Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 253, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EIGENVALUE PROBLEMS FOR p(x)-KIRCHHOFF TYPE
EQUATIONS
GHASEM A. AFROUZI, MARYAM MIRZAPOUR
Abstract. In this article, we study the nonlocal p(x)-Laplacian problem
“Z
”
1
−M
|∇u|p(x) dx div(|∇u|p(x)−2 ∇u) = λ|u|q(x)−2 u in Ω,
Ω p(x)
u = 0 on ∂Ω,
By means of variational methods and the theory of the variable exponent
Sobolev spaces, we establish conditions for the existence of weak solutions.
1. Introduction
The purpose of this article is to show the existence of solutions of the p(x)Kirchhoff type eigenvalue problem
Z
1
−M
|∇u|p(x) dx div(|∇u|p(x)−2 ∇u) = λ|u|q(x)−2 u in Ω,
(1.1)
Ω p(x)
u = 0 on ∂Ω,
where Ω ⊂ RN , N ≥ 3, is a bounded domain with smooth boundary ∂Ω, M :
R+ → R is a continuous function, p, q are continuous functions on Ω such that
1 < p(x) < N and q(x) > 1 for any x ∈ Ω and λ is a positive number. The study
of problems involving variable exponent growth conditions has a strong motivation
due to the fact that they can model various phenomena which arise in the study of
elastic mechanics [28], electrorheological fluids [1] or image restoration [6].
Equation (1.1) is called a nonlocal problem because of the the term M , which
implies that the equation in (1.1) is no longer a pointwise equation. This causes
some mathematical difficulties which make the study of such a problem particularly
interesting. Nonlocal differential equations are also called Kirchhoff-type equations
because Kirchhoff [23] investigated an equation of the form
Z L
∂u 2 ∂ 2 u
∂ 2 u ρ0
E
dx
ρ 2 −
+
= 0,
(1.2)
∂t
h
2L 0 ∂x
∂x2
which extends the classical D’Alembert’s wave equation, by considering the effect
of the changing in the length of the string during the vibration. A distinct feature
2000 Mathematics Subject Classification. 35J60, 35J35, 35J70.
Key words and phrases. p(x)-Kirchhoff type equations; variational methods;
boundary value problems.
c
2013
Texas State University - San Marcos.
Submitted August 17, 2013. Published November 20, 2013.
1
2
G. A. AFROUZI, M. MIRZAPOUR
EJDE-2013/253
R L ∂u 2
E
is that the (1.2) contains a nonlocal coefficient ρh0 + 2L
| | dx which depends
0 ∂x
R L ∂u 2
1
on the average 2L 0 ∂x dx, and hence the equation is no longer a pointwise
equation. The parameters in (1.2) have the following meanings: L is the length of
the string, h is the area of the cross-section, E is the Young modulus of the material,
ρ is the mass density and P0 is the initial tension. Lions [25] has proposed an
abstract framework for the Kirchhoff-type equations. After the work by Lions [25],
various equations of Kirchhoff-type have been studied extensively, see e.g. [3, 5] and
[9]-[14]. The study of Kirchhoff type equations has already been extended to the
case involving the p-Laplacian (for details, see [13, 14, 9, 10]) and p(x)-Laplacian
(see [4, 8, 11, 12, 22]). Motivated by the above papers and the results in [7, 26], we
consider (1.1) to study the existence of weak solutions.
2. Preliminaries
For the reader’s convenience, we recall some necessary background knowledge
and propositions concerning the generalized Lebesgue-Sobolev spaces. We refer the
reader to [15, 16, 18, 19] for details.
Let Ω be a bounded domain of RN , denote
C+ (Ω) = {p(x) : p(x) ∈ C(Ω), p(x) > 1, for all x ∈ Ω};
p− = min{p(x); x ∈ Ω};
Z
p(x)
L
(Ω) = u : u is a measurable real-valued function,
|u(x)|p(x) dx < ∞ ,
p+ = max{p(x) : x ∈ Ω},
Ω
with the norm
|u|Lp(x) (Ω) = |u|p(x) = inf λ > 0 :
Z
|
Ω
u(x) p(x)
|
dx ≤ 1
λ
becomes a Banach space [24]. We also define the space
W 1,p(x) (Ω) = {u ∈ Lp(x) (Ω) : |∇u| ∈ Lp(x) (Ω)},
equipped with the norm
kukW 1,p(x) (Ω) = |u(x)|Lp(x) (Ω) + |∇u(x)|Lp(x) (Ω) .
