# Existence of Solutions for a Robin Problem Involving thep

Authors: Mostafa Allaoui
Journal: Abstract and Applied Analysis
 (2016)

---

## Abstract

In this article we study the nonlinear Robin boundary-value problem-
Δ
p
(
x
)
u
=
f
(
x
,
u
)
i
n
Ω, |
∇
u
|
p
x
-
2
(
∂
u
/
∂
ν
)
+
β
(
x
)
u
p
(
x
)
-
2
u
=
0 on ∂
Ω. Using the variational method, under appropriate assumptions on f, we obtain results on existence and multiplicity of solutions.


---

## Body

## 1. Introduction

The aim of this article is to analyze the existence of solutions of the following problem:(1)
-
Δ
p
x
u
=
f
x
,
u
i
n
Ω
,
∇
u
p
x
-
2
∂
u
∂
ν
+
β
x
u
p
x
-
2
u
=
0
o
n
∂
Ω
,where Ω
⊂
R
N
(
N
≥
2
) is a bounded smooth domain, ∂
u
/
∂
ν is the outer unit normal derivative on ∂
Ω, p is a continuous function on Ω
¯ with p
-
≔
inf
x
∈
Ω
¯
⁡
p
(
x
)
>
1, and β
∈
L
∞
(
∂
Ω
) with β
-
≔
i
n
f
x
∈
∂
Ω
β
(
x
)
>
0 and f
:
Ω
×
R
→
R is a continuous function. 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 natural extension of the classical p-Laplace operator div
⁡
(
∇
u
p
-
2
∇
u
) obtained in the case when p is a positive constant. However, such generalizations are not trivial since the p
(
x
)-Laplace operator possesses a more complicated structure than p-Laplace operator; for example, it is inhomogeneous.We make the following assumptions on the functionf:(H
0):
f
:
Ω
×
R
→
R is a continuous function and there exist two constants C
1
,
C
2
≥
0 such that (2)
f
x
,
s
≤
C
1
+
C
2
s
q
x
-
1
∀
x
,
s
∈
Ω
×
R
,
whereq
(
x
)
∈
C
(
Ω
) and 1
<
q
(
x
)
<
p
∗
(
x
) for all x
∈
Ω.
(H
1):
the following limit holds uniformly for a.ex
∈
Ω:(3)
l
i
m
s
→
∞
f
x
,
s
s
s
p
+
=
+
∞
;
(H
2):
f
(
x
,
s
)
=
o
(
s
p
(
x
)
-
1
) as s
→
0 and uniformly for x
∈
Ω.
(H
3):
there exist two positive constantsc
1 and c
2 such that (4)
ζ
1
x
,
s
≤
c
1
ζ
1
x
,
t
≤
c
2
ζ
2
x
,
t
,
∀
0
≤
s
≤
t
,
whereζ
1
(
x
,
s
)
=
f
(
x
,
s
)
s
-
p
-
F
(
x
,
s
) and ζ
2
(
x
,
s
)
=
f
(
x
,
s
)
s
-
p
+
F
(
x
,
s
).By the famous Mountain Pass lemma we state the first result.Theorem 1.
Suppose that the conditions (H
0)–(H
3) with q
-
>
p
+ hold. Then problem (1) has at least a nontrivial weak solution.Assume the following hypotheses:(H
4):
(5)
lim
s
→
0
⁡
F
x
,
s
s
p
-
=
∞
;
(H
5):
f
(
x
,
0
)
=
0, and there exists δ
>
0 such that p
-
F
(
x
,
s
)
-
f
(
x
,
s
)
s
>
0 for every x
∈
Ω
¯ and s
≤
δ where F
(
x
,
s
)
=
∫
0
s
f
(
x
,
t
)
d
t.
(H
6):
f
(
x
,
-
s
)
=
-
f
(
x
,
s
), for x
∈
Ω, s
∈
R.We are now in the position to state our second theorem.Theorem 2.
Suppose thatp
:
Ω
¯
→
R is Lipschitz continuous function. Under the assumptions (H
0) and (H
4)–(H
6), problem (1) has a sequence of weak solutions u
n
n such that u
n
L
∞
(
Ω
)
→
0 as n
→
∞.For the next theorem we assume thatf satisfies the following conditions:(H
7):
f
(
x
,
s
)
≥
0 for all x
∈
Ω and s
≥
0 with f
(
x
,
0
)
≠
0.
(H
8):
f
(
x
,
s
) is nondecreasing with respect to s
≥
0, ∀
x
∈
Ω.Theorem 3.
Suppose that the conditions (H
0), (H
7), and (H
8) with p
∈
C
1
(
Ω
¯
) hold. Then problem (1) has a positive solution.Nonlinear boundary-value problems with variable exponent have received considerable attention in recent years. This is partly due to their frequent appearance in applications such as the modeling of electrorheological fluids [1–4] and image processing [5], but these problems are very interesting from a purely mathematical point of view as well. Many results have been obtained on this kind of problems; see for example [6–13]. In [9], the authors have studied the case f
(
x
,
u
)
=
u
p
(
x
)
-
2
u; they proved the existence of infinitely many eigenvalue sequences. Unlike the p-Laplacian case, for a variable exponent p
(
x
