Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 132, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A
STEKLOV PROBLEM INVOLVING THE P(X)-LAPLACE
OPERATOR
MOSTAFA ALLAOUI, ABDEL RACHID EL AMROUSS, ANASS OURRAOUI
Abstract. In this article we study the nonlinear Steklov boundary-value
problem
∆p(x) u = |u|p(x)−2 u
|∇u|
in Ω,
p(x)−2 ∂u
= λf (x, u) on ∂Ω.
∂ν
Using the variational method, under appropriate assumptions on f , we obtain
results on existence and multiplicity of solutions.
1. Introduction
Motivated by the developments in elastic mechanics, electrorheological fluids
and image restoration [4, 20, 22, 26, 27], the interest in variational problems and
differential equations with variable exponent has grown in recent decades; see for
example [5, 13, 14, 19]. We refer the reader to [3, 6, 7, 10, 11, 12, 18, 23, 24, 25]
for developments in p(x)-Laplacian equations.
The purpose of this article is to study the existence and multiplicity of solutions
for the Steklov problem involving the p(x)-Laplacian,
∆p(x) u = |u|p(x)−2 u
in Ω,
(1.1)
∂u
= λf (x, u) on ∂Ω,
∂ν
where Ω ⊂ RN (N ≥ 2) is a bounded smooth domain, ∂u
∂ν is the outer unit normal
derivative on ∂Ω, λ > 0 is a real number, p is a continuous function on Ω with
p− := inf x∈Ω p(x) > 1. The main interest in studying such problems arises from the
presence of the p(x)-Laplace operator div(|∇u|p(x)−2 ∇u), which is a generalization
of the classical p-Laplace operator div(|∇u|p−2 ∇u) obtained in the case when p
is a positive constant. Many authors have studied the inhomogeneous Steklov
problems involving the p-Laplacian [17]. The authors have studied this class of
inhomogeneous Steklov problems in the cases of p(x) ≡ p = 2 and of p(x) ≡ p > 1,
respectively.
|∇u|p(x)−2
2000 Mathematics Subject Classification. 35J48, 35J60, 35J66.
Key words and phrases. p(x)-Laplace operator; variable exponent Lebesgue space;
variable exponent Sobolev space; Ricceri’s variational principle.
c
2012
Texas State University - San Marcos.
Submitted February 24, 2012. Published August 15, 2012.
1
2
M. ALLAOUI, A. R. EL AMROUSS, A. OURRAOUI
EJDE-2012/132
We make the following assumptions on the function f :
(H0) f : ∂Ω × R → R satisfies the carathéodory condition and there exists a
constant C ≥ 0 such that:
|f (x, s)| ≤ C(1 + |s|β(x)−1 )
for all (x, s) ∈ ∂Ω × R,
where β(x) ∈ C(∂Ω), β(x) > 1 and β(x) < p∂ (x) for all x ∈ ∂Ω.
(H1) There exist R > 0, µ > p+ such that for all |s| ≥ R and x ∈ ∂Ω,
0 < µF (x, s) ≤ f (x, s)s.
p+ −1
(H2) f (x, s) = o(|s|
) as s → 0 and uniformly for x ∈ ∂Ω.
(H3) f (x, −s) = −f (x, s), x ∈ ∂Ω, s ∈ R.
The main results of this paper are as follows.
Theorem 1.1. If (H0), (H1), (H2) hold and β − > p+ , then for any λ ∈ (0, +∞),
(1.1) has at least a nontrivial weak solution.
Theorem 1.2. If (H0), (H1), (H3) hold and β − > p+ , then for any λ ∈ (0, +∞),
(1.1) has infinite many pairs of weak solutions.
For the next theorem we assume that f satisfies the following conditions:
(F1) |f (x, s)| ≤ a(x) + b|s|α(x)−1 , for all (x, s) ∈ ∂Ω × R, where a(x) is in
α(x)
L α(x)−1 (∂Ω), b ≥ 0 is a constant, α(x) ∈ C(∂Ω), 1 < α− := inf x∈Ω α(x) ≤
α+ := supx∈Ω α(x) < p− and p(x) > N .
(F2) f (x, t) < 0, when |t| ∈ (0, 1), f (x, t) ≥ m > 0, when t ∈ (t0 , ∞), t0 > 1.
