 p(r )−1 ,
e′1 = −
r 1 q(r )−1 ,
e′2 = −
N
N
and R
e1 =
r 1 p(r )−1 dr ,
e2 =
N
r
R
r 1 q(r )−1 dr . N
r
We denote e∗i = max0≤r ≤R ei (r ) for i = 1, 2, and let
1 1 (z1 , z2 ) = Ce1 , (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 g (Ce∗1 ) q− −1 e2 . By (H4)–(H5) we can choose C large enough so that 1
1
− C p −1 ≥ λ1 ∥a∥∞ f (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce∗1 )] q− −1 e∗2 + µ1 ∥α∥∞ h(Ce∗1 ).
(2.4)
1 1 g (Ce∗1 ) ≥ γ (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce∗1 )] q− −1 e∗2 .
(2.5)
Hence from (2.4) we have
p−2
Ω
|∇ z1 |
∇ z1 .∇ξ1 dx =
Ω
C p(r )−1 ξ1 dx
− −1
≥ Cp
Ω
≥ Ω
≥ Ω
ξ1 dx 1
1
λ1 ∥a∥∞ f (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce1 )] q− −1 e2 + µ1 ∥α∥∞ h(Ce1 ) ξ1 dx ∗
∗
∗
[λ1 a(x)f (z2 ) + µ1 α(x)h(z1 )]ξ1 dx,
for ξ1 ∈ W 1,p(x) (Ω ) with ξ1 ≥ 0. Also for ξ2 ∈ W 1,q(x) (Ω ) with ξ2 ≥ 0 from (2.5) we have
q(x)−2
Ω
|∇ z2 |
q(r )−1
∇ z2 · ∇ξ2 dx =
Ω
q(r )−1
(λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce∗1 )] q− −1 ξ2 dx
(λ2 ∥b∥∞ + µ2 ∥β∥∞ )g (Ce∗1 )ξ2 dx 1 1 ≥ λ2 b(x)g (Ce∗1 ) + µ2 β(x)γ (λ2 ∥b∥∞ + µ2 ∥β∥∞ ) q− −1 [g (Ce∗1 )] q− −1 e∗2 ξ2 dx
≥
Ω
Ω
≥ Ω
[λ2 b(x)g (z1 ) + µ2 β(x)γ (z2 )]ξ2 dx.
Hence (z1 , z2 ) is a supersolution of (1.3) such that zi ≥ ψi for C large, i = 1, 2. This completes the proof.
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