1,p(x)
We denote by W0
(Ω) the closure of C0∞ (Ω) in W 1,p(x) (Ω). Of course the norm
1,p(x)
kuk = |∇u|Lp(x) (Ω) is an equivalent norm in W0
(Ω). In this paper, we denote
1,p(x)
by X = W0
(Ω).
0
Proposition 2.1 ([15, 19]). (i) The conjugate space of Lp(x) (Ω) is Lp (x) (Ω), where
0
1
1
p(x)
(Ω) and v ∈ Lp (x) (Ω), we have
p(x) + p0 (x) = 1. For any u ∈ L
Z
1
1 |uv|dx ≤ − + 0− |u|p(x) |v|p0 (x) ≤ 2|u|p(x) |v|p0 (x)
p
p
Ω
(ii) If p1 (x), p2 (x) ∈ C+ (Ω) and p1 (x) ≤ p2 (x) for all x ∈ Ω, then Lp2 (x) (Ω) ,→
Lp1 (x) (Ω) and the embedding is continuous.
R
Proposition 2.2 ([20]). Set ρ(u) = Ω |∇u(x)|p(x) dx, then for u ∈ X and (uk ) ⊂
X, we have
(1) kuk < 1 (respectively= 1; > 1) if and only if ρ(u) < 1 (respectively= 1; > 1);
(2) for u 6= 0, kuk = λ if and only if ρ( λu ) = 1;
EJDE-2013/253
EIGENVALUE PROBLEMS
−
3
+
(3) if kuk > 1, then kukp ≤ ρ(u) ≤ kukp ;
+
−
(4) if kuk < 1, then kukp ≤ ρ(u) ≤ kukp ;
(5) kuk k → 0 (respectively → ∞) if and only if ρ(uk ) → 0 (respectively → ∞).
For x ∈ Ω, let us define
(
∗
p (x) =
N p(x)
N −p(x)
∞
if p(x) < N,
if p(x) ≥ N.
Proposition 2.3 ([19]). If q ∈ C+ (Ω) and q(x) ≤ p∗ (x) (q(x) < p∗ (x)) for x ∈ Ω,
then there is a continuous (compact) embedding X ,→ Lq(x) (Ω).
Lemma 2.4 ([21]). Denote
Z
1
I(u) =
|∇u|p(x) dx,
Ω p(x)
for all u ∈ X,
then I(u) ∈ C 1 (X, R) and the derivative operator I 0 of I is
Z
0
hI (u), vi =
|∇u|p(x)−2 ∇u∇v dx, for all u, v ∈ X,
Ω
and we have
(1) I is a convex functional;
(2) I 0 : X → X ∗ is a bounded homeomorphism and strictly monotone operator;
(3) I 0 is a mapping of type (S+ ), namely: un * u and lim supn→+∞ I 0 (un )(un −
u) ≤ 0, imply un → u.
Definition 2.5. A function u ∈ X is said to be a weak solution of (1.1) if
Z
Z
Z
1
M
|∇u|p(x)−2 ∇u∇v dx − λ
|u|q(x)−2 uv dx = 0,
|∇u|p(x) dx
Ω p(x)
Ω
Ω
for all v ∈ X.
The Euler-Lagrange functional associated to (1.1) is
Z
Z
1
1
c
Jλ (u) = M
|∇u|p(x) dx − λ
|u|q(x) dx,
p(x)
q(x)
Ω
Ω
Rt
c(t) = M (τ )dτ . Then
where M
0
Z
Z
Z
1
0
p(x)
p(x)−2
hJλ (u), vi = M
|∇u|
dx
|∇u|
∇u∇v dx − λ
|u|q(x)−2 uv dx,
Ω p(x)
Ω
Ω
for all u, v ∈ X, then we know that the weak solution of (1.1) corresponds to
the critical point of the functional Jλ . Hereafter M (t) is supposed to verify the
following assumptions:
(M1) There exists m2 ≥ m1 > 0 and β ≥ α > 1 such that m1 tα−1 ≤ M (t) ≤
m2 tβ−1 .
c(t) ≥ M (t)t.
(M2) For all t ∈ R+ , M
For simplicity, we use ci , to denote the general nonnegative or positive constant
(the exact value may change from line to line).
