Consider a class of
p
(
x
)Kirchhoff type equations of the form
where
p
(
x
),
q
(
x
)
with 1 <
p
^{−}
:= inf
_{Ω}
p
(
x
) ≤
p
^{+}
:= sup
_{Ω}
p
(
x
) <
N
,
M
: ℝ
^{+}
→ ℝ
^{+}
is a continuous function that may be degenerate at zero,
λ
is a positive parameter. Using variational method, we obtain some existence and multiplicity results for such problem in two cases when the weight function
V
(
x
) may change sign or not.
AMS Mathematics Subject Classification : 34B27, 35J60, 35B05.
1. Introduction
In this paper, we are concerned with the following
p
(
x
) Kirchhoff type equations
where Ω ⊂ ℝ
^{N}
is a smooth bounded domain with boundary ∂Ω,
with 1 <
p
^{−}
:= inf
_{Ω}
p
(
x
) ≤
p
^{+}
:= sup
_{Ω}
p
(
x
) <
N
,
M
: ℝ
^{+}
→ ℝ+ is a continuous function,
ƒ
is a Carathéodory function having special structures, and
λ
is a paramter.
Since the first equation in (1.1) contains an integral over Ω, it is no longer a pointwise identity, and therefore it is often called nonlocal problem. This problem models several physical and biological systems, where
u
describes a process which depends on the average of itself, such as the population density, see
[4]
. Problem (1.1) is related to the stationary version of the Kirchhoff equation
presented by Kirchhoff in 1883, see
[19]
. This equation is an extension of the classical d’Alembert’s wave equation by considering the effects of the changes in the length of the string during the vibrations. 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 thematerial,
ρ
is themass density, and
P
_{0}
is the initial tension.
In recent years, elliptic problems involving
p
Kirchhoff type operators have been studied in many papers, we refer to some interesting works
[2
,
5
,
9
,
21
,
22
,
25
,
26]
, in which the authors have used different methods to get the existence of solutions for (1.1) in the case when
p
(
x
) =
p
is a constant. To our knowledge, the study of
p
(
x
)Kirchhoff type problems was firstly done by G. Dai et al. in the papers
[11
,
12]
. It is not difficult to see that the
p
(
x
)Laplacian possesses more complicated nonlinearities than
p
Laplacian, for example it is inhomogeneous. The study of differential equations and variational problems involving
p
(
x
)growth conditions is a consequence of their applications. Materials requiring such more advanced theory have been studied experimentally since the middle of last century. In
[11]
, the authors established the existence of infinitely many distinct positive solutions for problem (1.1) in the special case
M
(
t
) =
a
+
bt
. In
[12]
, the authors considered the problem in the case when
M
: ℝ
^{+}
→ ℝ is a continuous and nondescreasing function, satisfying the wellknown condition:

(M0) there existsm0> 0 such thatM(t) ≥m0for allt≥ 0,
which plays an enssential role in the arguments, see further papers
[2
,
3
,
6
,
10
,
21
,
22]
. There have been some authors improving (
M
_{0}
) in the sense that the Kirchhoff function
M
may be degenerate at zero, see for example
[7
,
8
,
9
,
15]
. In this paper, we assume that the Kirchhoff function
M
satisfies the following hypotheses:

(M1) There existsm2≥m1> 0 and 1 <α≤β

for allt∈ ℝ+;

(M2) For allt∈ ℝ+, it holds that
Motivated by the ideas in
[7
,
8
,
9
,
15]
and the results in
[18
,
23]
for the
p
(
x
)Laplacian, i.e.,
M
(
t
) ≡ 1, in this paper, we consider problem (1.1) with
ƒ
(
x, u
) =
λV
(
x
) 
u

^{q(x)−2}
u
in two cases when the weight function
V
(
x
) may change sign or not. The results in this work suplement or complement our earlier ones in
[7]
, in which we studied the problem in the case when the concave and convex nonlinearities were combined and the weight function did not change sign.
First, we consider the case when the parameter
λ
= 1 and
ƒ
(
x, u
) =
_{V(x)uq(x)−2u}
in which the weight function
V
(
x
) does not change sign. Problem (1.1) then becomes
More exactly,
V
: Ω → [0,+∞) belongs to
L
^{∞}
(Ω) and satisfies

