Electronic Journal of Differential Equations, Vol. 2011 (2011), No. 137, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR NONLOCAL ELLIPTIC
SYSTEMS WITH NONSTANDARD GROWTH CONDITIONS
GUOWEI DAI
Abstract. This article concerns the existence and multiplicity of solutions for
a p(x)-Kirchhoff-type systems with Dirichlet boundary condition. By a direct
variational approach and the theory of the variable exponent Sobolev spaces,
under growth conditions on the reaction terms, we establish the existence and
multiplicity of solutions.
1. Introduction
In this article, we study the following nonlocal elliptic systems of gradient type
with nonstandard growth conditions
Z
∂F
1
−M1
|∇u|p(x) dx div |∇u|p(x)−2 ∇u =
(x, u, v) in Ω,
∂u
Ω p(x)
Z
∂F
1
(1.1)
−M2
|∇v|q(x) dx div |∇v|q(x)−2 ∇v =
(x, u, v) in Ω,
q(x)
∂v
Ω
u = 0, v = 0 on ∂Ω,
where Ω is a bounded domain in RN with a smooth boundary ∂Ω, p(x), q(x) ∈
C+ (Ω) with
1 < p− := min p(x) ≤ p+ := max p(x) < +∞,
Ω
Ω
1 < q − := min q(x) ≤ q + := max q(x) < +∞,
Ω
Ω
M1 (t), M2 (t) are continuous functions. We confine ourselves to the case where
M1 = M2 for simplicity. Notice that the results of this paper remain valid for
M1 6= M2 by adding some slight changes in the hypothesis (H4) and (H5). The
function F : Ω × R × R → R is assumed to be continuous in x ∈ Ω and of class C 1
in u, v ∈ R.
The operator − div(|∇u|p(x)−2 ∇u) is called the p(x)-Laplacian, and becomes
p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The
study of various mathematical problems with variable exponent growth condition
2000 Mathematics Subject Classification. 35D05, 35J60, 35J70.
Key words and phrases. Variational method; nonlinear elliptic systems; nonlocal condition.
c
2011
Texas State University - San Marcos.
Submitted July 9, 2011. Published October 19, 2011.
Supported by grants 11061030 from the NSFC, and NWNU-LKQN-10-21.
1
2
G. DAI
EJDE-2011/137
has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of
the most studied models leading to problem of this type is the model of motion
of electrorheological fluids, which are characterized by their ability to drastically
change the mechanical properties under the influence of an exterior electromagnetic
field [1, 34, 37]. Problems with variable exponent growth conditions also appear in
the mathematical modeling of stationary thermo-rheological viscous flows of nonNewtonian fluids and in the mathematical description of the processes filtration of
an ideal barotropic gas through a porous medium [5, 6]. Another field of application of equations with variable exponent growth conditions is image processing [9].
The variable nonlinearity is used to outline the borders of the true image and to
eliminate possible noise. We refer the reader to [13, 29, 35, 38, 39] for an overview
of and references on this subject, and to [2, 20, 21, 22, 23, 24, 25, 26] for the study
of the p(x)-Laplacian equations and the corresponding variational problems.
Problem (1.1) is related to the stationary version of a model introduced by
Kirchhoff [30]. More precisely, Kirchhoff proposed the model
Z L
∂2u
E
∂u
∂ 2 u ρ0
+
| |2 dx
= 0,
(1.2)
ρ 2 −
∂t
h
2L 0 ∂x
∂x2
where ρ, ρ0 , h, E, L are constants, which extends the classical D’Alembert’s wave
equation, by considering the effects of the changes in the length of the strings
during the vibrations. A distinguishing feature of equation (1.2) is that the equation
R L ∂u 2
E
contains a nonlocal coefficient ρh0 + 2L
| | dx which depends on the average
0 ∂x
R L ∂u 2
1
2L 0 | ∂x | dx, and hence the equation is no longer a pointwise identity. Some early
classical studies of Kirchhoff equations were Bernstein [7] and Pohožaev [33]. The
equation
Z
− a+b
|∇u|2 dx ∆u = f (x, u) in Ω,
(1.3)
Ω
u = 0 on ∂Ω,
is related to the stationary analogue of the equation (1.2). Equation (1.3) received
much attention only after Lions [31] proposed an abstract framework to the problem. Some important and interesting results can be found, for example, in [3, 8, 17].
More recently Alves et al. [4] and Ma and Rivera [32] obtained positive solutions
of such problems by variational methods. The study of Kirchhoff type equations
has already been extended to the case involving the p-Laplacian (for details, see
[10, 18, 19])and p(x)-Laplacian (see [12, 15, 17, 27]). In [12], by a direct variational approach, we establish conditions ensuring the existence and multiplicity of
solutions for the problem
Z
1
−M
|∇u|p(x) dx div(|∇u|p(x)−2 ∇u) = f (x, u) in Ω,
Ω p(x)
u = 0 on ∂Ω.
In [28], the author established that existence and multiplicity results for a class of
elliptic systems with nonstandard growth conditions.