Theorem 1.3. If (F1), (F2) hold, then there exist an open interval Λ ⊂ (0, ∞)
and a positive real number ρ such that each λ ∈ Λ, (1.1) has at least three solutions
whose norms are less than ρ.
The special features of the of the problems considered in this paper are that they
involve the variable exponent. To prove theorems (1.1)-(1.3) we use the theory of
variable exponent Sobolev spaces, established first by Kováčik and Rákosnı́k [16],
and some research results obtained recently for the p(x)-Laplacian equations. For
the proof of theorem (1.1), we will use the Mountain Pass Theorem. For the proof
of theorem (1.2), we will use the Fountain theorem. For the proof of theorem (1.3),
we will use Ricceri three-critical-points theorem.
This article is organized as follows. First, we will introduce some basic preliminary results and lemmas in Section 2. In Section 3, we will give the proofs of our
main results.
2. Preliminaries
For completeness, we first recall some facts on the variable exponent spaces
Lp(x) (Ω) and W k,p(x) (Ω). For more details, see [8, 9]. Suppose that Ω is a bounded
open domain of RN with smooth boundary ∂Ω and p ∈ C+ (Ω) where
C+ (Ω) = {p ∈ C(Ω)
and
inf p(x) > 1}.
x∈Ω
Denote by p− := inf x∈Ω p(x) and p+ := supx∈Ω p(x). Define the variable exponent
Lebesgue space Lp(x) (Ω) by
Z
p(x)
L
(Ω) = {u : Ω → R is a measurable and
|u|p(x) dx < +∞},
Ω
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EXISTENCE AND MULTIPLICITY OF SOLUTIONS
with the norm
3
Z
u
| |p(x) dx ≤ 1}.
Ω τ
Define the variable exponent Sobolev space W 1,p(x) (Ω) by
|u|p(x) = inf{τ > 0;
W 1,p(x) (Ω) = {u ∈ Lp(x) (Ω) : |∇u| ∈ Lp(x) (Ω)},
with the norm
u
∇u p(x)
|
+ | |p(x) )dx ≤ 1},
τ
τ
Ω
kuk = |∇u|p(x) + |u|p(x) .
Z
kuk = inf{τ > 0;
(|
We refer the reader to [8, 9] for the basic properties of the variable exponent
Lebesgue and Sobolev spaces.
Lemma 2.1 ([9]). Both (Lp(x) (Ω), | · |p(x) ) and (W 1,p(x) (Ω), k · k) are separable,
reflexive and uniformly convex Banach spaces.
Lemma 2.2 ([9]). Hölder inequality holds, namely
Z
|uv|dx ≤ 2|u|p(x) |v|q(x) ∀u ∈ Lp(x) (Ω), v ∈ Lq(x) (Ω),
where
1
p(x)
+
Ω
1
q(x)
= 1.
R
Lemma 2.3 ([9]). Let I(u) = Ω (|∇u|p(x) + |u|p(x) )dx, for u ∈ W 1,p(x) (Ω) we have
• kuk < 1(= 1, > 1) ⇔ I(u) < 1(= 1, > 1).
−
+
• kuk ≤ 1 ⇒ kukp ≤ I(u) ≤ kukp .
+
−
• kuk ≥ 1 ⇒ kukp ≤ I(u) ≤ kukp .
Lemma 2.4 ([8]). Assume that the boundary of Ω possesses the cone property and
p ∈ C(Ω) and 1 ≤ q(x) < p∗ (x) for x ∈ Ω, then there is a compact embedding
W 1,p(x) (Ω) ,→ Lq(x) (Ω), where
(
N p(x)
, if p(x) < N ;
∗
p (x) = N −p(x)
+∞,
if p(x) ≥ N.
Lemma 2.5 ([9]). If f : Ω × R → R is a carathéodory function and
p1 (x)
|f (x, s)| ≤ a(x) + b|s| p2 (x) ,
∀(x, s) ∈ Ω × R,
p2 (x)
where p1 (x), p2 (x) ∈ C(Ω), a(x) ∈ L
(Ω), p2 (x) > 1, p2 (x) > 1, a(x) ≥ 0 and
b ≥ 0 is a constant, then the Nemytskii operator from Lp1 (x) (Ω) to Lp2 (x) (Ω) defined
by Nf (u)(x) = f (x, u(x)) is a continuous and bounded operator.
