Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 139, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF MULTIPLE SOLUTIONS FOR A
p(x)-BIHARMONIC EQUATION
LIN LI, LING DING, WEN-WU PAN
Abstract. In this article, we show the existence of at least three solutions
to a Navier boundary problem involving the p(x)-biharmonic operator. The
technical approach is mainly base on a three critical points theorem by Ricceri.
1. Introduction and statement of the main result
In this article, we consider the fourth-order quasilinear elliptic equation
∆2p(x) u + |u|p(x)−2 u = λf (x, u) + µg(x, u),
u = 0,
∆u = 0,
in Ω,
on ∂Ω,
(1.1)
where ∆2p(x) u = ∆(|∆u|p(x)−2 ∆u) is the p(x)-biharmonic operator of fourth order,
λ, µ ∈ [0, ∞), Ω ⊂ RN (N > 1) is a nonempty bounded open set with a sufficient
smooth boundary
∂Ω. f , g : Ω × R →
Ru
R u R are Carathéodory functions. Next, let
F (x, u) = 0 f (x, s)ds and G(x, u) = 0 g(x, s)ds. For p ∈ C(Ω), denote 1 < p− =
minx∈Ω p(x) ≤ p+ = maxx∈Ω p(x) < +∞. Moreover,
(
N p(x)
p(x) < N2 ,
∗
p2 (x) = N −2p(x)
∞
p(x) ≥ N2 ,
is the critical exponent just as in many papers. Obviously, p(x) < p∗ (x) for all
1,p(x)
x ∈ Ω. In the sequel, X will denote the Sobolev space W 2,p(x) (Ω) ∩ W0
(Ω).
The energy functional corresponding to problem (1.1) is defined on X as
H(u) = Φ(u) + λΨ(u) + µJ(u),
(1.2)
where
Z
1
(|∆u|p(x) + |u|p(x) )dx,
Ω p(x)
Z
Ψ(u) = −
F (x, u)dx,
ZΩ
J(u) = −
G(x, u)dx.
Φ(u) =
Ω
2000 Mathematics Subject Classification. 35J65, 35J60, 47J30, 58E05.
Key words and phrases. p(x)-biharmonic equation; Navier boundary condition;
Multiple solutions; three critical points theorem; variational methods.
c
2013
Texas State University - San Marcos.
Submitted December 30, 2012. Published June 21, 2013.
1
(1.3)
(1.4)
(1.5)
2
L. LI, L. DING, W.-W. PAN
EJDE-2013/139
Let us recall that a weak solution of (1.1) is any u ∈ X such that
Z
(|∆u|p(x)−2 ∆u∆v + |u|p(x)−2 uv)dx
Ω
Z
Z
=λ
f (x, u)vdx + µ
g(x, u)vdx for all v ∈ X.
Ω
Ω
In recent years, the study of differential equations and variational problems with
p(x)-growth conditions has been an interesting topic, which arises from nonlinear
electrorheological fluids and elastic mechanics. In that context we refer the reader
to Ruzicka [14], Zhikov [19], and the references therein. Moreover, we point out that
elliptic equations involving the p(x)-biharmonic equations are not trivial generalizations of similar problems studied in the constant case since the p(x)-biharmonic
operator is not homogeneous and, thus, some techniques which can be applied in
the case of the p-biharmonic operators will fail in that new situation, such as the
Lagrange Multiplier Theorem.
Ricceri’s three critical points theorem is a powerful tool to study boundary
problem of differential equation (see, for example, [1, 2, 3, 4]). Particularly, Mihailescu [10] use three critical points theorem of Ricceri [12] study a particular
p(x)-Laplacian equation. He proved existence of three solutions for the problem.
Liu [9] study the solutions of the general p(x)-Laplacian equations with Neumann
or Dirichlet boundary condition on a bounded domain, and obtain three solutions
under appropriate hypotheses. Shi [15] generalizes the corresponding result of [10].