) (≠ constant), there does not exist a principal eigenvalue and the set of all eigenvalues is not closed under some assumptions. Finally, they presented some sufficient conditions that the infimum of all eigenvalues is zero and positive, respectively.In [14], the authors obtained results on existence and multiplicity of solutions for problem (1) in the case q
-
>
p
+, under (H
0) and the following Ambrosetti-Rabinowitz type condition:A
R
t
≥
M
⟹
0
≤
μ
F
x
,
t
≤
f
x
,
t
t
,
∃
μ
>
p
+
,
M
>
0
such that
x
∈
Ω
.Here, we notice that (H
3) is much weaker than the A
R condition in the constant exponent case.Very recently, the authors in [15] studied the following problem:(6)
-
Δ
p
x
u
∈
λ
∂
F
x
,
u
i
n
Ω
,
∇
u
p
x
-
2
∂
u
∂
ν
+
β
x
u
p
x
-
2
u
=
0
o
n
∂
Ω
,where λ is a positive parameter, F
(
x
,
t
) is locally Lipschitz function in the t-variable integrand, and ∂
F
(
x
,
t
) is the subdifferential with respect to the t-variable in the sense of Clarke. They claim that problem (6) admits at least two nontrivial solutions.In the first result, we consider problem (1) when the nonlinear term is superlinear at infinity but does not satisfy the A
R type condition, used in [14, 16], which is necessary to ensure the boundedness of the Palais-Smale (PS) type sequences of the associated functional. To overcome these difficulties, we will use the Mountain Pass Theorem [17] with Cerami condition (C) which is weaker than Palais-Smale (PS) condition.In the second result, a distinguishing feature is that we have assumed some conditions only at zero; however, there are no conditions imposed onf at infinity, which is necessary in many works. Finally, in Theorem 3, applying the subsuper solution method we get a positive solution of problem (1).This article is organized as follows. First, we will introduce some basic preliminary results and lemmas in Section2. 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 spacesL
p
(
x
)
(
Ω
) and W
1
,
p
(
x
)
(
Ω
). For more details, see [18, 19]. Suppose that Ω is a bounded open domain of R
N with smooth boundary ∂
Ω and p
∈
C
+
(
Ω
¯
), where (7)
C
+
Ω
¯
=
p
∈
C
Ω
¯
,
inf
x
∈
Ω
¯
⁡
p
x
>
1
.Denote by p
-
≔
i
n
f
x
∈
Ω
¯
p
(
x
) and p
+
≔
s
u
p
x
∈
Ω
¯
p
(
x
). Define the variable exponent Lebesgue space L
p
(
x
)
(
Ω
) by (8)
L
p
x
Ω
=
u
:
Ω
⟶
R
is  measurable  and
∫
Ω
u
p
x
d
x
<
+
∞
,with the norm (9)
u
p
x
=
inf
⁡
τ
>
0
;
∫
Ω
u
τ
p
x
d
x
≤
1
.Define the variable exponent Sobolev space W
1
,
p
(
x
)
(
Ω
) by (10)
W
1
,
p
x
Ω
=
u
∈
L
p
x
Ω
:
∇
u
∈
L
p
x
Ω
,with the norm (11)
u
=
inf
⁡
τ
>
0
;
∫
Ω
∇
u
τ
p
x
+
u
τ
p
x
d
x
≤
1
,
u
=
∇
u
p
x
+
u
p
x
.We refer the reader to [11, 18] for the basic properties of the variable exponent Lebesgue and Sobolev spaces.Lemma 4 (see [19]).
Both(
L
p
x
Ω
,
·
p
x
) and (
W
1
,
p
(
x
)
(
Ω
)
,
·
) are separable and uniformly convex Banach spaces.Lemma 5 (see [19]).
Hölder inequality holds, namely,(12)
∫
Ω
u
v
d
x
≤
2
u
p
x
v
p
′
x
∀
u
∈
L
p
x
Ω
,
v
∈
L
p
′
x
Ω
,where 1
/
p
x
+
1
/
p
′
x
=
1.Lemma 6 (see [18]).
Assume that the boundary ofΩ 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
)
↪
L
q
(
x
)
(
Ω
), where (13)
p
∗
x
=
N
p
x
N
-
p
x
,
i
f
p
x
<
N
;
+
∞
,
i
f
p
x
≥
N
.Now, we introduce a norm, which will be used later.Letβ
∈
L
∞
(
∂
Ω
) with β
-
≔
i
n
f
x
∈
∂
Ω
β
(
x
)
>
0 and, for u
∈
W
1
,
p
(
x
)
(
Ω
), define (14)
u
β
=
inf
⁡
τ
>
0
;
∫
Ω
∇
u
τ
p
x
d
x
+
∫
∂
Ω
β
x
u
τ
p
x
d
σ
≤
1
.Then, by Theorem  2.1 in [16], ·
β is also a norm on W
1
,
p
(
x
)
(
Ω
) which is equivalent to ·.An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mapping defined by the following.Lemma 7 (see [16]).
LetI
(
u
)
=
∫
Ω
|
∇
u
|
p
(
x
)
d
x
+
∫
∂
Ω
β
(
x
)
u
p
(
x
)
d
σ with β
-
>
0. For u
∈
W
1
,
p
(
x
)
(
Ω
) one has ⁡
u
β
<
1
(
=