Let a : ∂Ω → R be a measurable. Define the weighted variable exponent
Lebesgue space by
Z
p(x)
|a(x)||u|p(x) dσ < +∞},
La(x) (∂Ω) = {u : ∂Ω → R is measurable and
∂Ω
with the norm
Z
|u|(p(x),a(x)) = inf{τ > 0;
u
|a(x)| | |p(x) dσ ≤ 1},
τ
∂Ω
p(x)
where dσ is the measure on the boundary. Then La(x) (∂Ω) is a Banach space. In
p(x)
particular, when a ∈ L∞ (∂Ω), La(x) (∂Ω) = Lp(x) (∂Ω).
4
M. ALLAOUI, A. R. EL AMROUSS, A. OURRAOUI
Lemma 2.6 ([5]). Let ρ(u) =
EJDE-2012/132
p(x)
R
∂Ω
|a(x)||u|p(x) dσ for u ∈ La(x) (∂Ω) we have
−
+
+
−
• |u|(p(x),a(x)) ≥ 1 ⇒ |u|p(p(x),a(x)) ≤ ρ(u) ≤ |u|p(p(x),a(x)) .
• |u|(p(x),a(x)) ≤ 1 ⇒ |u|p(p(x),a(x)) ≤ ρ(u) ≤ |u|p(p(x),a(x)) .
For A ⊂ Ω, denote by p− (A) = inf x∈A p(x), p+ (A) = supx∈A p(x). Define
(
(N −1)p(x)
∂
∂
N −p(x) , if p(x) < N,
p (x) = (p(x)) :=
∞,
if p(x) ≥ N.
p∂r(x) (x) :=
r(x) − 1 ∂
p (x),
r(x)
where x ∈ ∂Ω, r ∈ C(∂Ω, R) and r(x) > 1.
Lemma 2.7 ([5]). Assume that the boundary of Ω possesses the cone property and
∂
(x)
p ∈ C(Ω) with p− > 1. Suppose that a ∈ Lr(x) (∂Ω), r ∈ C(∂Ω) with r(x) > p∂p(x)−1
for all x ∈ ∂Ω. If q ∈ C(∂Ω) and 1 ≤ q(x) < p∂r(x) (x), ∀x ∈ ∂Ω. Then there is
q(x)
a compact embedding W 1,p(x) (Ω) ,→ La(x) (∂Ω). In particular, there is a compact
embedding W 1,p(x) (Ω) ,→ Lq(x) (∂Ω), where 1 ≤ q(x) < p∂ (x), ∀x ∈ ∂Ω.
Lemma 2.8 ([2, 15, 21]). Let X be a separable and reflexive real Banach space,
φ : X → R is a continuous Gâteaux differentiable and sequentially weakly lower
semicontinuous functional whose Gâteaux derivative admits a continuous inverse
on X ∗ ; Ψ : X → R is a continuous Gâteaux differentiable functional whose Gâteaux
derivative is compact, assume that:
(i) limkukX →∞ (φ(u) + λψ(u)) = ∞ for all λ > 0,
(ii) there exist r ∈ R and u0 , u1 ∈ X such that φ(u0 ) < r < φ(u1 ),
(iii)
inf
u∈φ−1 (−∞,r]
ψ(u) >
(φ(u1 ) − r)ψ(u0 ) + (r − φ(u0 ))ψ(u1 )
.
φ(u1 ) − φ(u0 )
Then there exist an open interval Λ ⊂ (0, ∞) and a positive constant ρ > 0 such
that for any λ ∈ Λ the equation φ0 (u) + λψ 0 (u) = 0 has at least three solutions in
X whose norms are less than ρ.
Theorem 2.9. If f : ∂Ω × R → R is a carathéodory function and
(F1) |f (x, s)| ≤ a(x) + b|s|α(x)−1 , for all (x, s) ∈ ∂Ω × R,
α(x)
where a(x) ∈ L α(x)−1 (∂Ω) and b ≥ 0 is a constant, α(x) ∈ C+ (∂Ω) such that for
all x ∈ ∂Ω,
(
(N −1)p(x)
N −p(x) , if p(x) < N ;
α(x) <
(2.1)
+∞,
if p(x) ≥ N.