To our best of knowledge, there no result of multiple solutions of p(x)-biharmonic
equation under sublinear condition. The aim of this paper is to prove the following
result
Theorem 1.1. Assume that sup(x,s)∈Ω×R
|f (x,s)|
1+|s|t(x)−1
< +∞, where t ∈ C(Ω) and
t(x) < p∗ (x) for all x ∈ Ω and there exist two positive constants %, ϑ and a function
γ(x) ∈ C(Ω) with 1 < γ − ≤ γ + < p− , such that
(I1) F (x, s) > 0 for a.e. x ∈ Ω and all s ∈]0, %];
∗
(I2) there exist p1 (x) ∈ C(Ω) and p+ < p−
1 ≤ p1 (x) < p (x), such that
F (x, s)
< +∞;
p (x)
x∈Ω |s| 1
lim sup sup
s→0
(I3) |F (x, s)| ≤ ϑ(1 + |s|γ(x) ) for a.e. x ∈ Ω and all s ∈ R.
Then, there exist an open interval Λ ⊆ (0, +∞) and a positive real number ρ with
the following property: for each λ ∈ Λ and each function g(x, s) : Ω × R → R
satisfying
|g(x, s)|
sup
< +∞,
1
+
|s|p2 (x)−1
(x,s)∈Ω×R
where p2 ∈ C(Ω) and p2 (x) < p∗ (x) for all x ∈ Ω, there exists δ > 0 such that, for
each µ ∈ [0, δ], problem (1.1) has at least three weak solutions whose norms in X
are less than ρ.
Remark 1.2. The conclusion of Theorem 1.1 gives a precise information about
the p(x)-biharmonic equation (1.1) with parameter, namely, one can see that (1.1)
is stable with respect to small perturbations.
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EXISTENCE OF MULTIPLE SOLUTIONS
3
This article is divided into four sections. In Section 2, we recall some basic facts
about the variable exponent Lebesgue and Sobolev spaces. In the third section,
we present some important properties of the p(x)-biharmonic operator. In section
4, we recall B. Ricceri’s three critical points theorem at first, then prove our main
result.
2. Preliminaries
To study p(x)-biharmonic problems, we need some results on the spaces Lp(x) (Ω)
and W k,p(x) (Ω), and properties of p(x)-biharmonic operator, which we will use later.
Define the generalized Lebesgue space by
Z
Lp(x) (Ω) := u : Ω → R measurable and
|u(x)|p(x) dx < ∞ ,
Ω
where p(x) ∈ C+ (Ω) and
C+ (Ω) := p ∈ C(Ω) : p(x) > 1 , for any x ∈ Ω.
Denote
p+ = max p(x),
p− = min p(x),
x∈Ω
x∈Ω
and for any x ∈ Ω, k ≥ 1,
(
∗
p (x) :=
(
p∗k (x) :=
N p(x)
N −p(x)
if p(x) < N,
+∞
if p(x) ≥ N,
N p(x)
N −kp(x)
if kp(x) < N,
+∞
if kp(x) ≥ N.
One introduces in Lp(x) (Ω) the norm
Z
u(x) p(x)
|
dx ≤ 1 .
|u|p(x) = inf α > 0 :
|
α
Ω
The space (Lp(x) (Ω), | · |p(x) ) is a Banach space.
Proposition 2.1 ([8]). The space (Lp(x) (Ω), |·|p(x) ) is separable, uniformly convex,
reflexive and its conjugate space is Lq(x) (Ω) where q(x) is the conjugate function of
p(x); i.e.,
1
1
+
= 1,
p(x) q(x)
for all x ∈ Ω. For u ∈ Lp(x) (Ω) and v ∈ Lq(x) (Ω) we have
Z
1
1 u(x)v(x)dx ≤ − + − |u|p(x) |v|q(x) .
p
q
Ω
The Sobolev space with variable exponent W k,p(x) (Ω) is defined as
W k,p(x) (Ω) = u ∈ Lp(x) (Ω) : Dα u ∈ Lp(x) (Ω), |α| ≤ k ,
|α|
where Dα u = ∂xα1 ∂x∂α2 ...∂xαN u with α = (α1 , . . . , αN ) is a multi-index and |α| =
1
2
N
PN
k,p(x)
(Ω), equipped with the norm
i=1 αi . The space W
X
kukk,p(x) :=
|Dα u|p(x) ,
|α|≤k
4
L. LI, L. DING, W.-W. PAN
EJDE-2013/139
also becomes a Banach, separable and reflexive space. For more details, we refer
the reader to [5, 6, 7, 8].