4
G. A. AFROUZI, M. MIRZAPOUR
EJDE-2013/253
3. Main results and proofs
Theorem 3.1. Assume that M satisfies (M1) and (M2) and the function q ∈ C(Ω)
satisfies
βp+ < q − ≤ q + < p∗ (x).
(3.1)
Then for any λ > 0 problem (1.1) possesses a nontrivial weak solution.
Lemma 3.2. There exist η > 0 and α > 0 such that Jλ (u) ≥ α > 0 for any u ∈ X
with kuk = η.
Proof. First, we point out that
−
+
|u(x)|q(x) ≤ |u(x)|q + |u(x)|q ,
for all x ∈ Ω.
Using the above inequality and (M1), we find that
Z
Z
1
1
c
|∇u|p(x) dx − λ
|u|q(x) dx
Jλ (u) = M
p(x)
q(x)
Ω
ZΩ
α
−
+
1
m1 λ p(x)
≥
|∇u|
dx − − |u|qq− + |u|qq+ .
α
q
Ω p(x)
−
From the assumptions of Theorem 3.1, X is continuously embedded in Lq (Ω) and
+
Lq (Ω). Then, there exist two positive constants c1 and c2 such that
|u(x)|q− ≤ c1 kuk,
|u(x)|q+ ≤ c2 kuk,
for all u ∈ X.
Hence, for any u ∈ X with kuk < 1, we obtain
+
+
−
+
m1
λ −
Jλ (u) ≥
kukαp − − cq1 kukq + cq2 kukq .
+
α
α(p )
q
Since the function g : [0, 1] → R defined by
−
g(t) =
+
λcq1 q− −αp+ λcq2 q+ −αp+
m1
− −
t
− − t
,
+
α
α(p )
q
q
is positive in a neighborhood of the origin, the proof is complete.
Lemma 3.3. There exists e ∈ X with kek > η (where η is given in Lemma 3.2)
such that Jλ (e) < 0.
Proof. Let ψ ∈ C0∞ (Ω), ψ ≥ 0 and ψ 6= 0 and t > 1. By (M1) we have
Z
Z
1
1
p(x)
c
|∇tψ|
dx − λ
|tψ|q(x) dx
Jλ (tψ) = M
p(x)
q(x)
Ω
Ω
Z
− Z
β
m2 1
tq
≤
|∇tψ|p(x) dx − λ +
|ψ|q(x) dx
β
q
Ω p(x)
Ω
Z
− Z
β
m2 βp+ tq
p(x)
≤
t
|∇ψ|
dx − λ +
|ψ|q(x) dx.
β(p− )β
q
Ω
Ω
Since βp+ < q − , we obtain limt→∞ Jλ (tψ) = −∞. Then for t > 1 large enough, we
can take e = tψ such that kek > η and Jλ (e) < 0.
Proof of Theorem 3.1. By Lemmas 3.2–3.3 and the mountain pass theorem of Ambrosetti and Rabinowitz [2], we deduce the existence of a sequence (un ) ⊂ X such
that
Jλ (un ) → c3 > 0, Jλ0 (un ) → 0 as n → ∞.
(3.2)
EJDE-2013/253
EIGENVALUE PROBLEMS
5
We prove that (un ) is bounded in X. Arguing by contradiction. We assume that,
passing eventually to a subsequence, still denote by (un ), kun k → ∞ and kun k > 1
for all n. By (3.2) and (M1)-(M2), for n large enough, we have
1 + c3 + kun k
1
≥ Jλ (un ) − − hJλ0 (un ), un i
q
Z
Z
Z
1
1
1
≥M
|∇un |p(x) dx
|∇un |p(x) dx − λ
|un |q(x) dx
Ω q(x)
Ω p(x)
Ω p(x)
Z
Z
Z
1
1
λ
p(x)
p(x)
|un |q(x) dx
− −M
|∇un |
dx
|∇un |
dx + −
q
q
Ω
Ω p(x)
Ω
Z
1
1
m1
1 1 αp−
≥
− − kun k
+λ − −
|un |q(x) dx
α(p+ )α−1 p+
q
q
q(x) Ω
1
1
m1
1 1 αp−
q−
q+
≥
−
ku
k
+
λ
−
c
ku
k
+
c
ku
k
.
n
1
n
2
n
α(p+ )α−1 p+
q−
q−
q(x)
−
Dividing the above inequality by kun kαp , taking into account (3.1) holds true and
passing to the limit as n → ∞, we obtain a contradiction. It follows that (un ) is
bounded in X. This information, combined with the fact that X is reflexive, implies
that there exists a subsequence, still denote by (un ) and u1 ∈ X such that (un )
converges weakly to u1 in X. Note that Proposition 2.3 yields that X is compactly
embedded in Lq(x) (Ω), it follows that (un ) converges strongly to u1 in Lq(x) (Ω).