Ru
R
Set X = W 1,p(x) (Ω), F (x, u) = 0 f (x, t)dt, ψ(u) = − ∂ΩR F (x, u(x))dσ,
then ψ(u) ∈ C 1 (X, R) and Dψ(u, ϕ) =< ψ 0 (u), ϕ >= − ∂Ω f (x, u(x))ϕdσ, moreover, the operator ψ 0 : X → X ∗ is compact.
EJDE-2012/132
EXISTENCE AND MULTIPLICITY OF SOLUTIONS
5
Proof. By the Mean-value theorem, we have
ψ(u + tϕ) − ψ(u)
t
Z
F (x, u(x) + tϕ(x)) − F (x, u(x)
dσ
= − lim
t→0 ∂Ω
t
Z
= − lim
f (x, u(x) + tθϕ(x))ϕ(x)dσ,
Dψ(u, ϕ) = lim
t→0
t→0
(2.2)
∂Ω
where 0 ≤ θ = θ(u(x), tϕ(x)) ≤ 1. If u, ϕ ∈ X, then by condition (2.1) and the
embedding theorem (lemma2.7), we have u, ϕ ∈ Lα(x) (∂Ω). Then there is some
constant C such that
kwkLα(x) (∂Ω) ≤ CkwkX
∀w ∈ X.
(2.3)
By (F1) and Young’s inequality, we have
|f (x, u(x) + tθϕ(x))ϕ(x)|
≤ [a(x) + b|u(x) + tθϕ(x)|α(x)−1 ]|ϕ(x)|
(2.4)
α(x)
1
α(x) − 1
[a(x) + b|u(x) + tθϕ(x)|α(x)−1 ] α(x)−1 +
|ϕ(x)|α(x) .
≤
α(x)
α(x)
Using the inequality
(a + b)p ≤ 2p−1 (|a|p + |b|p ),
p ≥ 1,
we have
α(x)
α(x) − 1
1
[a(x) + b|u(x) + tθϕ(x)|α(x)−1 ] α(x)−1 +
|ϕ(x)|α(x)
α(x)
α(x)
α(x)
α(x)
1
1
(α(x) − 1) α(x)−1
2
[(a(x)) α(x)−1 + b α(x)−1 |u(x) + tθϕ(x)|α(x) ] +
|ϕ(x)|α(x)
≤
α(x)
α(x)
α(x)
α(x)
1
(α(x) − 1) α(x)−1
2
≤
[(a(x)) α(x)−1 + 2α(x)−1 b α(x)−1 [|u(x)|α(x) + |ϕ(x)|α(x) ]]
α(x)
1
|ϕ(x)|α(x) ,
+
α(x)
for |t| ≤ 1. Note that the right hand side of the above inequality is independent of
t and integrable on ∂Ω, then by the Lebesgue dominated convergence theorem, we
have
Z
Dψ(u, ϕ) = −
f (x, u(x))ϕ(x)dσ.
(2.5)
∂Ω
Obviously the operator Dψ(u, ϕ) is a linear operator for a given u. We know that
the Nemytskii operator Nf : u(x) 7→ f (x, u(x)) is a continuous bounded operator
α(x)
from Lα(x) (∂Ω) into L α(x)−1 (∂Ω). Then by (2.3) and (2.5) we have
Z
f (x, u(x))ϕ(x)dσ ≤ 2Ckf (x, u)k α(x)
Dψ(u, ϕ) = −
kϕ(x)kX .
L α(x)−1 (∂Ω)
∂Ω
So Dψ(u, ϕ) is a linear bounded functional, therefore the Gâteaux derivative of the
linear bounded functional ψ(u) exists and
Z
Dψ(u, ϕ) =< Dψ(u), ϕ >= −
f (x, u(x))ϕ(x)dσ ∀u, ϕ ∈ X.