Motivated by above, we consider the nonlocal elliptic systems (1.1). We establish
the existence and multiplicity of solutions for system (1.1). Local elliptic systems
with standard growth conditions have been the subject of a sizeable literature. We
EJDE-2011/137
EXISTENCE OF SOLUTIONS
3
refer to the excellent survey article by De Figueiredo [14]. We also refer to [11]
about nonlocal elliptic systems of p-Kirchhoff-type.
This paper is organized as follows. In Section 2, we present some necessary
preliminary knowledge on variable exponent Sobolev spaces. In Sections 3, we give
some existence results of weak solutions of problem (1.1) and their proofs.
2. Preliminaries
1,p(x)
(Ω) which is called
To discuss problem (1.1), we need some theory on W0
variable exponent Sobolev space. Firstly we state some basic properties of spaces
1,p(x)
(Ω) which will be used later (for details, see [25]). Denote by S(Ω) the set
W0
of all measurable real functions defined on Ω. Two functions in S(Ω) are considered
as the same element of S(Ω) when they are equal almost everywhere. Write
C+ (Ω) = {h : h ∈ C(Ω), h(x) > 1 for any x ∈ Ω},
−
h := min h(x),
h+ := max h(x)
Ω
for every h ∈ C+ (Ω).
Ω
Define
Lp(x) (Ω) = {u ∈ S(Ω) :
Z
|u(x)|p(x) dx < +∞ for p ∈ C+ (Ω)}
Ω
with the norm
Z
|u|Lp(x) (Ω) = |u|p(x) = inf{λ > 0 :
|
Ω
u(x) p(x)
|
dx ≤ 1},
λ
and
W 1,p(x) (Ω) = {u ∈ Lp(x) (Ω) : |∇u| ∈ Lp(x) (Ω)}
with the norm
kukW 1,p(x) (Ω) = |u|Lp(x) (Ω) + |∇u|Lp(x) (Ω) .
Denote by
1,p(x)
(Ω)
W0
the closure of C0∞ (Ω) in W 1,p(x) (Ω).
1,p(x)
(Ω) are sepProposition 2.1 ([25]). The spaces Lp(x) (Ω), W 1,p(x) (Ω) and W0
arable and reflexive Banach spaces.
R
Proposition 2.2 ([25]). Set ρ(u) = Ω |u(x)|p(x) dx. For any u ∈ Lp(x) (Ω), then
(1) for u 6= 0, |u|p(x) = λ if and only if ρ( λu ) = 1;
(2) |u|p(x) < 1 (= 1; > 1) if and only if ρ(u) < 1 (= 1; > 1);
−
+
+
−
(3) if |u|p(x) > 1, then |u|pp(x) ≤ ρ(u) ≤ |u|pp(x) ;
(4) if |u|p(x) < 1, then |u|pp(x) ≤ ρ(u) ≤ |u|pp(x) ;
(5) limk→+∞ |uk |p(x) = 0 if and only if limk→+∞ ρ(uk ) = 0;
(6) limk→+∞ |uk |p(x) = +∞ if and only if limk→+∞ ρ(uk ) = +∞.
1,p(x)
Proposition 2.3 ([25]). In W0
(Ω) the Poincaré inequality holds; that is, there
exists a positive constant C0 such that
|u|Lp(x) (Ω) ≤ C0 |∇u|Lp(x) (Ω) ,
1,p(x)
∀u ∈ W0
(Ω).
1,p(x)
So, |∇u|Lp(x) (Ω) is a norm equivalent to the norm kuk in the space W0
(Ω).
We will use the equivalent norm in the following discussion and write kukp =
|∇u|Lp(x) (Ω) for simplicity.
4
G. DAI
EJDE-2011/137
Proposition 2.4 ([22, 25]). If q ∈ C+ (Ω) and q(x) ≤ p∗ (x) (q(x) < p∗ (x)) for
1,p(x)
x ∈ Ω, then there is a continuous (compact) embedding W0
(Ω) ,→ Lq(x) (Ω),
where
(
N p(x)
if p(x) < N,
∗
p (x) = N −p(x)
+∞
if p(x) ≥ N.
Proposition 2.5 ([23, 25]). The conjugate space of Lp(x) (Ω) is Lq(x) (Ω), where
1
1
p(x)
(Ω) and v ∈ Lq(x) (Ω), we have
q(x) + p(x) = 1 holds a.e. in Ω. For any u ∈ L
the following Hölder-type inequality
Z
1
1
uv dx ≤ ( − + − )|u|p(x) |v|q(x) .
p
q
Ω
We write
Z
I(u) =
Ω
1
|∇u|p(x) dx.
p(x)
Proposition 2.6 ([23]). The functional I : X → R is convex. The mapping
I 0 : X → X ∗ is a strictly monotone, bounded homeomorphism, and is of (S+ ) type,
namely
un * u and lim sup I 0 (un )(un − u) ≤ 0 implies un → u,
n→+∞
where X =
1,p(x)
(Ω),
W0
∗
X is the dual space of X.
1,p(x)
1,q(x)
For every (u, v) and (ϕ, ψ) in W := W0
(Ω) × W0
Z
F(u, v) :=
F (x, u, v) dx.