Then by Hölder inequality we deduce
Z
|un |q(x)−2 un (un − u1 )dx = 0.
(3.3)
lim
n→∞
Ω
Using (3.2), we infer that
lim hJλ0 (un ), un − u1 i = 0.
n→∞
(3.4)
Since (un ) is bounded in X, passing to a subsequence, if necessary, we may assume
that
Z
1
|∇un |p(x) dx → t0 ≥ 0 as n → ∞.
Ω p(x)
If t0 = 0 then (un ) converges strongly to u1 = 0 in X and the proof is complete. If
t0 > 0 then since the function M is continuous, we obtain
Z
1
M
|∇un |p(x) dx → M (t0 ) ≥ 0 as n → ∞.
Ω p(x)
Thus, by (M1), for sufficiently large n, we have
Z
1
0 < c4 ≤ M
|∇un |p(x) dx ≤ c5 .
Ω p(x)
From (3.3)-(3.5), we deduce that
Z
lim
|∇un |p(x)−2 ∇un (∇un − ∇u1 )dx = 0.
n→∞
(3.5)
(3.6)
Ω
Using Lemma 2.4, we deduce that actually (un ) converges strongly to u1 in X.
Then by relation (3.2) we have
Jλ (u1 ) = c3 > 0,
Jλ0 (u1 ) = 0;
6
G. A. AFROUZI, M. MIRZAPOUR
EJDE-2013/253
that is, u1 ia a nontrivial weak solution of (1.1).
Theorem 3.4. If we assume that (M1)–(M2) hold and q ∈ C+ (Ω) satisfies
1 < q − ≤ q + < αp− ,
(3.7)
then there exists λ∗ > 0 such that for any λ > λ∗ , problem (1.1) possesses a
nontrivial weak solution.
Under the theorem’s conditions, we want to construct a global minimizer of the
functional. We start with the following auxiliary result.
Lemma 3.5. The functional Jλ is coercive on X.
Proof. By Theorem 3.1 and Proposition 2.2, we deduce that for all u ∈ X,
Z
λ m1 p(x)
α
q−
q+
|∇u|
dx
)
−
c
kuk
+
c
kuk
.
Jλ (u) ≥
1
2
α(p+ )α Ω
q−
Now we set kuk > 1, then
Jλ (u) ≥
−
+
λ m1
kukαp − − c1 kukq− + c2 kukq .
+
α
α(p )
q
Since by relation (3.7) we have αp− > q + ≥ q − , we infer that Jλ (u) → ∞ as
kuk → ∞. In other words, Jλ is coercive in X.
Proof of Theorem 3.4. Jλ (u) is a coercive functional and weakly lower semi-continuous on X. These two facts enable us to apply [27, Theorem 1.2] in order to
find that there exists uλ ∈ X a global minimizer of Jλ and thus a weak solution of
problem (1.1).
We show uλ is not trivial for λ large enough. Letting t0 > 1 be a constant and
Ω1 be an open subset of Ω with |Ω1 | > 0, we assume that v0 ∈ C0∞ (Ω) is such that
v0 (x) = t0 for any x ∈ Ω1 and 0 ≤ v0 (x) ≤ t0 in Ω\Ω1 . We have
Z
Z
1
1
p(x)
c
Jλ (v0 ) = M
|∇v0 |
dx − λ
|v0 |q(x) dx
p(x)
q(x)
Ω
Ω
Z
λ
λ −
≤ c6 − +
|v0 |q(x) dx ≤ c6 − + tq0 |Ω1 |.
q
q
Ω
So there exists λ∗ > 0 such that Jλ (v0 ) < 0 for any λ ∈ [λ∗ , +∞). It follows
that for any λ ≥ λ∗ , uλ is a nontrivial weak solution of problem (1.1) for λ large
enough.
Theorem 3.6. If q ∈ C+ (Ω) with
1 < q(x) < p(x) < p∗ (x),
(3.8)
then there exists λ∗∗ > 0 such that for any λ ∈ (0, λ∗∗ ), problem (1.1) possesses a
nontrivial weak solution.