1
,
>


1
)
⇔
I
(
u
)
<
1
(
=


1
,
>


1
);
⁡
u
β
≤
1
⇒
u
β
p
+
≤
I
(
u
)
≤
u
β
p
-;
⁡
u
β
≥
1
⇒
u
β
p
-
≤
I
(
u
)
≤
u
β
p
+;
⁡
u
n
-
u
β
→
0
⇔
I
(
u
n
-
u
)
→
0.We recall the definition of the following condition (C), see [20].Definition 8 (see [20]).
LetX be a Banach space and J
∈
C
1
(
X
,
R
). Given c
∈
R, one says that J satisfies the Cerami c condition (one denotes condition (C
c)) if (i)
any bounded sequenceu
n
⊂
X such that J
(
u
n
)
→
c and J
′
(
u
n
)
→
0 has a convergent subsequence;
(ii)
there exist constantsδ
,
R
,
β
>
0 such that (15)
J
′
u
u
≥
β
∀
u
∈
J
-
1
c
-
δ
,
c
+
δ
with
u
≥
R
.If J
∈
C
1
(
X
,
R
) satisfies condition (C
c) for every c
∈
R, one says that J satisfies condition (C).Note that condition (C) is weaker than the (PS) condition. However, it was shown in [17] that from condition (C) it is possible to obtain a deformation lemma, which is fundamental in order to get some min-max theorems.Theorem 9 (see [17]).
LetX a Banach space, J
∈
C
1
(
X
,
R
), e
∈
X, and r
>
0, such that e
>
r and (16)
b
≔
inf
u
=
r
⁡
J
u
>
J
0
≥
J
e
.If J satisfies the condition (
C
c
) with (17)
c
≔
inf
γ
∈
Γ
⁡
max
t
∈
0,1
⁡
J
γ
t
,
Γ
≔
γ
∈
C
0,1
,
X
∣
γ
0
=
0
,
γ
1
=
e
.Then c is a critical value of J.Here, problem (1) is stated in the framework of the generalized Sobolev space X
≔
W
1
,
p
(
x
)
(
Ω
).The Euler-Lagrange functional associated with (1) is defined as J
:
X
→
R in (18)
J
u
=
∫
Ω
1
p
x
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
p
x
u
p
x
d
σ
-
∫
Ω
F
x
,
u
d
x
.One says that u
∈
X is a weak solution of (1) if (19)
∫
Ω
∇
u
p
x
-
2
∇
u
∇
v
d
x
+
∫
∂
Ω
β
x
u
p
x
-
2
u
v
d
σ
-
∫
Ω
f
x
,
u
v
d
x
=
0
,for all v
∈
X.Standard arguments imply thatJ
∈
C
1
(
X
,
R
) and (20)
J
′
u
,
v
=
∫
Ω
∇
u
p
x
-
2
∇
u
∇
v
d
x
+
∫
∂
Ω
β
x
u
p
x
-
2
u
v
d
σ
-
∫
Ω
f
x
,
u
v
d
x
,for all u
,
v
∈
X. Thus, the weak solutions of (1) coincide with the critical points of J.


