# Existence of Weak Solutions for Nonlinear Time-Fractionalp-Laplace Problems

Authors: Meilan Qiu; Liquan Mei
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

The existence of weak solution forp-Laplace problem is studied in the paper. By exploiting the relationship between the Nehari manifold and fibering maps and combining the compact imbedding theorem and the behavior of Palais-Smale sequences in the Nehari manifold, the existence of weak solutions is established. By means of the Arzela-Ascoli fixed point theorem, some existence results of the corresponding time-fractional equations of the p-Laplace problem are obtained.


---

## Body

## 1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that can be noninteger. The increasing interest of fractional equations is motivated by their applications in various fields of science such as physics, fluid mechanics, heat conduction with memory, chemistry, and engineering [1]. In consequence, the subject of fractional differential equations is gaining diverse and continuous attention. For example, for fractional initial value problems, the existence and multiplicity of solutions (or positive solutions) were discussed in [2–4].In this paper, we consider the following semilinear boundary value problem:(1)
-
div
⁡
(
a
(
x
)
|
∇
u
(
x
)
|
p
-
2
∇
u
(
x
)
)
=
λ
|
u
(
x
)
|
p
-
2
u
(
x
)
+
b
(
x
)
|
u
(
x
)
|
α
-
1
u
(
x
)
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhh
in
Ω
,
u
(
x
)
=
0
,
on
⁡
∂
Ω
,
where Ω is a bounded region with smooth boundary in R
N and a
(
x
) is a positive weight function with positive measure of the Sobolev space W
0
1
,
p
(
a
(
x
)
,
Ω
), u
(
x
)
∈
W
0
1
,
p
(
a
(
x
)
,
Ω
), b
(
x
)
:
Ω
→
R is a smooth function which may change sign, and λ is a real positive parameter and assume throughout that α is a fixed number such that 1
<
α
<
p
-
1
(
2
<
p
≤
p
*
(
p
*
=
n
p
/
(
n
-
p
)
)
). Thus, we will study a sublinear perturbation of a linear problem.The problem (1) is an important and basic mathematical model, widely used in many fields. As for the specific theoretical implicity of the above model, one can see Drábek et al. [5], Adams and Fournier [6], and so on.Similar problems have been studied by Brown and Zhang [7, 8] (when a
(
x
)
≡
1, p
≡
2, with α
>
2) and by Brown [9] (when a
(
x
)
≡
1, p
≡
2, but with 1
<
α
<
2) by using variational viewpoint of the Nehari manifold. When a
(
x
)
≡
1, p
=
2, Amann and Lopez-Gomez [10] have studied the existence of the equation by using global bifurcation theory and Binding et al. [11, 12] used variational methods. Huang and Pu in [13] have studied the following problem:
(2)
-
div
⁡
(
a
(
x
)
|
∇
u
(
x
)
|
p
-
2
∇
u
(
x
)
)
=
λ
b
(
x
)
|
u
(
x
)
|
p
-
2
u
(
x
)
+
c
(
x
)
|
u
(
x
)
|
α
-
1
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
x
∈
Ω
,
u
(
x
)
=
0
,
x
∈
∂
Ω
,
where Ω is a bounded region with smooth boundary in R
N, λ is a real positive parameter, b
(
x
) is a nonnegative function and satisfies b
(
x
)
∈
L
q
/
(
q
-
p
)  (
p
<
q
) or b
(
x
)
∈
L
∞
(
Ω
), and c
(
x
) is a smooth function which may change sign in Ω, and the existence of multiple positive solutions and the properties on Nehari manifold for (2) have been established under the assumption that p
-
1
<
α
<
p
s
*
-
1, where p
s
=
p
s
/
(
s
+
1
), p
s
*
=
N
p
s
/
(
N
-
p
s
), s
∈
(
N
/
p
,
∞
)
∩
[
1
/
(
p
-
1
)
,
∞
), p
s
<
N
(
s
+
1
).In the above, we investigated thep-Laplace Dirichlet problem. Next, we will switch our viewpoint to consider the existence of weak solutions for the corresponding nonlinear time-fractional differential equation of the problem (1). Consider
(
E
λ
,
b
,
t
)
D
β
u
(
x
,
t
)
=
div
⁡
(
a
(
x
)
|
∇
u
(
x
,
t
)
|
p
-
2
∇
u
(
x
,
t
)
)
+
λ
|
u
(
x
,
t
)
|
p
-
2
u
(
x
,
t
)
+
b
(
x
)
|
u
(
x
,
t
)
|
α
-
1
u
(
x
,
t
)
,
in
Ω
T
,
u
(
x
,
t
)
=
0
,
on
⁡
∂
Ω
T
,
u
(
x
,
0
)
=
ϕ
(
x
)
,
in
Ω
,
u
t
(
x
,
0
)
=
ψ
(
x
)
,
in
Ω
,
where Ω
T
=
Ω
×
[
0
,
T
], D
β denotes the Caputo fractional derivatives [14], 0
<
β
≤
1 is a parameter describing the order of the fractional time, and ϕ
(
x
), ψ
(
x
)
∈
H
0
1
(
Ω
) are given real-valued functions.Recently, the subject of fractional differential equations has emerged as an important area of investigation. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, porous media, engineering, and so forth. For some recent developments on the subject, one can see [14–21]. As far as we know, no contributions exist concerning the existence of weak solutions for the problems as we stated above.