(Ω), let
Ω
Then
F 0 (u, v)(ϕ, ψ) = D1 F(u, v)(ϕ) + D2 F(u, v)(ψ),
where
Z
∂F
(x, u, v)ϕ dx,
∂u
ZΩ
∂F
D2 F(u, v)(ψ) =
(x, u, v)ψ dx.
Ω ∂v
D1 F(u, v)(ϕ) =
The Euler-Lagrange functional associated to (1.1) is given by
Z
Z
1
1
p(x)
c
c
|∇u|
dx + M (
|∇v|q(x) dx) − F(u, v),
J(u, v) := M
Ω p(x)
Ω q(x)
R
c(t) := t M (τ ) dτ . It is easy to verify that J ∈ C 1 (W, R) is weakly lower
where M
0
semi-continuous and (u, v) ∈ W is a weak solution of (1.1) if and only if (u, v) is a
critical point of J. Moreover, we have
J 0 (u, v)(ϕ, ψ) = D1 J(u, v)(ϕ) + D2 J(u, v)(ψ),
(2.1)
where
Z
1
|∇u|p(x) dx
|∇u|p(x)−2 ∇u∇ϕ dx − D1 F(u, v)(ϕ),
Ω p(x)
Ω
Z
Z
1
q(x)
D2 J(u, v)(ψ) = M
|∇v|
dx
|∇v|q(x)−2 ∇v∇ψ dx − D2 F(u, v)(ψ).
Ω q(x)
Ω
D1 J(u, v)(ϕ) = M
Z
EJDE-2011/137
EXISTENCE OF SOLUTIONS
5
Let us choose on W the norm k · k defined by
k(u, v)k := max{kukp , kvkq }.
The dual space of W will be denoted by W ∗ and k · k∗ will stand for its norm.
Therefore
kJ 0 (u, v)k∗ = kD1 J(u, v)k∗,p + kD2 J(u, v)k∗,q
1,p(x)
where k · k∗,p (respectively k · k∗,q ) is the norm of (W0
1,q(x)
(W0
(Ω))∗ ).
(Ω))∗ (respectively
3. Existence of solutions
In this section we discuss the existence of weak solutions of (1.1). For simplicity,
we use c, ci , i = 1, 2, . . . to denote the general positive constant (the exact value
may change from line to line).
Before stating our results, we introduce some natural growth hypotheses on the
right-hand side of (1.1) and the nonlocal coefficient M (t). These hypotheses will
ensure the mountain pass geometry and the Palais-Smale condition for the EulerLagrange functional J.
(H1) For all (x, s, t) ∈ Ω × R2 , we assume
|F (x, s, t)| ≤ c 1 + |s|p1 (x) + |t|q1 (x) + |s|α(x) |t|β(x) ,
where c is a positive constant, (p1 (x), q1 (x), α(x), β(x)) ∈ (C+ (Ω))4 such
that
2α(x) 2β(x)
p1 (x) < p∗ (x), q1 (x) < q ∗ (x),
+ ∗
< 1 in Ω,
p∗ (x)
q (x)
p−
1,
2α− > p+ ,
+
q1− ,
2β − > q + .
+
p
q
(H2) There exist M > 0, θ1 > 1−µ
, θ2 > 1−µ
such that for all x ∈ Ω, and all
2
θ1
θ2
(s, t) ∈ R with |s| + |t| ≥ 2M , one has
s ∂F
t ∂F
0 < F (x, s, t) ≤
(x, s, t) +
(x, s, t),
θ1 ∂s
θ2 ∂t
where µ comes from (H5) below.
+
+
(H3) F (x, s, t) = o(|s|p + |t|q ) as (s, t) → (0, 0) uniformly with respect to to
x ∈ Ω.
(H4) There exists m0 > 0, such that M (t) ≥ m0 .
c(t) ≥ (1 − µ)M (t)t.
(H5) There exists 0 < µ < 1 such that M
As an example, we let M (t) = a + bt : R+ → R with a, b are two positive
constants. It is clear that M (t) ≥ a > 0. Taking µ = 1/2, we have
Z t
1
1
c(t) =
M
M (s) ds = at + bt2 ≥ (a + bt)t = (1 − µ)M (t)t.
2
2
0
So conditions (H4), (H5) are satisfied.
Theorem 3.1. If M satisfies (H4) and
|F (x, s, t)| ≤ c1 (1 + |s|α1 + |t|β1 ),
where α1 , β1 are two constants with 1 ≤ α1 < min{p− , q − }, 1 ≤ β1 < min{p− , q − }
then (1.1) has a weak solution.