(2.6)
∂Ω
6
M. ALLAOUI, A. R. EL AMROUSS, A. OURRAOUI
EJDE-2012/132
We will prove that ψ 0 : X → X ∗ is completely continuous. For u, v, ϕ ∈ X, from
(2.5) and (2.6), we obtain
|hDψ(u) − Dψ(v), ϕi| ≤ 2Ckf (x, u) − f (x, v)k
α(x)
L α(x)−1 (∂Ω)
kϕkX .
Then
kDψ(u) − Dψ(v)kX ∗ ≤ 2Ckf (x, u) − f (x, v)k
.
α(x)
L α(x)−1 (∂Ω)
α(x)
The above inequality shows that the operator T : L α(x)−1 (∂Ω) → X ∗ defined by
T (f (x, u)) = Dψ(u) is continuous. Then the composite operator Dψ = T oNf oI :
u → Dψ(u) from X into X ∗ is continuous. Therefore, ψ is Frèchet differentiable
1
and its Frèchet derivative ψ 0 (u)
R = Dψ(u) . This shows0 that ψ(u) ∗∈ C (X, R),
0
Dψ(u, ϕ) =< ψ (u), ϕ >= − ∂Ω f (x, u(x))ϕ(x)dσ and ψ : X → X is compact.
We say that u ∈ X is a weak solution of (1.1) if
Z
Z
Z
|∇u|p(x)−2 ∇u∇v dx +
|u|p(x)−2 uv dx = λ
Ω
Ω
f (x, u)vdσ
for all v ∈ X.
∂Ω
Let
Z
1
(|∇u|p(x) + |u|p(x) )dx,
Ω p(x)
Z
ψ(u) = −
F (x, u)dσ,
φ(u) =
∂Ω
J(u) = φ(u) + λψ(u),
where F (x, t) =
Rt
f (x, s)ds. Then we have
Z
(φ0 (u), v) = (|∇u|p(x)−2 ∇u∇v + |u|p(x)−2 uv)dx,
Ω
Z
0
(ψ (u), v) = −
f (x, u)vdσ.
0
∂Ω
3. Proof of main results
For the proof of theorem 1.1, we will use the Mountain Pass Theorem. We start
with the following lemmas.
Lemma 3.1. If (H0), (H1) hold, then for any λ ∈ (0, +∞) the functional J satisfies
the Palais Smale condition (PS).
Proof. Suppose that (un ) ⊂ X is a (PS) sequence; i.e.,
sup |J(un )| ≤ M, J 0 (un ) → 0
as n → ∞.
Let us show that (un ) is bounded in X. Using hypothesis (H1), since J(un ) is
bounded, we have for n large enough:
1
1
M + 1 ≥ J(un ) − hJ 0 (un ), un i + hJ 0 (un ), un i
µ
µ
Z
Z
1
=
(|∇un |p(x) + |un |p(x) )dx − λ
F (x, un )dσ
Ω p(x)
∂Ω
Z
Z
i 1
1h
−
(|∇un |p(x) + |un |p(x) )dx − λ
f (x, un )un dσ + hJ 0 (un ), un i
µ Ω
µ
∂Ω
EJDE-2012/132
EXISTENCE AND MULTIPLICITY OF SOLUTIONS
7
−
1
1
1
− )kun kp − kJ 0 (un )kX ∗ kun k − C
+
p
µ
µ
1
C1
1
p−
kun k − C,
≥ ( + − )kun k −
p
µ
µ
where C and C1 are two constants. From the inequality above, we know that un is
bounded in X since µ > p+ . The proof is complete.
≥(
Lemma 3.2. There exist r1 , C 0 > 0 such that J(u) ≥ C 0 for all u ∈ X such that
kuk = r1 .
Proof. Conditions (H0) and (H2) assure that
+
|F (x, s)| ≤ ε|s|p + C(ε)|s|β(x)
for all (x, s) ∈ ∂Ω × R.
For kuk small enough, we have
Z
+
1
F (x, u)dσ
J(u) ≥ + kukp − λ
p
∂Ω
Z
+
+
1
(ε|u|p + C(ε)|u|β(x) )dσ.
≥ + kukp − λ
p
∂Ω
(3.1)
Since p+ < β − ≤ β(x) < p∂ (x), for all x ∈ ∂Ω, we have
+
W 1,p(x) (Ω) ,→ Lp (∂Ω),
with a continuous and compact embedding, which implies the existence of C3 > 0
such that
|u|Lp+ (∂Ω) ≤ C3 kuk, ∀u ∈ X.