Proposition 2.2 ([8]). For p, r ∈ C+ (Ω) such that r(x) ≤ p∗k (x) for all x ∈ Ω,
there is a continuous and compact embedding
W k,p(x) (Ω) ,→ Lr(x) (Ω).
k,p(x)
We denote by W0
(Ω) the closure of C0∞ (Ω) in W k,p(x) (Ω).
3. Properties of the p(x)-biharmonic operator
Note that the weak solutions of (1.1) are considered in the generalized Sobolev
space
1,p(x)
X := W 2,p(x) (Ω) ∩ W0
(Ω),
equipped with the norm
Z o
n
u(x) p(x) ∆u(x) p(x)
|
+|
|
dx ≤ 1 .
kuk = inf α > 0 :
|
α
α
Ω
Remark 3.1. (1) According to [17], the norm k · k2,p(x) , cited in the preliminaries,
is equivalent to the norm |∆ · |p(x) in the space X. Consequently, the norms k ·
k2,p(x) , k · k and |∆ · |p(x) are equivalent.
(2) By the above remark and Proposition 2.2, there is a continuous and compact
embedding of X into Lq(x) (Ω), where q(x) < p∗2 (x) for all x ∈ Ω.
We consider the functional
Z
Φ(u) =
Ω
1
|∆u|p(x) + |u|p(x) dx,
p(x)
It is well known that Φ(u) is well defined and continuous differentiable in X. Now
we give the following fundamental proposition.
Proposition 3.2. For u ∈ X we have
(1)
(2)
(3)
(4)
(5)
kuk < (=; >)1 ⇔ Φ(u) < (=; >)1,
+
−
kuk ≤ 1 ⇒ kukp ≤ Φ(u) ≤ kukp ,
−
+
kuk ≥ 1 ⇒ kukp ≤ Φ(u) ≤ kukp , for all un ∈ X we have
kun k → 0 ⇔ Φ(un ) → 0,
kun k → ∞ ⇔ Φ(un ) → ∞.
The proof of this proposition is similar to the proof in [8, Theorem 1.3]. Moreover, the operator T := Φ0 : X → X 0 defined as
Z
hT (u), vi = (|∆u|p(x)−2 ∆u∆v + |u|p(x)−2 uv)dx for any u, v ∈ X,
Ω
satisfies the assertions of the following theorem.
Theorem 3.3. The following statements hold:
(1) T is continuous, bounded and strictly monotone.
(2) T is of (S+ ) type.
(3) T is a homeomorphism.
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EXISTENCE OF MULTIPLE SOLUTIONS
5
Proof. (1) Since T is the Fréchet derivative of Φ, it follows that T is continuous
and bounded. Let us define the sets
Up = {x ∈ Ω : p(x) ≥ 2},
Vp = {x ∈ Ω : 1 < p(x) < 2}.
Using the elementary inequalities [16]
|x − y|γ ≤ 2γ (|x|γ−2 x − |y|γ−2 y)(x − y) if γ ≥ 2,
1
(|x| + |y|)2−γ (|x|γ−2 x − |y|γ−2 y)(x − y) if 1 < γ < 2,
|x − y|2 ≤
(γ − 1)
for all (x, y) ∈ RN × RN , we obtain for all u, v ∈ X such that u 6= v,
hT (u) − T (v), u − vi > 0,
which means that T is strictly monotone.
(2) Let (un )n be a sequence of X such that
un * u weakly in X
lim suphT (un ), un − ui ≤ 0.
and
n→+∞
From Proposition 3.2, it suffices to shows that
Z
(|∆un − ∆u|p(x) + |un − u|p(x) )dx → 0.