6
G. DAI
EJDE-2011/137
c(t) ≥ m0 t. For (un , vn ) ∈ W such that k(un , vn )k →
Proof. From (H4) we have M
+∞, we have
J(un , vn )
Z
Z
Z
1
1
p(x)
q(x)
c
c
= M(
|∇un |
dx) + M (
|∇vn |
dx) −
F (x, un , vn ) dx
p(x)
q(x)
Ω
ZΩ
ZΩ
1
1
|∇un |p(x) dx + m0
|∇vn |q(x) dx
≥ m0
p(x)
q(x)
Ω
Ω
Z
Z
− c1
|un |α1 dx − c1
|vn |β1 dx − c1 |Ω|
Ω
≥
Ω
−
−
m0
m0
β1
1
kun kpp + + kvn kqq − c3 kun kα
p − c2 kvn kq − c1 |Ω|,
p+
q
where |Ω| denotes the measure of Ω. Without loss of generality, we may assume
kun kp ≥ kvn kq . Hence,
−
m0
β1
1
J(un , vn ) ≥ + kun kpp − c3 kun kα
(3.1)
p − c2 kun kp − c1 |Ω|,
p
By the definition of norm on W , we have k(un , vn )k = kun kp → +∞. In view of
(3.1) and the assumptions on α1 and β1 , we can easily see that J(un , vn ) → +∞
as n → +∞; i.e., J is a coercive functional. Since J also is weakly lower semicontinuous, J has a minimum point (u, v) in W , and (u, v) is a weak solution pair
which may be trivial of (1.1). The proof is completed.
Lemma 3.2. Let (un , vn ) be a Palais-Smale sequence for the Euler-Lagrange functional J. If (H2), (H4), (H5) are satisfied then (un , vn ) is bounded.
Proof. Let (un , vn ) be a Palais-Smale sequence for the functional J. This means
that J(un , vn ) is bounded and kJ 0 (un , vn )k∗ → 0 as n → +∞. Then, there is a
positive constant c0 such that
c0 ≥ J(un , vn )
Z
Z
Z
1
1
p(x)
q(x)
c
c
|∇un |
dx + M
|∇vn |
dx −
F (x, un , vn ) dx
=M
Ω q(x)
Ω
Ω p(x)
Z
Z
1
1
≥ (1 − µ)M
|∇un |p(x) dx
|∇un |p(x) dx
p(x)
p(x)
Ω
Ω
Z
Z
un ∂F
1
−
(x, un , vn ) dx + (1 − µ)M
|∇vn |q(x) dx
θ1 ∂u
Ω q(x)
ZΩ
Z
1
v
∂F
n
×
|∇vn |q(x) dx −
(x, un , vn ) dx − c4 ,
Ω q(x)
Ω θ2 ∂v
where c4 is some positive constant. Then
c0 ≥ J(un , vn )
Z
Z
1
1 1
1−µ
p(x)
−
M
|∇un |
dx
|∇un |p(x) dx + D1 J(un , vn )(un )
≥
p+
θ1
p(x)
θ
1
Ω
Ω
Z
Z
1−µ
1 1
q(x)
q(x)
−
M
|∇vn |
dx
|∇vn |
dx
+
q+
θ2
Ω q(x)
Ω
1
+ D2 J(un , vn )(vn ) − c4
θ2
EJDE-2011/137
EXISTENCE OF SOLUTIONS
7
Z
Z
1−µ
1
1
1−µ
p(x)
−
m
−
m
|∇u
|
dx
+
|∇vn |q(x) dx
0
0
n
p+
θ1
q+
θ2
Ω
Ω
1
1
− kD1 J(un , vn )k∗,p kun kp − kD2 J(un , vn )k∗,q kvn kq − c4 .
θ1
θ2
Now, suppose that the sequence (un , vn ) is not bounded. Without loss of generality,
we may assume kun kp ≥ kvn kq .
Therefore, for n large enough, we have
1
1
1−µ
1
p−
m
ku
k
−
kD
J(u
,
v
)k
+
kD
J(u
,
v
)k
kun kp .
c5 ≥
−
0
n
1
n
n
∗,p
2
n
n
∗,q
p
p+
θ1
θ1
θ2
≥
But, this cannot hold true since p− > 1. Hence, {k(un , vn )k} is bounded.
In the following lemma, we show every bounded Palais-Smale sequence for the
functional J contains a convergence subsequence.
Lemma 3.3. Let (un , vn ) be a bounded Palais-Smale sequence for the Euler-Lagrange functional J. If (H1), (H4) are satisfied, then (un , vn ) contains a convergent
subsequence.
Proof. Let (un , vn ) be a bounded Palais-Smale sequence for the functional J. Then
there is a subsequence still denoted by (un , vn ) which converges weakly in W .
Without loss of generality, we assume that (un , vn ) * (u, v), then J 0 (un , vn )(un −
u, vn − v) → 0. Thus, we have
J 0 (un , vn )(un − u, vn − v)
Z
Z
1
|∇un |p(x) dx
|∇un |p(x)−2 ∇un (∇un − ∇u) dx
=M
Ω
Ω p(x)
Z
Z
1
q(x)
+M
|∇vn |
dx
|∇vn |q(x)−2 ∇vn (∇vn − ∇v) dx
Ω q(x)
Ω
Z
Z
∂F
∂F
−
(x, un , vn )(un − u) dx −
(x, un , vn )(vn − v) dx → 0.
∂u
Ω
Ω ∂v
On the other hand, let α
e, βe be two continuous and positive functions on Ω such
that
e
2α(x) + α
e(x) 2β(x) + β(x)
+
= 1, ∀x ∈ Ω.