(3.2)
From (3.1) and (3.2), we have for kuk small enough
+
+
−
1
J(u) ≥ + kukp − λεC3 kukp − λC(ε)C3 kukβ .
p
Choose ε > 0 small enough that 0 < λεC3 <
1
2p+ ,
we obtain
+
−
1
kukp − C(λ, ε)C3 kukβ
+
2p
−
+
+
1
≥ kukp ( + − C(λ, ε)C3 kukβ −p ).
2p
J(u) ≥
−
+
Since p+ < β − , the function t 7→ ( 2p1+ − C(λ, ε)C3 tβ −p ) is strictly positive in a
neighborhood of zero. It follows that there exist r1 > 0 and C 0 > 0 such that
J(u) ≥ C 0
∀u ∈ X : kuk = r1 .
The proof is complete.
Proof of theorem 1.1. To apply the Mountain Pass Theorem, we must prove that
J(tu) → −∞ as t → +∞, for a certain u ∈ X. From condition (H1), we obtain
F (x, s) ≥ c|s|µ
for all (x, s) ∈ ∂Ω × R.
Let u ∈ X and t > 1, we have
Z p(x)
Z
t
J(tu) =
[|∇u|p(x) + |u|p(x) ]dx − λ
F (x, tu)dσ
Ω p(x)
∂Ω
Z
Z
1
p+
p(x)
p(x)
µ
≤t
[|∇u|
+ |u|
]dx − ct λ
|u|µ dσ.
Ω p(x)
∂Ω
8
M. ALLAOUI, A. R. EL AMROUSS, A. OURRAOUI
EJDE-2012/132
The fact µ > p+ , implies for any λ ∈ (0, +∞) J(tu) → −∞ as t → +∞.
It follows that there exists e ∈ X such that kek > r1 and J(e) < 0. According to
the Mountain Pass Theorem, J admits a critical value τ ≥ C 0 which is characterized
by
τ = inf sup J(h(t))
h∈Γ t∈[0,1]
where
Γ = {h ∈ C([0, 1], X) : h(0) = 0 and h(1) = e}.
This completes the proof.
Since X is a separable and reflexive Banach space [3, 8], there exist {en }∞
n=1 ⊂ X
∗
and {fn }∞
n=1 ⊂ X such that
(
1, if n = m;
fn (em ) = δn,m =
0, if n 6= m.
∗
X ∗ = spanw {fn : n = 1, 2, . . . }.
X = span{en : n = 1, 2, . . . },
For k = 1, 2, . . . denote by
Xn = span{en },
Yn = ⊕nj=1 Xj ,
Zn = ⊕∞
j=n Xj .
Lemma 3.3 ([5, 11]). For β(x) ∈ C+ (∂Ω), β(x) < p∂ (x) and x ∈ ∂Ω, let βk =
sup{|u|Lβ(x) (∂Ω) : kuk = 1, u ∈ Zk }. Then limk→∞ βk = 0.
Proof of theorem 1.2. We use the Fountain theorem [1]. Obviously, J is an even
functional and satisfies the (PS) condition. We will prove that if k is large enough,
then there exist ρk > rk > 0 such that:
(A1) bk := inf{J(u)/u ∈ Zk , kuk = rk } → +∞ as k → +∞,
(A2) ak := max{J(u)/u ∈ Yk , kuk = ρk } ≤ 0 as k → +∞.
(A1): For u ∈ Zk such that kuk = rk > 1, by condition (H0), we have
Z
Z
i
1 h
p(x)
p(x)
|∇u|
+ |u|
dx − λ
F (x, u)dσ
J(u) =
∂Ω
Ω p(x)
Z
−
1
≥ + kukp − λ
C(1 + |u|β(x) )dσ
p
∂Ω
+
−
−
1
≥ + kukp − λC max{|u|βLβ(x) (∂Ω) , |u|βLβ(x) (∂Ω) } − C1 .
p
It follows that
(
J(u) ≥
≥
1
p−
p+ kuk
1
p−
p+ kuk
− (C2 (λ) + C1 )
+
− C2 (λ)(βk kuk)β − C1
if |u|Lβ(x) (∂Ω) ≤ 1
if |u|Lβ(x) (∂Ω) > 1
−
+
1
kukp − C2 (λ)(βk kuk)β − C3 .