(3.1)
Ω
In view of the monotonicity of T , we have
hT (un ) − T (u), un − ui ≥ 0,
and since un * u weakly in X, it follows that
lim suphT (un ) − T (u), un − ui = 0.
(3.2)
n→+∞
Put
ϕn (x) = (|∆un |p(x)−2 ∆un − |∆u|p(x)−2 ∆u)(∆un − ∆u),
ψn (x) = (|un |p(x)−2 un − |u|p(x)−2 u)(un − u).
By the compact embedding of X into Lp(x) (Ω), it follows that
un → u in Lp(x) (Ω),
|un |p(x)−2 un → |u|p(x)−2 u in Lq(x) (Ω),
where 1/q(x) + 1/p(x) = 1 for all x ∈ Ω. It results that
Z
ψn (x)dx → 0.
(3.3)
Ω
It follows by (3.2) and (3.3) that
Z
lim sup
n→+∞
ϕn (x)dx = 0.
Thanks to the above inequalities,
Z
Z
+
|∆un − ∆uk |p(x) dx ≤ 2p
Up
Z
Up
(3.4)
Ω
ϕn (x)dx,
Up
|un − uk |p(x) dx ≤ 2p
+
Z
ψn (x)dx.
Up
6
L. LI, L. DING, W.-W. PAN
EJDE-2013/139
Then
Z
|∆un − ∆u|p(x) + |un − u|p(x) dx → 0
as n → +∞.
(3.5)
Up
On the other hand, in Vp , setting δn = |∆un | + |∆u|, we have
Z
Z
p(x)
p(x)
1
p(x)
|∆un − ∆u|
dx ≤ −
(ϕn ) 2 (δn ) 2 (2−p(x)) dx .
p
−
1
Vp
Vp
For d > 0, by Young’s inequality,
Z
Z
p(x)
p(x)
d
|∆un − ∆u|p(x) dx ≤
[d(ϕn ) 2 ](δn ) 2 (2−p(x)) dx,
Vp
Vp
Z
2
≤
(3.6)
Z
(δn )p(x) dx.
ϕn (d) p(x) dx +
Vp
Vp
From (3.4) and since ϕn ≥ 0, one can consider that
Z
ϕn dx < 1.
0≤
Vp
If
R
Vp
p(x)
R
ϕn dx = 0 then
Vp
R
|∆un − ∆u|
dx = 0. If 0 < Vp ϕn dx < 1, we choose
−1/2
Z
ϕn (x)dx
> 1,
d=
Vp
and the fact that 2/p(x) < 2, inequality (3.6) becomes
Z
Z
Z
1
p(x)
2
|∆un − ∆u|
dx ≤
ϕn d dx +
δnp(x) dx ,
d Vp
Ω
Vp
Z
Z
1/2 ≤
ϕn dx
1+
δnp(x) dx .
Vp
Note that,
R
Ω
p(x)
dx
δ
Ω n
is bounded, which implies
Z
|∆un − ∆u|p(x) dx → 0 as n → +∞.
Vp
A similar method gives
Z
|un − u|p(x) dx → 0
as n → +∞.
Vp
Hence, it result that
Z
(|∆un − ∆u|p(x) + |un − u|p(x) )dx → 0
as n → +∞.
(3.7)
Vp
Finally, (3.1) is given by combining (3.5) and (3.7).
(3) Note that the strict monotonicity of T implies its injectivity. Moreover, T is
a coercive operator. Indeed, since p− − 1 > 0, for each u ∈ X such that kuk ≥ 1
we have
−
hT (u), ui
Φ(u)
=
≥ kukp −1 → ∞ as kuk → ∞.
kuk
kuk
Consequently, thanks to Minty-Browder theorem [18], the operator T is an surjection and admits an inverse mapping. It suffices then to show the continuity of T −1 .
EJDE-2013/139
EXISTENCE OF MULTIPLE SOLUTIONS
7
Let (fn )n be a sequence of X 0 such that fn → f in X 0 . Let un and u in X such
that
T −1 (fn ) = un and T −1 (f ) = u.