## 3. Proof of Main Results

For simplicity, we useC
i
,
i
=
1,2
,
…, to denote the general positive constants whose exact values may change from line to line.Noting thatJ is the sum of (S
+) type map and a weakly-strongly continuous map, so J
′ is of (S
+) type. To see that Cerami condition (C) holds, it is enough to verify that any Cerami sequence is bounded.Proof of Theorem1.
We check the assumption of compactness of the Mountain Pass Theorem as in the following lemma.
Lemma  10.Suppose that (
H
0
)–(
H
3
) hold. If
c
∈
R
, then any
(
C
)
c
sequence of
J
is bounded.
Proof. Let u
n be a (
C
)
c sequence of J. If u
n is unbounded, up to a subsequence we may assume that(21)
J
u
n
⟶
c
,
u
n
β
⟶
∞
,
J
′
u
n
u
n
β
⟶
0
.
Letw
n
=
u
n
β
-
1
u
n, then w
n is bounded in X; up to a subsequence we have (22)
w
n
⇀
w
in
X
,
w
n
⟶
w
in
L
q
x
Ω
,
w
n
x
⟶
w
x
a.e
x
∈
Ω
.If w
≠
0, we have J
′
(
u
n
)
u
n
=
0; that is,(23)
∫
Ω
∇
u
n
p
x
d
x
+
∫
∂
Ω
β
x
u
n
p
x
d
σ
-
∫
Ω
f
x
,
u
n
u
n
d
x
=
0
.Dividing (23) by u
n
β
p
+, we get (24)
∫
Ω
f
x
,
u
n
u
n
u
n
β
p
+
d
x
<
∞
.On the other side, using (H
1) and lemma of Fatou we obtain (25)
∫
Ω
f
x
,
u
n
u
n
u
n
β
p
+
d
x
=
∫
Ω
f
x
,
u
n
u
n
w
n
p
+
u
n
p
+
d
x
⟶
∞
;we obtain a contradiction.
Ifw
≡
0, since w
n
→
0 in L
q
(
x
)
(
Ω
) and F
x
,
t
≤
C
(
1
+
t
q
(
x
)
), by the continuity of the Nemitskii operator, we see that F
(
·
,
w
n
)
→
0 in L
1
(
Ω
) as n
→
+
∞; therefore,(26)
lim
n
→
∞
⁡
∫
Ω
F
x
,
w
n
d
x
=
0
.We choose a sequence t
n
⊂
[
0,1
] such that (27)
J
t
n
u
n
=
max
t
∈
0,1
⁡
J
t
u
n
.Given m
>
0, since for n large enough we have u
n
β
-
1
(
2
m
p
+
)
1
/
p
-
∈
(
0,1
), using (26) with R
=
(
2
m
p
+
)
1
/
p
-, we obtain(28)
J
t
n
u
n
≥
J
R
u
n
β
u
n
=
J
R
w
n
=
∫
Ω
R
p
x
p
x
∇
w
n
p
x
d
x
+
∫
∂
Ω
R
p
x
p
x
β
x
w
n
p
x
d
σ
-
∫
Ω
F
x
,
R
w
n
d
x
≥
R
p
-
p
+
-
∫
Ω
F
x
,
R
w
n
d
x
≥
m
.That is, J
(
t
n
u
n
)
→
+
∞, but J
(
0
)
=
0, J
(
u
n
)
→
c; we see that t
n
∈
(
0,1
) and J
′
(
t
n
u
n
)
,
t
n
u
n
=
t
n
(
d
/
d
t
)
∣
t
=
t
n
J
(
t
u
n
)
=
0
. It yields (29)
J
t
n
u
n
-
1
p
-
J
′
t
n
u
n
·
t
n
u
n
⟶
+
∞
.Therefore,(30)
∫
Ω
1
p
x
-
1
p
-
t
n
∇
u
n
p
x
d
x
+
∫
∂
Ω
1
p
x
-
1
p
-
β
x
t
n
u
n
p
x
d
σ
+
∫
Ω
1
p
-
f
x
,
t
n
u
n
t
n
u
n
-
F
x
,
t
n
u
n
d
x
⟶
+
∞