∗
∗
p (x)
q (x)
Using the Young inequality, we obtain
α(x)p∗ (x)
β(x)q ∗ (x)
e
e
|s|α(x) |t|β(x) ≤ |s| 2α(x)+α(x)
+ |t| 2β(x)+β(x)
= |s|p2 (x) + |t|q2 (x) ,
where p2 (x) :=
α(x)p∗ (x)
2α(x)+e
α(x)
< p∗ (x), q2 (x) :=
β(x)q ∗ (x)
e
2β(x)+β(x)
< q ∗ (x). From (H1), we
can obtain that there exist p3 (x), q3 (x) ∈ C+ (Ω) with p3 (x) < p∗ (x), q3 (x) < q ∗ (x)
in Ω such that
|F (x, s, t)| ≤ c6 1 + |s|p3 (x) + |t|q3 (x) .
From this inequality, Propositions 2.4 and 2.5, we can easily obtain
Z
∂F
(x, un , vn )(un − u) dx → 0
Ω ∂u
and
Z
∂F
(x, un , vn )(vn − v) dx → 0.
(3.2)
Ω ∂v
8
G. DAI
EJDE-2011/137
Therefore, we have
Z
Z
1
M
|∇un |p(x) dx
|∇un |p(x)−2 ∇un (∇un − ∇u) dx → 0,
p(x)
Ω
Ω
Z
Z
1
q(x)
|∇vn |
dx
M
|∇vn |q(x)−2 ∇vn (∇vn − ∇v) dx → 0.
Ω
Ω q(x)
In view of (H4), we have
Z
|∇un |p(x)−2 ∇un (∇un − ∇u) dx → 0,
Ω
Z
|∇vn |q(x)−2 ∇vn (∇vn − ∇v) dx → 0.
Ω
1,p(x)
1,q(x)
Using Proposition 2.6, we have un → u in W0
(Ω) and vn → v in W0
which implies that (un , vn ) → (u, v) in W . This completes the proof.
(Ω),
Theorem 3.4. If hypotheses (H1)–(H5) hold, then (1.1) has at least one weak
solution.
Proof. Let us show that J satisfies the conditions of Mountain Pass Theorem (see
[36, Theorem 2.10]). By Lemmas 3.2 and 3.3, J satisfies Palais-Smale condition in
W.
2β(x)
For k(u, v)k < 1, using the Young’s inequality, the fact 2α(x)
p∗ (x) + q ∗ (x) < 1 in Ω,
Propositions 2.2 and 2.4, we obtain
Z
Z
Z
−
−
1
1
|u|α(x) |v|β(x) dx ≤
|u|2α(x) dx +
|v|2β(x) dx ≤ c7 (kuk2α
+ kvk2β
).
p
q
2 Ω
2 Ω
Ω
+
1,p(x)
1,q(x)
On the other hand, assuming (H1), W0
(Ω) ,→ Lp (Ω), and W0
q+
L (Ω). Then there exists c8 , c9 > 0 such that
|u|p+ ≤ c8 kukp
|v|q+ ≤ c9 kvkq
1,p(x)
for u ∈ W0
for v ∈
(Ω) ,→
(Ω)
1,q(x)
W0
(Ω),
where |·|r denote the norm on Lr(x) (Ω) with r ∈ C+ (Ω). Let ε > 0 be small enough
+
q+
m0
m0
such that εcp8 ≤ 2p
≤ 2q
+ and εc9
+ . By the assumptions (H1) and (H3), we have
+
+
|F (x, s, t)| ≤ ε |s|p + |t|q + c(ε)(|s|p1 (x) + |t|q1 (x) + |s|α(x) |t|β(x) )
for all (x, s, t) ∈ Ω × R2 . In view of (H4) and and the above inequality, for k(u, v)k
sufficiently small, noting Proposition 2.2, we have
Z
Z
Z
Z
+
+
m0
m0
|∇u|p(x) dx + +
|∇v|q(x) dx − ε
|u|p dx − ε
|v|q dx
J(u, v) ≥ +
p
q
Ω
Ω
Ω
Ω
Z − c(ε)
|u|p1 (x) + |v|q1 (x) + |u|α(x) |v|β(x) dx
Ω
+
+
+
+
+
+
m0
m0
≥ + kukpp − εcp8 kukpp + + kvkqq − εcq9 kvkqq
p
q
−
p−
q1−
2β −
1
+
c
kvk
− c(ε) kukp + kvkq + c7 kuk2α
7
p
q
−
+
+
−
−
m
m0
p
q−
0
≥ + kukpp + + kvkqq − c(ε) kukp1 + kvkq1 + c7 kuk2α
+ c7 kvk2β
.
p
q
2p
2q
EJDE-2011/137
EXISTENCE OF SOLUTIONS
9
−
−
+
−
Since p−
> q + , there exist r > 0, δ > 0 such that J(u) ≥
1 , 2α > p and q1 , 2β
δ > 0 for every k(u, v)k = r.