By the coercivity of T , one deducts that the sequence (un ) is bounded in the
reflexive space X. For a subsequence, we have un * u
b in X, which implies
lim hT (un ) − T (u), un − u
bi = lim hfn − f, un − u
bi = 0.
n→+∞
n→+∞
It follows by the second assertion and the continuity of T that
un → u
b in X
and T (un ) → T (b
u) = T (u)
in X 0 .
Moreover, since T is an injection, we conclude that u = u
b.
4. Proof of main theorem
For the reader’s convenience, we recall the revised form of Ricceri’s three critical
points theorem [13, Theorem 1] and [11, Proposition 3.1].
Theorem 4.1 ([13, Theorem 1]). Let X be a reflexive real Banach space. Φ : X →
R is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X 0 and
Φ is bounded on each bounded subset of X; Ψ : X → R is a continuously Gâteaux
differentiable functional whose Gâteaux derivative is compact; I ⊆ R an interval.
Assume that
lim (Φ(x) + λΨ(x)) = +∞
(4.1)
kxk→+∞
for all λ ∈ I, and that there exists h ∈ R such that
sup inf (Φ(x) + λ(Ψ(x) + h)) < inf sup(Φ(x) + λ(Ψ(x) + h)).
λ∈I x∈X
x∈X λ∈I
(4.2)
Then, there exists an open interval Λ ⊆ I and a positive real number ρ with the
following property: for every λ ∈ Λ and every C 1 functional J : X 7→ R with
compact derivative, there exists δ > 0 such that, for each µ ∈ [0, δ] the equation
Φ0 (x) + λΨ0 (x) + µJ 0 (x) = 0
has at least three solutions in X whose norms are less than ρ.
Proposition 4.2 ([11, Proposition 3.1]). Let X be a non-empty set and Φ, Ψ two
real functions on X. Assume that there are r > 0 and x0 , x1 ∈ X such that
Φ(x0 ) = −Ψ(x0 ) = 0,
Φ(x1 ) > r,
sup
−Ψ(x) < r
x∈Φ−1 (]−∞,r])
−Ψ(x1 )
.
Φ(x1 )
Then, for each h satisfying
sup
−Ψ(x) < h < r
x∈Φ−1 (]−∞,r])
−Ψ(x1 )
,
Φ(x1 )
one has
sup inf (Φ(x) + λ(h + Ψ(x))) < inf sup(Φ(x) + λ(h + Ψ(x))).
λ≥0 x∈X
x∈X λ≥0
Now we can give the proof of our main result.
8
L. LI, L. DING, W.-W. PAN
EJDE-2013/139
Proof Theorem 1.1. Set Φ(u), Ψ(u) and J(u) as (1.3), (1.4) and (1.5). So, for each
u, v ∈ X, one has
Z
0
hΦ (u), vi = (|∆u|p(x)−2 ∆u∆v + |u|p(x)−2 uv) dx,
Ω
Z
0
hΨ (u), vi = −
f (x, u)v dx,
ZΩ
hJ 0 (u), vi = −
g(x, u)v dx.
Ω
From Theorem 3.3, of course, Φ is a continuous Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a
continuous inverse on X 0 , moreover, Ψ and J are continuously Gâteaux differentiable functionals whose Gâteaux derivative is compact. Obviously, Φ is bounded
on each bounded subset of X under our assumptions.
From Proposition 3.2, we have: if kuk ≥ 1, then
−
+
1
1
kukp ≤ Φ(u) ≤ − kukp .
+
p
p
(4.3)
Meanwhile, for each λ ∈ Λ,
Z
λΨ(u) = −λ
F (x, u)dx
ZΩ
≥ −λ
ϑ(1 + |u|γ(x) )dx
Ω
+
≥ −λϑ(|Ω| + |u|γγ(x) )
+
≥ −C2 (1 + |u|γγ(x) )
+
≥ −C3 (1 + kukγ )
for any u ∈ X, where C2 and C3 are positive constants. Here, we use condition (I3)
and (ii) of Proposition 2.1. Combining the two inequalities above, we obtain
Φ(u) + λΨ(u) ≥
−
+
1
kukp − C3 (1 + kukγ ),
+
p
because of γ + < p− , it follows that
lim
(Φ(u) + λΨ(u)) = +∞
kuk→+∞
∀u ∈ X,
λ ∈ [0, +∞).