,so we get (31)
∫
Ω
1
p
-
f
x
,
t
n
u
n
t
n
u
n
-
F
x
,
t
n
u
n
d
x
⟶
+
∞
.Appropriately, we have (32)
J
u
n
=
J
u
n
-
1
p
+
J
′
u
n
·
u
n
=
∫
Ω
1
p
x
-
1
p
+
∇
u
n
p
x
d
x
+
∫
∂
Ω
1
p
x
-
1
p
+
β
x
u
n
p
x
d
σ
+
∫
Ω
1
p
+
f
x
,
u
n
u
n
-
F
x
,
u
n
d
x
≥
∫
Ω
1
p
+
f
x
,
u
n
u
n
-
F
x
,
u
n
d
x
.
From (H
3), there exist two constants c
1 and c
2 such that(33)
J
u
n
≥
∫
Ω
1
p
+
f
x
,
u
n
u
n
-
F
x
,
u
n
d
x
≥
c
1
∫
Ω
1
p
-
f
x
,
u
n
u
n
-
F
x
,
u
n
d
x
≥
c
1
c
2
∫
Ω
1
p
-
f
x
,
t
n
u
n
t
n
u
n
-
F
x
,
t
n
u
n
d
x
.Hence, J
(
u
n
)
→
+
∞, which is impossible and thus (
u
n
)
n is bounded in X.
We will show thatJ possesses the Mountain Pass geometry.
Lemma11.Under the conditions (
H
0
)–(
H
2
), there exist
r
>
0
and
τ
such that
J
(
u
)
>
τ
when
u
β
=
r.
Proof. In view of (H
0) and (H
2), there exists C
1
>
0 such that (34)
F
x
,
t
≤
1
2
p
+
t
p
x
+
C
1
t
q
x
,
for
x
,
t
∈
Ω
×
R
.Therefore, for u
β
≤
1 we have(35)
J
u
≥
1
p
+
∫
Ω
∇
u
p
x
d
x
+
1
p
+
∫
∂
Ω
β
x
u
p
x
d
σ
-
1
2
p
+
∫
Ω
u
p
x
d
x
-
C
1
∫
Ω
u
q
x
d
x
≥
C
2
2
p
+
∫
Ω
∇
u
p
x
d
x
+
C
2
2
p
+
∫
∂
Ω
β
x
u
p
x
d
σ
-
C
1
∫
Ω
u
q
x
d
x
≥
C
2
2
p
+
u
β
p
+
-
C
3
u
β
q
-
≥
u
β
p
+
C
2
2
p
+
-
C
3
u
β
q
-
-
p
+
.Since p
+
<
q
-, the function t
↦
(
C
2
/
2
p
+
-
C
3
t
q
-
-
p
+
) is strictly positive in a neighborhood of zero. It follows that there exist r
>
0 and τ
>
0 such that(36)
J
u
≥
τ
∀
u
∈
X
:
u
=
r
.
To apply the Mountain Pass Theorem, it suffices to show that(37)
J
t
u
⟶
-
∞
as
t
⟶
+
∞for a certain u
∈
X.
Letu
∈
X
∖
0; by (H
1), we can choose a constant A
>
∫
Ω
(
1
/
p
x
)
∇
u
p
x
d
x
+
∫
∂
Ω
(
β
(
x
)
/
p
(
x
)
)
|
u
|
p
x
d
σ
/
∫
Ω
|
u
|
p
+
d
x, such that (38)
F
x
,
t
≥
A
t
p
+
uniformly in
x
∈
Ω
.
Lett
>
1 be large enough; we have(39)
J
t
u
=
∫
Ω
t
p
x
p
x
∇
u
p
x
d
x
+
∫
∂
Ω
t
p
x
β
x
p
x
u
p
x
d
σ
-
∫
Ω
F
x
,
t
u
d
x
≤
t
p
+
∫
Ω
1
p
x
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
p
x
u
p
x
d
σ
-
∫
t
u
>
C
A
F
x
,
t
u
d
x
-
∫
t
u
≤
C
A
F
x
,
t
u
d
x
≤
t
p
+
∫
Ω
1
p
x
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
p
x
u
p
x
d
σ
-
A
t
p
+
∫
Ω
u
p
+
d
x
-
∫
t
u
≤
C
A
F
x
,
t
u
d
x
+
A
t
p
+
∫
t
u
≤
C
A
u
p
+
d
x
≤
t
p
+
∫
Ω
1
p
x
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
p
x
u
p
x
d
σ
-
A
t
p
+
∫
Ω
u
p
+
d
x
+