(V1) There exist anx0∈ Ω and two positive constantsrandRwith 0
and the function
q
is assumed to satisfy

where the numbersαandβare given by (M1).
Definition 1.1.
A function
is said to be a weak solution of problem (1.3) if and only if
for all
v
∈
X
.
Our main result concerning problem (1.3) is given by the following theorem.
Theorem 1.2.
Assume that the conditions
(
M
_{1}
)(
M
_{2}
), (
V
_{1}
)
and
(
Q
_{1}
)(
Q
_{2}
)
are satisfied. Then there exists a positive constant
ε
_{0}
such that problem (1.3) has at least two nontrivial nonnegative weak solutions, provided that

V

_{L∞(Ω)}
<
ε
_{0}
.
It should be noticed that Theorem 1.2 is only true when
q
(
x
) is a nonconstant function while
p
(
x
) may be a constant. If
p
(
x
) =
p
is a constant then it follows from (
Q
_{2}
) that
α
=
β
.
Next, we consider problem (1.1) in the case when
ƒ
(
x, u
) =
λV
(
x
)
u

^{q(x)−2}
u
, in which
V
(
x
) is a sign changing weight function, that is,
More exactly, we study the existence of solutions for (1.4) under the hypotheses (
M
_{1}
), (
M
_{2}
) and

(V2)andV(x) > 0 in Ω0⊂ Ω with Ω0 > 0,αis given by (M1);
and the function
is assumed to satisfy the following condition
As we shall see in Section 4, due to the hypothesis (
Q
_{3}
), we cannot use the mountain pass theorem
[1]
in order to get the solutions for problem (1.4) as in Theorem 1.2. We emphasize that this is the main different point between two problems (1.3) and (1.4).
Definition 1.3.
A function
is said to be a weak solution of problem (1.4) if and only if
for all
v
∈
X
.
Our main result concerning problem (1.4) in this case is given by the following theorem.
Theorem 1.4.
Assume that the conditions
(
M
_{1}
)(
M
_{2}
), (
V
_{2}
)
and
(
Q
_{3}
)
are satisfied. Then there exists a positive constant λ^{∗} such that for any λ
∈ (0,
λ
^{∗}
),
problem (1.4) has at least one nontrivial nonnegative weak solution, i.e., any λ
∈ (0,
λ
^{∗}
)
is an eigenvalue of eigenvalue problem (1.4).
Our paper is organized as follows. In the next section, we shall recall some useful concepts and properties on the generalized LebesgueSobolev spaces. Section 3 is devoted to the proof of Theorem 1.2 while we shall present the proof of Theorem 1.4 in Section 4.
2. Preliminaries
We recall in what follows some definitions and basic properties of the generalized LebesgueSobolev spaces
L
^{p(x)}
(Ω) and
W
^{1,p(x)}
(Ω) where Ω is an open subset of ℝ
^{N}
. In that context, we refer to the book of Musielak
[24]
and the papers of Kováčik and Rákosník
[20]
and Fan et al.
[16
,
17]
. Set
For any
we define
h
^{+}
= sup
_{x∈Ω}
h
(
x
) and
h
^{−}
= inf
_{x∈Ω}
h
(
x
). For any
, we define the variable exponent Lebesgue space
We recall the following socalled
Luxemburg norm
on this space defined by the formula
Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1 <
p
^{−}
≤
p
^{+}
< ∞ and continuous functions are dense if
p
^{+}
< ∞. The inclusion between Lebesgue spaces also generalizes naturally: if 0 < Ω < ∞ and
p
_{1}
,
p
_{2}
are variable exponents so that
p
_{1}
(
x
) ≤
p
_{2}
(
x
) a.e.
x
∈ Ω then there exists a continuous embedding
the conjugate space of
L
^{p(x)}
(Ω), where
For any
u
∈
L
^{p(x)}
(Ω) and
v
∈
L
^{p′(x)}
(Ω) the Holder inequality
holds true.
Moreover, if
h
_{1}
,
h
_{2}
and
h
_{3}
:
are three Lipschitz continuous functions such that
then for any
u
∈
L
^{h1(x)}
(Ω),
v
∈
L
^{h2(x)}
(Ω) and
w
∈
L
^{h3(x)}
(Ω), the following inequality holds:
An important role in manipulating the generalized LebesgueSobolev spaces is played by the
modular
of the
L^{p(x)}
(Ω) space, which is the mapping
ρ_{p(x)}
:
L^{p(x)}
(Ω) → ℝ defined by
Proposition 2.1
(
[17]
).
If u
∈
L
^{p(x)}
(Ω)
and p^{+}
<
∞ then the following relations hold
provided that