We plan to apply Ekeland variational principle [17] to get a nontrivial solution
to problem (1.1). We start with two auxiliary results.
Lemma 3.7. There exists λ∗∗ > 0 such that for any λ ∈ (0, λ∗∗ ) there are ρ, a > 0
such that Jλ (u) ≥ a > 0 for any u ∈ X with kuk = ρ.
EJDE-2013/253
EIGENVALUE PROBLEMS
7
Proof. Under the assumption of Theorem 3.6, X is continuously embedded in
Lq(x) (Ω). Thus, there exists a positive constant c7 such that
|u|q(x) ≤ c7 kuk
for all u ∈ X.
(3.9)
Now, Let us assume that kuk < min{1, c17 }, where c7 is the positive constant from
above. Then we have |u|q(x) < 1. Using Proposition 2.2 we obtain
Z
−
|u|q(x) dx ≤ |u|qq(x) for all u ∈ X with kuk = ρ ∈ (0, 1).
(3.10)
Ω
Relations (3.9) and (3.10) imply
Z
−
−
|u|q(x) dx ≤ cq7 kukq
for all u ∈ X with kuk = ρ.
(3.11)
Ω
Using the hypotheses (M1) and (3.11), we deduce that for any u ∈ X with kuk = ρ,
the following hold
Z
Z
1
1
c
|∇u|p(x) dx − λ
|u|q(x) dx
Jλ (u) = M
p(x)
q(x)
Ω
Ω
m1
λ q−
αp+
q−
≥
kuk
− − c7 kuk
α(p+ )α
q
m
+
−
λ q− 1
αp −q −
ρ
c
−
.
= ρq
α(p+ )α
q− 7
By (3.8) we have q − ≤ q + < p− ≤ p+ < αp+ . So, if we take
λ∗∗ =
m1 q − αp+ −q−
ρ
,
2α(p+ )α
then for any λ ∈ (0, λ∗∗ ) and u ∈ X with kuk = ρ, there exists a =
that Jλ (u) ≥ a > 0.
(3.12)
+
ραp
2α(p+ )α
such
Lemma 3.8. For any λ ∈ (0, λ∗∗ ) given by (3.12), there exists ϕ ∈ X such that
ϕ ≥ 0, ϕ 6= 0 and Jλ (tϕ) < 0 for all t > 0 small enough.
Proof. Assumption (3.8) implies that q(x) < βp(x). Let 0 > 0 such that q − + 0 <
βp− . Since q ∈ C(Ω), there exists an open set Ω0 ⊂ Ω such that |q(x) − q − | < 0
for all x ∈ Ω0 . It follows that q(x) < q − + 0 < βp− for all x ∈ Ω0 .
Let ϕ ∈ C0∞ (Ω) be such that supp(ϕ) ⊃ Ω0 , ϕ(x) = 1 for all x ∈ Ω0 and
0 ≤ ϕ ≤ 1 in Ω. Then for any t ∈ (0, 1), we have
Z
Z
1
1
c
Jλ (tϕ) = M
|∇tϕ|p(x) dx − λ
|tϕ|q(x) dx
Ω p(x)
Ω q(x)
Z
Z
β
m2 1
1 q(x) q(x)
≤
|∇tϕ|p(x) dx − λ
t
|ϕ|
dx
β
Ω p(x)
Ω q(x)
Z
Z
β
m2 βp− λ −
p(x)
≤
t
|∇ϕ|
dx
− + tq +0
|ϕ|q(x) dx < 0,
−
β
β(p )
q
Ω
Ω0
1
for all t < δ βp− −q− −0 with
R
n λβ(p− )β
|ϕ|q(x) dx o
Ω0
0 < δ < min 1,
β .
R
m2 q +
|∇ϕ|p(x) dx
Ω
8
G. A. AFROUZI, M. MIRZAPOUR
EJDE-2013/253
Proof of Theorem 3.6. Let λ∗∗ be defined as in (3.12) and λ ∈ (0, λ∗∗ ). By Lemma
3.7, it follows that on the boundary of the ball centered at the origin and of radius
ρ in X, we have
inf Jλ (u) > 0.
∂Bρ (0)
On the other hand, by Lemma 3.8, there exists ϕ ∈ X such that
Jλ (tϕ) < 0
for t > 0 small enough.
Moreover, for u ∈ Bρ (0),
Jλ (u) ≥
+
−
m1
λ −
kukαp − − cq7 kukq .