On the other hand, we have known that the assumption (H2) implies the following assertion: for every x ∈ Ω, s, t ∈ R, the inequality
F (x, s, t) ≥ c10 (|s|θ1 + |t|θ2 − 1)
(3.3)
holds; see [28]. When t > t0 , from (H5) we can easily obtain that
c
c(t) ≤ M (t0 ) t1/(1−µ) := c11 t1/(1−µ) ,
M
1/(1−µ)
t0
where t0 is an arbitrarily positive constant. For (e
u, ve) ∈ W \ {(0, 0)} and t > 1, we
have
Z
Z
1
1
c
c
J(te
u, te
v) = M
|t∇e
u|p(x) dx + M
|t∇e
v |q(x) dx
p(x)
Ω q(x)
Z Ω
−
F (x, te
u, te
v ) dx
Ω
Z
Z
p(x)
1/(1−µ)
θ1
≤ c12 ( |t∇e
u|
dx)
− c10 t
|e
u|θ1 dx
Ω
Ω
Z
Z
1/(1−µ)
q(x)
θ2
+ c13
|t∇e
v|
dx
− c10 t
|e
v |θ2 dx − c14
Ω
Ω
Z
Z
1/(1−µ)
p+
|∇e
u|p(x) dx
− c10 tθ1
|e
u|θ1 dx
≤ c12 t 1−µ
Ω
Ω
Z
Z
1/(1−µ)
q+
q(x)
θ
2
|∇e
v|
dx
− c10 t
+ c13 t 1−µ
|e
v |θ2 dx − c14
Ω
→ −∞,
+
Ω
as t → +∞,
+
q
p
and θ2 > 1−µ
. Since J(0, 0) = 0, considering Lemmas 3.2 and 3.3,
due to θ1 > 1−µ
we see that J satisfies the conditions of Mountain Pass Theorem. So J admits at
least one nontrivial critical point.
Next we will prove under some symmetry condition on the function F that (1.1)
possesses infinitely many nontrivial weak solutions.
Theorem 3.5. Assume (H1), (H2), (H4), (H5), and that F (x, u, v) is even in u, v.
Then (1.1) has a sequence of solutions {(±uk , ±vk )}∞
k=1 such that J(±uk , ±vk ) →
+∞ as k → +∞.
1,p(x)
1,q(x)
Because W0
and W0
are a reflexive and separable Banach space, then
W and W ∗ are too. There exist {ej } ⊂ W and {e∗j } ⊂ W ∗ such that
W = span{ej : j = 1, 2, . . . },
and
hei , e∗j i
W ∗ = span{e∗j : j = 1, 2, . . . },
(
1, i = j,
=
0, i =
6 j,
where h·, ·i denotes the duality product between W and W ∗ . For convenience, we
write Xj = span{ej }, Yk = ⊕kj=1 Xj , Zk = ⊕∞
j=k Xj . We will use the following
“Fountain theorem” to prove Theorem 3.5.
Lemma 3.6 ([36]). Assume
10
G. DAI
(A1)
(A2)
(A2)
(A3)
(A4)
Then I
EJDE-2011/137
X is a Banach space, I ∈ C 1 (X, R) is an even functional.
For each k = 1, 2, . . . , there exist ρk > rk > 0 such that
inf u∈Zk ,kuk=rk I(u) → +∞ as k → +∞.
maxu∈Yk ,kuk=ρk I(u) ≤ 0.
I satisfies Palais-Smale condition for every c > 0.
has a sequence of critical values tending to +∞.
For every a > 1, u, v ∈ La (Ω), we define
|(u, v)|a := max{|u|a , |v|a }.
Set
a := max{2α(x), 2β(x), p1 (x), q1 (x)} > min{p− , q − },
x∈Ω
b := min{2α(x), 2β(x), p1 (x), q1 (x)} > 0.
x∈Ω
Then we have the following result.
Lemma 3.7 ([28]). Denote
βk = sup{|(u, v)|a : k(u, v)k = 1, (u, v) ∈ Zk }.
Then limk→+∞ βk = 0.
Proof of Theorem 3.5. According to the assumptions on F , Lemmas 3.2 and 3.3,
J is an even functional and satisfies Palais-Smale condition. We will prove that if
k is large enough, then there exist ρk > rk > 0 such that (A2 ) and (A3 ) holding.
Thus, the conclusion can be obtained from Fountain theorem.