u

_{p(x)}
> 1
while
provided that

u

_{p(x)}
< 1
and
Proposition 2.2
(
[18]
).
Let p and q be measurable functions such that p
∈
L
^{∞}
(Ω)
and
1 ≤
p(x)q(x)
≤ ∞
for a.e. x
∈
Ω. Let u
∈
L^{q(x)}
(Ω),
u
≠ 0.
Then the following relations hold
provided that

u

_{p(x)}
≤ 1
while
provided that

u

_{p(x)}
≥ 1
In particular, if p(x) = p is a constant, then
In this paper, we assume that
is the space of all the functions of
which are
logarithmic Höolder continuous
, that is, there exists
R
> 0 such that for all
x, y
∈ Ω with
see
[13
,
16]
. We define the space
under the norm
Proposition 2.3
(
[17
,
18]
).
The space
is a separable and Banach space. Moreover, if
then the embedding
is compact and continuous, where
3. Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2, which is essentially based on the mountain pass theorem
[1]
combined with the Ekeland variational principle
[14]
.
Let us define the functional
by the formula
where
where
Then, the functional
J
associated with problem (1.1) is well defined and of
C
^{1}
class on
X
. Moreover, we have
for all
u, v
∈
X
. Thus, weak solutions of problem (1.3) are exactly the ciritical points of the functional
J
. Due to the conditions (
M
_{1}
) and (
Q
_{1}
), we can show that
J
is weakly lower semicontinuous in
X
. The following lemma plays an essential role in our arguments.
Lemma 3.1.
The following assertions hold:

(i) There exists ε0> 0such that for anyVL∞(Ω)<ε0,there exist ρ1,γ1> 0for which J(u) ≥γ1,∀u∈X withu =ρ1;

(ii) There exists φ1∈X, φ1≥ 0,φ1≠ 0such thatlimt→∞J(tφ1) = −∞

(iii) There exists ψ1∈X, ψ1≥ 0,ψ1≠ 0such that J(tψ1) < 0for all t> 0small enough.
Proof
. We shall prove Lemma 3.1 in details for the case
the remaining case is similarly proved.
(
i
) Let us define the function
We denote
By the conditions (
Q
_{1}
) and (
Q
_{2}
),
which helps us to deduce that
X
is continuously embedded in
Then there exists a positive constant
c
_{1}
such that
From (3.4), there exist two positive constants
c
_{2}
,
c
_{3}
such that
and
Using the hypothesis (
M
_{1}
), relations (3.5) and (3.6) give us
for all
u
∈
X
with 
u
 < 1. Since the function
g
: [0, 1] → ℝ defined by
is positive in a neighbourhood of the origin, it follows that there exists
ρ
_{1}
∈ (0, 1) such that
g
(
ρ
_{1}
) > 0. On the other hand, defining
we deduce that, for any 
V

_{L∞(Ω)}
<
ε
_{0}
, there exists
γ
_{1}
> 0 such that for any
u
∈
X
with ∥
u
∥ =
ρ
_{1}
we have
J
(
u
) ≥
γ
_{1}
. (
ii
) Let
and there exist
x
_{1}
∈ Ω\
B_{R}
(
x
_{0}
) and
ε
> 0 such that for any
x
∈
B_{ε}
x
_{1}
) ⊂ (Ω\
B_{R}
(
x
_{0}
)) we have
ψ
_{1}
(
x
) > 0. For any
t
> 1, we have
Since
we infer that lim
_{t→ ∞}
J
(
tψ
_{1}
) = −∞. (
iii
) Let
φ
_{1}
≥ 0 and there exist
x
_{2}
∈
B_{r}
(
x
_{0}
) and
ε
> 0 such that for any
x
∈
B_{ε}
(
x
_{2}
) ⊂
B_{r}
(
x
_{0}
) we have
φ
_{1}
(
x
) > 0. Letting 0 <
t
< 1 we find
Obviously, we have
J
(
tφ
_{1}
) < 0 for any
where
The proof of Lemma 3.1 is complete.
Lemma 3.2.
The functional J satisfies the PalaisSmale condition in X
.
Proof
. Let {
u_{m}
} ⊂
X
be such that
where
X
^{∗}
is the dual space of
X
.
We shall prove that {
u_{m}
} is bounded in
X
. In order to do that, we assume by contradiction that passing if necessary to a subsequence, still denoted by {
u_{m}
}, we have ∥
u_{m}
∥ → ∞ as
m
→ ∞. By (3.12) and (
M
_{1}
)(
M
_{2}
), for
m
large enough and 
V