C
1
,where C
1
>
0 is a constant, which implies that(40)
J
t
u
⟶
-
∞
as
t
⟶
+
∞
.It follows that there exists e
∈
X such that e
β
>
r and J
(
e
)
<
0. According to the Mountain Pass Theorem, J admits a critical value c which is characterized by(41)
c
=
i
n
f
γ
∈
Γ
sup
t
∈
0,1
⁡
J
γ
t
,where(42)
Γ
=
γ
∈
C
0,1
,
X
:
γ
0
=
0
,
γ
1
=
e
.This completes the proof.Proof of Theorem2.
The main idea (developed by Wang [21]) is to extend f
∈
C
(
Ω
×
(
-
ϵ
,
ϵ
)
,
R
) to an appropriate function f
~
∈
C
(
Ω
×
R
,
R
) in order to prove for the associated modified functional J
¯ the existence of a sequence of weak solutions tending to zero in L
∞ norm. Therefore, it is worth recalling the following proposition.
Proposition12 (see [22]).Let
J
∈
C
1
(
X
,
R
)
, where
X
is a Banach space. Assume that
J
satisfies the (
P
S
) condition and is even and bounded from below, and
J
(
0
)
=
0
. If for any
n
∈
N
, there exists a
k
-dimensional subspace
X
n
and
ρ
n
>
0
such that
(43)
sup
X
n
∩
S
ρ
k
⁡
J
<
0
,
where
S
ρ
:
=
u
∈
X
:
u
=
ρ
, then
J
has a sequence of critical values
c
n
<
0
satisfying
c
n
→
0
as
n
→
∞
.
We need to state the following results.
Claim 1. When J
u
=
J
′
u
·
u
=
0, then u
=
0.
Indeed, suppose thatu
≠
0. Thus (44)
∫
Ω
F
x
,
u
d
x
=
∫
Ω
1
p
x
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
p
x
u
p
x
d
σ
,
∫
Ω
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
u
p
x
d
σ
=
∫
Ω
f
x
,
u
u
d
x
.Then we obtain(45)
p
-
∫
Ω
F
x
,
u
d
x
=
p
-
∫
Ω
1
p
x
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
p
x
u
p
x
d
σ
≤
∫
Ω
f
x
,
u
u
d
x
,which contradicts the assumption (H
5).
Claim 2. There exist δ
>
0 and f
~
∈
C
(
Ω
×
R
) such that f
~ is odd and(46)
f
x
,
t
=
f
~
x
,
t
for
t
<
δ
,
(47)
p
-
F
~
x
,
t
-
f
~
x
,
t
t
≥
0
,
∀
x
,
t
∈
Ω
×
R
,
(48)
p
-
F
~
x
,
t
-
f
~
x
,
t
t
=
0
,
for
t
>
δ
or
t
=
0
,where F
~
(
x
,
t
)
=
∫
0
t
f
~
(
x
,
s
)
d
s.
In fact, let us defineF
~
(
x
,
t
)
=
h
(
t
)
F
(
x
,
t
)
+
C
1
(
1
-
h
(
t
)
)
t
p
- where C
1 is a positive constant and h is a cut-off function presented as follows:(49)
h
t
=
1
,
if
t
≤
δ
2
;
0
,
if
t
≥
δ
,
h
′
t
t
≤
0
,
h
′
t
≤
4
δ
.For |
t
|
≤
δ
/
2, (46) easily holds.
On the other hand, we have(50)
f
~
x
,
t
=
∂
∂
t
F
~
x
,
t
=
h
′
t
F
x
,
t
+
h
t
f