+
p
+
1
For rk = (C2 (λ)β + βkβ ) p− −β+ , we have
−
J(u) ≥ rkp (
1
1
− + ) − C3 .
+
p
β
Since βk → 0 and p+ < β + , we have rk → +∞
J(u) → +∞
as k → +∞. Consequently,
as kuk → +∞, u ∈ Zk .
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EXISTENCE AND MULTIPLICITY OF SOLUTIONS
9
So (A1) holds.
(A2): Condition (H1) implies
F (x, s) ≥ C1 |s|µ − C2 ,
∀(x, s) ∈ ∂Ω × R.
Let u ∈ Yk be such that kuk = ρk > rk > 1. Then
Z
+
1
C1 |s|µ − C2 dσ
J(u) ≤ − kukp − λ
p
∂Ω
Z
1
p+
|u|µ dσ + C3 .
≤ − kuk − λC1
p
∂Ω
Note that the space Yk has finite dimension, then all norms are equivalents and we
obtain
+
1
J(u) ≤ − kukp − λC2 kukµ + C3 .
p
Finally,
J(u) → −∞ as kuk → +∞, u ∈ Yk
because µ > p+ . The assertion (A2) is then satisfied and the proof of theorem 1.2
is complete.
Proof of theorem 1.3. For proving our result we use lemma 2.8. It is well known
that φ is a continuous convex functional, then it is weakly lower semicontinuous
and its inverse derivative is continuous, from theorem 2.9 the precondition of lemma
2.8 is satisfied. In following we must verify that the conditions (i), (ii) and (iii) in
lemma 2.8 are fulfilled.
For u ∈ X such that kukX ≥ 1, we have
Z
Z Z u(x)
ψ(u) = −
F (x, u)dσ = −
[
f (x, t)dt]dσ
∂Ω 0
Z ∂Ω
b
≤
[a(x)|u(x)| +
|u|α(x) ]dσ
α(x)
∂Ω
Z
b
kukLα(x) (∂Ω) + −
|u|α(x) dσ
≤ 2kak α(x)
α
L α(x)−1 (∂Ω)
∂Ω
Z
b
≤ 2Ckak α(x)
kukX + −
|u|α(x) dσ.
α
L α(x)−1 (∂Ω)
∂Ω
By the embedding theorem, we have u ∈ Lα(x) (∂Ω); therefore,
Z
+
α−
0
α+
|u|α(x) dσ ≤ max{kukα
Lα(x) (∂Ω) , kukLα(x) (∂Ω) } ≤ C kukX .
∂Ω
Then
|ψ(u)| ≤ 2Ckak
α(x)
L α(x)−1 (∂Ω)
kukX +
+
b 0
C kukα
X .
α−
On the other hand,
Z
φ(u) =
Ω
−
1
1
(|∇u|p(x) + |u|p(x) )dx ≥ + kukpX .
p(x)
p
Which implies that for any λ > 0,
φ(u) + λψ(u) ≥
+
1
λbC 0
p−
kuk
kuk
−
kukα
−
2λCkak
α(x)
X
X .
X
−
α(x)−1
p+
α
L
(∂Ω)
10
M. ALLAOUI, A. R. EL AMROUSS, A. OURRAOUI
EJDE-2012/132
For p− > α+ we have
(φ(u) + λψ(u)) = ∞,
lim
kukX →∞
then (i) of lemma 2.8 is verified.
lt remains to show (ii) and (iii) of this lemma (Ricceri). By (F2), it is clear
that F(x,t) is increasing for t ∈ (t0 , ∞) and decreasing for t ∈ (0, 1) uniformly
for x ∈ ∂Ω, and F (x, 0) = 0 is obvious, F (x, t) → +∞ when t → +∞ because
(F (x, t) ≥ mt uniformly on x). Then, there exists a real number δ > t0 such that
F (x, t) ≥ 0 = F (x, 0) ≥ F (x, τ ) ∀u ∈ X, t > δ, τ ∈ (0, 1).