_{L∞(Ω)}
<
ε
_{0}
, we have
Dividing the above inequality by ∥
u_{m}
∥
^{αp−}
taking into account that (3.3) holds true and passing to the limit as
m
→ ∞ we obtain a contradiction. It follows that {
u_{m}
} is bounded in
X
. Thus, there exists
u
_{1}
∈
X
such that passing to a subsequence, still denoted by {
u_{m}
}, it converges weakly to
u
_{1}
in
X
. Then {∥
u_{m}
−
u
∥} is bounded. By (3.3), the embedding from
X
to the space
L
^{q(x)}
(Ω) is compact. Then, using the Hölder inequality, Propositions 2.12.3, we have
This fact and relation (3.12) yield
Since {
u_{m}
} is bounded in
X
, passing to a subsequence, if necessary, we may assume that
If
t
_{0}
= 0 then {
u_{m}
} converges strongly to
u
= 0 in
X
and the proof is finished. If
t
_{0}
> 0 then we deduce by the continuity of
M
that
Thus, by (
M
_{1}
), for sufficiently large
m
, we have
From (3.15), (3.16), it follows that
Thus, {
u_{m}
} converges strongly to
u
in
X
and the functional
J
satisfies the PalaisSmale condition.
Proof of Theorem 1.2.
By Lemmas 3.1 and 3.2, all assumptions of the mountain pass theorem in
[1]
are satisfied. Then we deduce
u
_{1}
as a nontrivial critical point of the functional
J
with
J
(
u
_{1}
) =
and thus a nontrivial weak solution of problem (1.3).
We now prove that there exists a second weak solution
u
_{2}
∈
X
such that
u
_{2}
≠
u
_{1}
. Indeed, let
ε
_{0}
as in the proof of Lemma 3.1(i) and assume that 
V