(A2): For any (uk , vk ) ∈ Zk , kuk kp ≥ 1, kvk kq ≥ 1 and k(uk , vk )k = rk (rk will
be specified below), we have
J(uk , vk )
Z
c
=M
Z
Z
1
1
c
|∇uk |p(x) dx + M
|∇vk |q(x) dx −
F (x, uk , vk ) dx
Ω p(x)
Ω q(x)
Ω
Z
Z
Z
1
1
|∇uk |p(x) dx + m0
|∇vk |q(x) dx −
F (x, uk , vk ) dx
≥ m0
p(x)
Ω q(x)
Ω
ZΩ
Z
m0
m0
≥ +
|∇uk |p(x) dx + +
|∇vk |q(x) dx
p
q
Ω
Ω
Z −c
1 + |uk |p1 (x) + |vk |q1 (x) + |uk |α(x) |vk |β(x) dx
Ω
−
−
m0
m0
p (ξ k )
q (ξ k )
≥ + kuk kpp + + kvk kqq − c|uk |p11 (x)1 − c|vk |q11 (x)2
p
q
2α(η k )
2β(η k )
− c15 |uk |2α(x)1 − c15 |vk |2β(x)2 − c|Ω|,
where ξ1k , ξ2k , η1k , η2k ∈ Ω. Therefore,
J(uk , vk )
m0
p1 (ξ1k )
q1 (ξ2k )
min{p− ,q − }
≥
k(u
,
v
)k
−
c|u
|
−
c|v
|
a
a
k
k
k
k
max{p+ , q + }
2α(η k )
2β(η k )
− c|uk |a 1 − c|vk |a 2 − c|Ω|
− −
k
k
m0
≥
k(uk , vk )kmin{p ,q } − c(βk k(uk , vk )k)p1 (ξ1 ) − c(βk k(uk , vk )k)q1 (ξ2 )
max{p+ , q + }
EJDE-2011/137
EXISTENCE OF SOLUTIONS
k
11
k
− c(βk k(uk , vk )k)2α(η1 ) − c(βk k(uk , vk )k)2β(η2 ) − c|Ω|
− −
m0
k(uk , vk )kmin{p ,q } − c16 βkb k(uk , vk )ka − c|Ω|,
≥
+
+
max{p , q }
where a, b are defined above. At this stage, we fix rk as follows:
1/(a−min{p− ,q− })
m0
rk :=
→ +∞ as k → +∞.
2c16 max{p+ , q + }βkb
Consequently, if k(uk , vk )k = rk then
− −
m0
J(uk , vk ) ≥
k(uk , vk )kmin{p ,q } − c|Ω| → +∞
2 max{p+ , q + }
as k → +∞.
(A3): From (H2), we have F (x, u, v) ≥ c10 (|u|θ1 + |v|θ2 − 1) for every x ∈ Ω and
u, v ∈ R. Therefore, for any (u, v) ∈ Yk with k(u, v)k = 1 and 1 < ρk = tk with
tk → +∞, we have
J(tk u, tk v)
Z
Z
Z
1
1
p(x)
q(x)
c
c
|tk ∇u|
dx + M
|tk ∇v|
dx −
F (x, tk u, tk v) dx.
=M
Ω q(x)
Ω
Ω p(x)
Z
Z
1/(1−µ)
1/(1−µ)
|tk ∇v|q(x) dx
|tk ∇u|p(x) dx
+ c18
≤ c17
Ω
Ω
Z
Z
θ1
θ2
θ1
θ2
− c10 tk
|u| dx − c10 tk
|v| dx + c19 ,
Ω
Ω
Z
Z
1/(1−µ)
p+ θ1
1−µ
p(x)
≤ c17 tk
|∇u|
dx
− c10 tk
|u|θ1 dx
Ω
Ω
Z
Z
1/(1−µ)
q+ θ2
1−µ
q(x)
+ c18 tk
|∇v|
dx
− c10 tk
|v|θ2 dx + c19 .
Ω
p+
1−µ ,
Ω
q+
1−µ
θ2 >
and dimYk = k, it is easy to see that J(tk u, tk v) → −∞ as
By θ1 >
k(tk u, tk v)k → +∞ for (u, v) ∈ Yk .
The proof of Theorem 3.5 is completed by the Fountain theorem.
Acknowledgments. The author wishes to express his gratitude to the anonymous
referee for reading the original manuscript carefully and making several corrections
and remarks.
References
[1] E. Acerbi and G. Mingione; Regularity results for stationary electro-rheologicaluids, Arch.
Ration. Mech. Anal., 164 (2002), 213–259.
[2] E. Acerbi, G. Mingione; Gradient estimate for the p(x)-Laplacean system, J. Reine Angew.
Math., 584 (2005), 117–148.
[3] A. Arosio, S. Panizzi; On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc.
348 (1996) 305–330.
[4] C. O. Alves, F. J. S. A. Corrêa, T. F. Ma; Positive solutions for a quasilinear elliptic equation
of Kirchhoff type, Comput. Math. Appl. 49 (2005) 85–93.
[5] S. N. Antontsev, S. I. Shmarev; A Model Porous Medium Equation with Variable Exponent
of Nonlinearity: Existence, Uniqueness and Localization Properties of Solutions, Nonlinear
Anal. 60 (2005), 515–545.
[6] S. N. Antontsev , J. F. Rodrigues; On Stationary Thermo-rheological Viscous Flows, Ann.
Univ. Ferrara, Sez. 7, Sci. Mat. 52 (2006), 19–36.
[7] S. Bernstein; Sur une classe d’équations fonctionnelles aux dérivées partielles, Bull. Acad.
Sci. URSS. Ser. Math. [Izv. Akad. Nauk SSSR] 4 (1940) 17–26.
12
G. DAI
EJDE-2011/137
[8] M. M. Cavalcanti, V. N. Cavalcanti, J.A. Soriano; Global existence and uniform decay rates
for the Kirchhoff-Carrier equation with nonlinear dissipation, Adv. Differential Equations 6
(2001) 701–730.
[9] Y. Chen, S. Levine, M. Rao; Variable exponent, linear growth functionals in image restoration, SIAM J. Appl.Math. 66 (4) (2006), 1383–1406.