Let a, b be two real numbers such that 0 < a < min{1, c1 } where c1 is a constant
which satisfies
kukC(Ω) ≤ c1 kukX ,
kukC(Ω) = sup |u(x)| .
x∈Ω
This inequality is well defined due to compactly embedding from W 1,p(x) (Ω) to
C(Ω) (because N < p− ).
−
We choose b > δ satisfying bp |Ω| > 1. When t ∈ [0, a] we have
F (x, t) ≤ F (x, 0) = 0.
Then
Z
Z
sup F (x, t)dσ ≤
∂Ω 0<t<a
F (x, 0)dσ = 0.
∂Ω
Furthermore, since b > δ we have
Z
F (x, b)dσ > 0.
∂Ω
Moreover,
+
1 ap
.
p+ bp−
c1
Z
F (x, b)dσ > 0.
∂Ω
Which implies
+
1 ap
sup F (x, t)dσ ≤ 0 < p+ p−
c1 b
∂Ω 0<t<a
Z
Z
F (x, b)dσ.
∂Ω
Let u0 , u1 ∈ X , u0 (x) = 0 and u1 (x) = b for any x ∈ Ω. We define r =
Clearly r ∈ (0, 1), φ(u0 ) = ψ(u0 ) = 0,
Z
1 p(x)
1 −
1
1 a +
φ(u1 ) =
b
dx ≥ + bp |Ω| > + 1 > + ( )p = r,
p
p
p c1
Ω p(x)
and
Z
Z
ψ(u1 ) = −
F (x, u1 (x))dσ =
F (x, b)dσ < 0.
∂Ω
∂Ω
So we have φ(u0 ) < r < φ(u1 ). Then (ii) of lemma 2.8 is verified.
On the other hand, we have
(φ(u1 ) − r)ψ(u0 ) + (r − φ(u0 ))ψ(u1 )
ψ(u1 )
−
= −r
φ(u1 ) − φ(u0 )
φ(u1 )
R
F (x, b)dσ
= r R ∂Ω 1 p(x)
> 0.
b
dx
Ω p(x)
1
a p+
p+ ( c1 ) .
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EXISTENCE AND MULTIPLICITY OF SOLUTIONS
11
R
Let u ∈ X be such that φ(u) ≤ r < 1. Set I(u) = Ω (|∇u|p(x) + |u|p(x) )dx. Since
1
1,p(x)
(Ω), we obtain
p+ I(u) ≤ φ(u) ≤ r, for u ∈ W
a +
I(u) ≤ p+ .r = ( )p < 1.
c1
It follows that kukX < 1 by lemma 2.3. We have
+
1
1
kukp ≤ + I(u) ≤ φ(u) ≤ r.
p+
p
Then
1
|u(x)| ≤ c1 kukX ≤ c1 (p+ .r) p+ = a ∀u ∈ X, x ∈ Ω, φ(u) ≤ r.
The above inequality shows that
Z
−
inf
ψ(u) =
sup
−ψ(u) ≤
sup F (x, t)dσ ≤ 0.
−1
u∈φ
(−∞,r]
u∈φ−1 (−∞,r]
∂Ω 0<t<a
Then
(φ(u1 ) − r)ψ(u0 ) + (r − φ(u0 ))ψ(u1 )
.
φ(u1 ) − φ(u0 )
Which means that condition (iii) in lemma 2.8 is obtained. Since the assumptions
of lemma 2.8 are verified, there exist an open interval Λ ⊂ (0, ∞) and a positive
constant ρ > 0 such that for any λ ∈ Λ the equation φ0 (u) + λψ 0 (u) = 0 has at least
three solutions in X whose norms are less than ρ.
inf
u∈φ−1 (−∞,r]
ψ(u) >
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Mostafa Allaoui
University Mohamed I, Faculty of sciences, Department of Mathematics, Oujda, Morocco
E-mail address: allaoui19@hotmail.com
Abdel Rachid El Amrouss
University Mohamed I, Faculty of sciences, Department of Mathematics, Oujda, Morocco
E-mail address: elamrouss@hotmail.com
Anass Ourraoui
University Mohamed I, Faculty of sciences, Department of Mathematics, Oujda, Morocco
E-mail address: anas.our@hotmail.com