x
,
t
+
C
1
1
-
h
t
t
p
-
′
-
C
1
h
′
t
t
p
-
.It is easy to check that for |
t
|
≥
δ we have (51)
p
-
F
~
x
,
t
=
C
1
p
-
t
p
-
.Hence, (48) is satisfied. In the rest, from (H
4), we can choose δ
>
0 small enough to get F
(
x
,
t
)
≥
C
1
t
p
- when t
∈
[
δ
/
2
,
δ
] and the formula (47) holds since h
′
(
t
)
t
≤
0.
Claim 3. The associated modified functional J
¯ satisfies the Palais-Smale condition.
In fact, by Claim2, it is easy to see that J
¯ is even and J
¯
∈
C
1
(
X
,
R
). For u
β
>
1, we have(52)
J
¯
=
∫
Ω
1
p
x
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
p
x
u
p
x
d
σ
-
∫
Ω
F
~
x
,
u
d
x
≥
1
p
+
∫
Ω
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
u
p
x
d
σ
-
A
∫
Ω
u
p
-
d
x
.Because X
↪
L
p
-
(
Ω
) with A is a positive constant, J
¯ is coercive, that is, J
¯
→
+
∞ as u
β
→
+
∞. Hence, to verify that J
¯ satisfies (P
S) condition on X, it is enough to verify that any (P
S) sequence is bounded. Hence, by the coercivity of J
¯, any (P
S) sequence is bounded in X.
Next, we modify and extendf
(
x
,
u
) to get f
~
(
x
,
u
)
∈
C
(
Ω
¯
×
R
) satisfying the assertions of Proposition 12.
For anyk
∈
N we have k independent smooth functions e
i for i
=
1,2
,
…
,
k, and define the subspace X
k
:
=
span
⁡
e
1
,
…
,
e
k.
From Claim2, for u
β
<
1 we can obtain(53)
J
¯
≤
1
p
-
∫
Ω
∇
u
p
x
d
x
+
∫
∂
Ω
β
x
u
p
x
d
σ
-
C
∫
Ω
u
p
-
d
x
≤
1
p
-
u
β
p
-
-
C
∫
Ω
u
p
-
d
x
.By (53) and as it is well known that all norms in X
k are equivalent, for sufficiently small ρ
k and suitable positive constant C we obtain (54)
sup
X
k
∩
S
ρ
n
⁡
J
¯
<
0
.As a consequence of this fact, we observe that the conditions of Proposition 12 hold and thus there exists a sequence of negative critical values c
k for the functional J
¯ such that c
k
→
0 as k
→
∞.
Afterwards, for anyu
k
∈
X satisfying J
¯
(
u
k
)
=
c
k and J
¯
′
(
u
k
)
=
0, u
k
k is (
P
S
)
0 sequence of J
¯. Passing, if necessary, to a subsequence still denoted by u
k
k, we may suppose that u
k
k has a limit.
From Claims1 and 2 it is clear that 0 is the only critical point when the energy is zero and thus u
k
k converges to 0. It follows from [23, 24] that(55)
u
k
k
∈
C
Ω
¯
,
u
k
L
∞
Ω
⟶
0
as
k
⟶
∞
.So in view of Claim 2, we have |
u
k
|
C
(
Ω
)
≤
δ