_{L∞(Ω)}
<
ε
_{0}
. By Lemma 3.1(i), it follows that on the boundary of the ball centered at the origin and of radius
ρ
_{1}
in
X
, denoted by
Bρ
_{1}
(0) = {
u
∈
X
: ∥
u
∥ <
ρ
_{1}
}, we have
On the other hand, by Lemma 3.1(ii), there exists
φ
_{1}
∈
X
such that
J
(
tφ
_{1}
) < 0 for all
t
> 0 small enough. Moreover, from (3.7), the functional
J
is bouned from below on
B
_{ρ1}
(0). It follows that
Applying the Ekeland variational principle in
[14]
to the functional
it follows that there exists
such that
By Lemma 3.1, we have
Let us choose
ε
> 0 such that
Then,
J
(
u_{ε}
) < inf
_{u∈∂Bρ1(0)}
J
(
u
) and thus,
u_{ε}
∈
B
_{ρ1}
(0).
Now, we define the functional
It is clear that
u_{ε}
is a minimum point of
I
and thus
for all
t
> 0 small enough and all
v
∈
B
_{ρ1}
(0). The above information shows that
Letting
t
→ 0
^{+}
, we deduce that
It should be noticed that −
v
also belongs to
B
_{ρ1}
(0), so replacing
v
by −
v
, we get
or
which helps us to deduce that ∥
J
′(
u_{ε}
)∥
_{X∗}
≤
ε
. Therefore, there exists a sequence {
u_{m}
} ⊂
B
_{ρ1}
(0) such that
From Lemma 3.2, the sequence {
u_{m}
} converges strongly to
u
_{2}
as
m
→ ∞. Moreover, since
J
∈
C
^{1}
(
X
,ℝ), by (3.17) it follows that
Thus,
u
_{2}
is a nontrivial weak solution of problem (1.2).
Finally, we point out the fact that
u
_{1}
≠
u
_{2}
since
Moreover, since
J
(
u
) =
J
(
u
), problem (1.3) has at least two nontrivial nonnegative weak solutions. The proof of Theorem 1.2 is complete.
4. Proof of Theorem 1.4
In this section, assume that we are under the hypotheses of Theorem 1.4, we shall prove Theorem 1.4 using the Ekeland variational principle
[14]
. For each
λ
∈ ℝ, define the functional
where
From (
V
_{2}
), (2.4) and (2.5), it is clear that for all
u
∈
X
,
On the other hand, by (
V
_{2}
) and (
Q
_{3}
), we have
and thus the embeddings
are continuous and compact. For these reasons, we can use the similar arguments as in
[18, Proposition 2]
in order to show that the functional
J_{λ}
is welldefined. Moreover,
J_{λ}
is of
C
^{1}
class in
X
and
for all
u, v
∈
X
. Thus, weak solutions of problem (1.4) are exactly the ciritical points of the functional
J_{λ}
.
Lemma 4.1
.
For any ρ
_{2}
∈ (0, 1),
there exist λ
^{∗}
> 0
and γ
_{2}
> 0
such that for all u
∈
X with
∥
u
∥ =
ρ
_{2}
,
Proof
. Since the embedding
is continuous, there exists a positive constant
c
_{7}
such that
Now, let us assume that
where
c
_{7}
is the positive constant from above. Then we have
Using relations (2.2), (4.2), the condition (
M
_{1}
) and the Hölder inequality, we deduce that for any
u
∈
X
with ∥
u
∥ =
ρ
_{2}
∈ (0, 1) the following inequalities hold true
By (
Q
_{3}
) we have
q
^{−}
≤
q
^{+}
<
p
^{−}
≤
p
^{+}
<
αp
^{+}
. So, if we take
then for any
λ
∈ (0,
λ
^{∗}
) and
u
∈
X
with ∥
u
∥ =
ρ
_{2}
, there exists
γ
_{2}
> 0 such that
J_{λ}
(
u
) ≥
γ
_{2}
> 0. The proof of the Lemma 4.1 is complete.
Lemma 4.2.
For any λ
∈ (0,
λ
^{∗}
),
where λ^{∗} is given by (4.4), there exists ψ
_{2}
∈
X such that ψ
_{2}
≥ 0,
ψ
_{2}
≠ 0
and J_{λ}
(
tψ
_{2}
) < 0
for all t
> 0
smaller than a certain value depending on λ
.
Proof.
From (
Q
_{3}
) we have
q
(
x
) <
βp
(
x
) for all
where Ω
_{0}
is given by (
V
_{2}
). In the sequel, we use the notation
Let
δ
_{0}
> 0 be such that
there exists an open set Ω
_{1}
⊂ Ω
_{0}
such that
for all
x
∈ Ω
_{1}
. It follows that
Let
such that supp(
ψ
_{2}
) ⊂ Ω
_{1}
⊂ Ω
_{0}
,
ψ
_{2}
= 1 in a subset
Then, using (
M
_{1}
) we have
Therefore
Finally, we shall point that
In fact, due to the choice of
ψ
_{2}
, if
Using (2.3), we deduce that ∇
ψ
_{2}
 = 0 and consequently
ψ
_{2}
= 0 in Ω, which is a contradiction. The proof of Lemma 4.2 is complete.
Proof of Theorem 1.4.
Let
λ
^{∗}
> 0 be defined by (4.4) and
λ
∈ (0,
λ
^{∗}
). By Lemma 4.1, it follows that on the boundary of the ball centered at the origin and of radius
ρ
_{2}
in
X
, denoted by
B
_{ρ2}
(0), we have
On the other hand, by Lemma 4.2, there exists
ψ
_{2}
∈
X
such that
J_{λ}
(
tψ
_{2}
) < 0 for all
t
> 0 small enough. Moreover, relation (4.3) implies that for any
u
∈
B
_{ρ2}
(0) we have
It follows that
Using the Ekeland variational principle
[14]
and the similar arguments as those used in the proof of Theorem 1.1, we can deduce that there exists a sequence {
u_{m}
} ⊂
B
_{ρ2}
(0) such that
It is clear that {
u_{m}
} is bounded in
X
. Thus, there exists
u
∈
X
such that, up to a subsequence, {
u_{m}
} converges weakly to
u
in
X
. Since
we deduce that
X
is compactly embedded in
hence the sequence {
u_{m}
} converges strongly to
u
in
Using the Hölder inequality, we have
Now, if
The compact embedding
ensures that
Relation (4.6) yields
Using the above information, we also obtain relation (3.15) and thus, {
u_{m}
} converges strongly to some
u
in
X
. So, by (4.6),
It is clear that
J_{λ}
(
u
) =
J_{λ}
(
u
). Therefore,
u
is a nontrivial nonnegative weak solution of problem (1.4). Theorem 1.4 is completely proved.
Remark 4.3.
We cannot use the mountain pass argument in the proof of Theorem 1.4 since the functional
J_{λ}
does not satisfy the geometry of the mountain pass theorem. More exactly, we cannot find a function
φ
_{2}
≥ 0 such that
J_{λ}
(
tφ
_{2}
) → −∞ as
t
→ ∞ as in Lemma 3.1.
Acknowledgements
The author would like to thank the referees for their helpful comments and suggestions which improved the presentation of the original manuscript. This paper was done when the author was working at the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore as a Research Fellow.
BIO
Nguyen Thanh Chung
Dep. Science Management & International Cooperation, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam.
email: ntchung82@@yahoo.com
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