[10] F. J. S. A. Corrêa, G. M. Figueiredo; On a elliptic equation of p-kirchhoff type via variational
methods, Bull. Austral. Math. Soc. Vol. 74 (2006) 263–277.
[11] F. J. S. A. Corrêa, R.G. Nascimento; On a nonlocal elliptic system of p-Kirchhoff-type under
Neumann boundary condition, Mathematical and Computer Modelling 49 (2009) 598–604.
[12] G. Dai, R. Hao; Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal.
Appl. 359 (2009) 275–284.
[13] L. Diening, P. Hästö, A. Nekvinda; Open problems in variable exponent Lebesgue and Sobolev
spaces, in: P. Drábek, J. Rákosnı́k, FSDONA04 Proceedings, Milovy, Czech Republic, 2004,
pp.38–58.
[14] G. Dai, D. Liu; Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, J. Math.
Anal. Appl. 359 (2009) 704–710.
[15] G. Dai, J. Wei; Infinitely many non-negative solutions for a p(x)-Kirchhoff-type problem with
Dirichlet boundary condition, Nonlinear Analysis 73 (2010) 3420–3430.
[16] P. D’Ancona, S. Spagnolo; Global solvability for the degenerate Kirchhoff equation with real
analytic data, Invent. Math. 108 (1992) 247–262.
[17] D. G. De Figueiredo; Semilinear elliptic systems: a survey of superlinear problems, Resenhas
2 (1996) 373–391.
[18] M. Dreher; The Kirchhoff equation for the p-Laplacian, Rend. Semin. Mat. Univ. Politec.
Torino 64 (2006) 217–238.
[19] M. Dreher; The wave equation for the p-Laplacian, Hokkaido Math. J. 36 (2007) 21–52.
[20] X. L. Fan; On the sub-supersolution methods for p(x)-Laplacian equations, J. Math. Anal.
Appl. 330 (2007), 665–682.
[21] X. L. Fan and X. Y. Han; Existence and multiplicity of solutions for p(x)-Laplacian equations
in RN , Nonlinear Anal. 59 (2004), 173–188.
[22] X. L. Fan, J. S. Shen, D. Zhao; Sobolev embedding theorems for spaces W k,p(x) (Ω), J. Math.
Anal. Appl. 262 (2001), 749–760.
[23] X. L. Fan, Q. H. Zhang; Existence of solutions for p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 52 (2003), 1843–1852.
[24] X. L. Fan, Q. H. Zhang and D. Zhao; Eigenvalues of p(x)-Laplacian Dirichlet problem, J.
Math. Anal. Appl. 302 (2005), 306–317.
[25] X.L. Fan, D. Zhao; On the Spaces Lp(x) and W m,p(x) , J. Math. Anal. Appl. 263 (2001),
424–446.
[26] X. L. Fan, Y. Z. Zhao, Q. H. Zhang; A strong maximum principle for p(x)-Laplace equations,
Chinese J. Contemp. Math. 24 (3) (2003) 277–282.
[27] X. L. Fan; On nonlocal p(x)-Laplacian Dirichlet problems, Nonlinear Anal. 72 (2010) 3314–
3323.
[28] A. El Hamidi; Existence results to elliptic systems with nonstandard growth conditions, J.
Math. Anal. Appl. 300 (2004) 30–42.
[29] P. Harjulehto, P. Hästö; An overview of variable exponent Lebesgue and Sobolev spaces, in
Future Trends in Geometric Function Theory (D. Herron (ed.), RNC Work-shop), Jyväskylä,
2003, 85–93.
[30] G. Kirchhoff; Mechanik, Teubner, Leipzig, 1883.
[31] J. L. Lions; On some equations in boundary value problems of mathematical physics, in:
Contemporary Developments in Continuum Mechanics and Partial Differential Equations
(Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), NorthHolland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346.
[32] T. F. Ma, J. E. Muñoz Rivera; Positive solutions for a nonlinear nonlocal elliptic transmission
problem, Appl. Math. Lett. 16 (2003) 243–248.
[33] S. I. Pohožaev; A certain class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 96 (1975)
152–166, 168.
[34] M. Ru̇z̆ic̆ka; Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag,
Berlin, 2000.
EJDE-2011/137
EXISTENCE OF SOLUTIONS
13
[35] S. Samko; On a progress in the theory of Lebesgue spaces with variable exponent Maximal
and singular operators, Integral Transforms Spec. Funct. 16 (2005), 461–482.
[36] M. Willem; Minimax Theorems, Birkhäuser, Boston, 1996.
[37] V. V. Zhikov; Averaging of functionals of the calculus of variations and elasticity theory,
Math. USSR. Izv. 9 (1987), 33–66.
[38] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik; Homogenization of Differential Operators and
Integral Functionals, Translated from the Russian by G.A. Yosifian, Springer-Verlag, Berlin,
1994.
[39] V. V. Zhikov; On some variational problems, Russian J. Math. Phys. 5 (1997), 105–116.
Guowei Dai
Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China
E-mail address: daiguowei@nwnu.edu.cn