/
2. Thereby, the sequences u
k
k are solutions of problem (1).Proof of Theorem3.
Firstly, we recall the definition of subsupersolution of problem (1) as follows. We call u
∈
X a subsolution (resp. supersolution) of (1) if, for every v
∈
X with v
≥
0, (56)
∫
Ω
∇
u
p
x
-
2
∇
u
∇
v
d
x
+
∫
∂
Ω
β
x
u
p
x
-
2
u
v
d
σ
≤
resp
.
≥
∫
Ω
f
x
,
u
v
d
x
.
Lemma13.Let
p
∈
C
1
(
Ω
¯
)
. Suppose that
f
satisfies the subcritical growth condition (
H
0
) and the function
f
(
x
,
t
)
is nondecreasing in
t
∈
R
. If there exist a subsolution
u
0
∈
W
1
,
p
(
x
)
(
Ω
)
∩
L
∞
(
Ω
)
and a supersolution
v
0
∈
W
1
,
p
(
x
)
(
Ω
)
∩
L
∞
(
Ω
)
of (1) such that
u
0
≤
v
0
, then (1) has a minimal solution
u
_
and a maximal solution
v
¯
in the order interval
[
u
0
,
v
0
]
(i.e.,
u
0
≤
u
_
≤
v
¯
≤
v
0
).
The proof of Lemma13 is built on the fixed point theory for the increasing operator on the order interval (see e.g., [25]) and is similar to that given in [26] for the p
(
x
)-Laplacian case.
According to Proposition2.2 in [15], the mapping I
′
:
X
→
X
∗ such that for all u, v in X; 〈
I
′
(
u
)
,
v
〉
=
∫
Ω
∇
u
p
(
x
)
-
2
∇
u
∇
v
d
x
+
∫
∂
Ω
β
(
x
)
|
u
|
p
(
x
)
-
2
u
v
d
σ is a strictly monotone, bounded homeomorphism, and consequently we have the following.
Proposition14.Let
q
∈
C
(
Ω
¯
)
with
1
<
q
(
x
)
<
p
∗
(
x
)
for
x
∈
Ω
¯
, then for
g
∈
L
(
q
x
)
/
(
q
x
-
1
)
(
Ω
)
(or
g
∈
C
0
,
α
(
Ω
)
), the problem
P
1
-
Δ
p
x
u
=
g
x
,
i
n
Ω
;
∇
u
p
x
-
2
∂
u
∂
ν
+
β
x
u
p
x
-
2
u
=
0
,
o
n
∂
Ω
,
has a unique solution
u
in
X
.
Let us consider the following problem:P
2
-
Δ
p
x
u
=
M
,
in
Ω
;
∇
u
p
x
-
2
∂
u
∂
ν
+
β
x
u
p
x
-
2
u
=
0
,
on
∂
Ω
,with M
>
0. By Proposition 14, the strong maximum principle [27] and the result of regularity in [28], problem P
2 has a unique positive solution u
1 such that u
1
(
x
)
>
0 for each x
∈
Ω
¯.
TakingM
=
s
u
p
x
∈
Ω
¯
f
(
x
,
t
), for any v
∈
X with v
≥
0 we have(57)
∫
Ω
∇
u
1
p
x
-
2
∇
u
1
∇
v
d
x
+
∫
∂
Ω
β
x
u
1
p
x
-
2
u
1
v
d
σ
=
∫
Ω
M
v
d
x
≥
∫
Ω
f
x
,
u
1
v
d
x
.Hence, u
1 is a positive supersolution of problem (1).
Obviously 0 is a subsolution of (1). By Lemma 13, (1) has a solution u
∈
[
0
,
u
1
].



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*Source: 2349172-2016-06-13.xml*