Then assumption (4.1) of Theorem 4.1 is satisfied.
Next, we will prove that assumption (4.2) is also satisfied. It suffices to verify
the conditions of Proposition 4.2. Let u0 = 0, we can easily have
Φ(u0 ) = −Ψ(u0 ) = 0.
Now we claim that (4.2) is satisfied.
From (I2), exist η ∈ [0, 1], C4 > 0, such that
−
F (x, s) < C4 |s|p1 (x) < C4 |s|p1
∀s ∈ [−η, η], a.e. x ∈ Ω.
Then, from (I3), we can find a constant M such that
−
F (x, s) < M |s|p1
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EXISTENCE OF MULTIPLE SOLUTIONS
9
for all s ∈ R and a.e. x ∈ Ω. Consequently, by the Sobolev embedding theorem
−
(X ,→ Lp1 (Ω) is continuous), we have (for suitable positive constant C5 , C6 )
Z
Z
−
−
−
+
−Ψ(u) =
F (x, u)dx < M
|u|p1 dx ≤ C5 kukp1 ≤ C6 rp1 /p ,
Ω
p+
when kuk
Ω
+
/p ≤ r. Hence, being
lim+
p−
1
> p+ , it follows that
supkukp+ /p+ ≤r −Ψ(u)
= 0.
(4.4)
r
Let u1 ∈ C 2 (Ω) be a function positive in Ω, with u1 |∂Ω = 0 and maxΩ u1 ≤ d.
Then,
of course, u1 ∈ X and Φ(u1 ) > 0. In view of (i1 ) we also have −Ψ(u1 ) =
R
F
(x,
u1 (x))dx > 0. Therefore, from (4.4), we can find r ∈ 0, min{Φ(u1 ), p1+ }
Ω
such that
−Ψ(u1 )
.
sup
(−Ψ(u)) < r
Φ(u1 )
kukp+ /p+ ≤r
R
Now, let u ∈ Φ−1 ((−∞, r]). Then, Ω (|∆u|p(x) + |u|p(x) )dx ≤ rp+ < 1 which, by
Proposition 3.2, implies kuk < 1. Consequently,
Z
1
1
p+
kuk ≤
(|∆u|p(x) + |u|p(x) )dx < r.
p+
p(x)
Ω
n
o
+
Therefore, we infer that Φ−1 ((−∞, r]) ⊂ u ∈ X : p1+ kukp < r , and so
r→0
−Ψ(u) < r
sup
u∈Φ−1 (]−∞,r])
−Ψ(u1 )
.
Φ(u1 )
At this point, conclusion follows from Proposition 4.2 and Theorem 4.1.
Acknowledgments. The authors are very grateful to the anonymous referees for
their knowledgeable reports, which helped us to improve our manuscript.
The first and the third author were supported by grant XDJK2013D007 from
the Fundamental Research Funds for the Central Universities, grant 2011KY03
from the Scientific Research Fund of SUSE, and grant 12ZB081 from the Scientific
Research Fund of SiChuan Provincial Education Department.
The second author was supported by grant 11101347 from the National Natural
Science Foundation of China, grant 2012M510363 from the Postdoctor Foundation
of China, and grants D20112605, D20122501 from the Key Project in Science and
Technology Research Plan of the Education Department of Hubei Province.
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Lin Li
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
E-mail address: lilin420@gmail.com
Ling Ding
School of Mathematics and Computer Science, Hubei University of Arts and Science,
Hubei 441053, China
E-mail address: 591517149@qq.com
Wen-Wu Pan
Department of Science, Sichuan University of Science and Engineering, Zigong 643000,
China
E-mail address: 23973445@qq.com