α(p+ )α
q
It follows that
−∞ < c8 = inf Jλ (u) < 0.
Bρ (0)
We let now 0 < ε < inf ∂Bρ (0) Jλ − inf Bρ (0) Jλ . Applying Ekeland variational principle [17] to the functional Jλ : Bρ (0) → R, we find uε ∈ Bρ (0) such that
Jλ (uε ) < inf Jλ + ε
Bρ (0)
Jλ (uε ) < Jλ (u) + εku − uε k,
u 6= uε .
Since
Jλ (uε ) ≤ inf Jλ + ε ≤ inf Jλ + ε < inf Jλ ,
Bρ (0)
Bρ (0)
∂Bρ (0)
we deduce that uε ∈ Bρ (0). Now, we define Kλ : Bρ (0) → R by Kλ (u) = Jλ (u) +
εku − uε k. It is clear that uε is a minimum point of Kλ and thus
Kλ (uε + tv) − Kλ (uε )
≥ 0,
t
for small t > 0 and v ∈ Bρ (0). The above relation yields
Jλ (uε + tv) − Jλ (uε )
+ εkvk ≥ 0.
t
Letting t → 0 it follows that hJλ0 (uε ), vi + εkvk > 0 and we infer that kJλ0 (uε )k ≤ ε.
We deduce that there exists a sequence (vn ) ⊂ B1 (0) such that
Jλ (vn ) → c8 ,
Jλ0 (vn ) → 0.
(3.13)
It is clear that (vn ) is bounded in X. Thus, there exists u2 ∈ X such that, up to a
subsequence, (vn ) converges weakly to u2 in X. Actually, with similar arguments
as those used in the proof Theorem 3.1, we can show that vn → u2 in X. Thus, by
relation (3.13),
Jλ (u2 ) = c8 < 0,
Jλ0 (u2 ) = 0;
i.e., u2 is a nontrivial weak solution for problem (1.1).
EJDE-2013/253
EIGENVALUE PROBLEMS
9
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10
G. A. AFROUZI, M. MIRZAPOUR
EJDE-2013/253
4. Corrigendum posted in August 27, 2014
A reader pointed out that no function M (t) can satisfy both hypotheses (M1)
and (M2). In response, we present a proof of our results by adding the following
assumption
m1 q − (p− )α > m2 αp− (p+ )α .
(4.1)
and without assumption (M2).
Modified assumptions. We delete the assumption (M2) and re-state the following:
(M1) There exist m2 ≥ m1 > 0 and α > 1 such that
m1 tα−1 ≤ M (t) ≤ m2 tα−1 ,
∀t ∈ R+
(The original (M1) implies α = β, so we rename constant α.);
In the proof of Theorem 3.1, By (3.2) and (M1), for n large enough, we can write
1 + c3 + kun k
1
≥ Jλ (un ) − − hJλ0 (un ), un i
q
Z
Z
1
1
c
=M
|∇un |p(x) dx − λ
|un |q(x) dx
p(x)
q(x)
Ω
Ω
Z
Z
1
λ
1
p(x)
|∇un |
dx + −
|un |q(x) dx
− −M
q
q
Ω p(x)
Ω
Z
α m Z
α−1 Z
m1 1
1
2
|∇un |p(x) dx
≥
|∇un |p(x) dx − −
|∇un |p(x) dx
α
q
Ω
Ω p(x)
Ω p(x)
Z 1
1 +λ
−
|un |q(x) dx
−
q(x)
Ω q
m
m2
1
αp−
q−
q+
−
ku
k
+
λ
c
ku
k
+
c
ku
k
.
≥
n
1
n
2
n
α(p+ )α
q − (p− )α−1
−
Dividing the above inequality by kun kαp , taking into account (3.1) and (4.1) hold
true and passing to the limit as n → ∞, we obtain a contradiction. It follows that
(un ) is bounded in X.
Theorem 3.6 remains unchanged. However, Theorems 3.1 and 3.4 need to be
stated without assumption (M2). Relation (3.1) need to be changed by αp+ <
q − ≤ q + < p∗ (x). The proofs of Theorems and Lemmas are similar to the original
proofs, but replacing β by α.
The authors would like to thank anonymous reader and the editor for allowing
us to correct our mistake.
Ghasem Alizadeh Afrouzi
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: afrouzi@umz.ac.ir
Maryam Mirzapour
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
E-mail address: mirzapour@stu.umz.ac.ir