## 2. Notations and Preliminary Results

LetW
0
1
,
p
(
a
(
x
)
,
Ω
) be a weighted Sobolev space with a positive measurable weight function a
(
x
); its norm is defined as ∥
u
∥
=
(
∫
Ω
‍
(
a
(
x
)
|
∇
u
(
x
)
|
p
)
d
x
)
1
/
p, and the function a
(
x
) satisfies ν
(
x
)
/
c
≤
a
(
x
)
≤
c
ν
(
x
), x
∈
Ω, where c
≥
1, ν
(
x
) is also a weight function (see [5, 6, 13]) and satisfies ν
(
x
)
∈
L
loc
⁡
1
(
Ω
), ν
(
x
)
-
1
/
(
p
-
1
)
∈
L
loc
⁡
1
(
Ω
), ν
(
x
)
-
s
∈
L
1
(
Ω
); here, s
∈
(
N
/
p
,
∞
)
∩
[
1
/
(
p
-
1
)
,
∞
). Throughout this paper, we denote by S
p the best Sobolev constant for the imbedding of W
0
1
,
p
(
a
(
x
)
,
Ω
) in L
p
(
Ω
). In particular, ∥
u
∥
L
p
(
Ω
)
≤
S
p
∥
u
∥, for all u
∈
W
0
1
,
p
(
a
(
x
)
,
Ω
). For simplicity, we will denote W
0
1
,
p
(
a
(
x
)
,
Ω
) by X and denote  ∥
·
∥
X by ∥
·
∥, and unless otherwise stated, integrals are over Ω.Letλ
1 denote the positive principal eigenvalue of the problem:
(3)
-
div
⁡
(
a
(
x
)
|
∇
u
(
x
)
|
p
-
2
∇
u
(
x
)
)
=
λ
|
u
(
x
)
|
p
-
2
u
(
x
)
,
for
x
∈
Ω
,
u
(
x
)
=
0
,
for
x
∈
∂
Ω
,
with corresponding positive principal eigenfunction ϕ
1. The Euler functional associated with (1) is
(4)
J
λ
(
u
)
=
1
p
∫
‍
a
|
∇
u
|
p
d
x
-
λ
p
∫
|
u
|
p
d
x
-
1
α
+
1
∫
‍
b
(
x
)
|
u
|
α
+
1
d
x
,
for
u
∈
X
.The next lemma shows the behavior of functionalJ
λ on X.Lemma 1.
(i) Suppose thatλ
<
λ
1; then J
λ is bounded below on X.
(ii) Ifλ
>
λ
1, then J
λ is no longer bounded below on X.Proof.
(i) By the spectral theorem, we have(5)
∫
‍
a
|
∇
u
|
p
d
x
-
λ
∫
‍
|
u
|
p
d
x
≥
(
λ
1
-
λ
)
∫
‍
|
u
|
p
d
x
,
∀
u
∈
X
,
and so
(6)
J
λ
(
u
)
≥
1
p
(
λ
1
-
λ
)
∫
‍
|
u
|
p
d
x
-
b
-
α
+
1
∫
‍
|
u
|
α
+
1
d
x
≥
1
p
(
λ
1
-
λ
)
∫
‍
|
u
|
p
d
x
-
b
-
α
+
1
|
Ω
|
1
-
(
α
+
1
)
/
p
(
∫
‍
|
u
|
p
d
x
)
(
α
+
1
)
/
p
,
where b
-
=
sup
⁡
x
∈
Ω
b
(
x
). Hence, J
λ is bounded below on X when λ
<
λ
1.
(ii) Ifλ
>
λ
1, then lim
⁡
t
→
∞
⁡
J
λ
(
t
ϕ
1
)
=
-
∞, so J
λ is unbounded below on X.In order to obtain existence results in the case ofλ
<
λ
1, we introduce the Nehari manifold:
(7)
N
λ
=
{
u
∈
X
:
〈
J
λ
′
(
u
)
,
u
〉
=
0
}
,
where 〈
·
,
·
〉 denotes the usual duality. Thus, u
∈
N
λ if and only if
(8)
∫
‍
a
|
∇
u
|
p
d
x
-
λ
∫
‍
|
u
|
p
d
x
-
∫
‍
b
(
x
)
|
u
|
α
+
1
d
x
=
0
.Obviously,N
λ is a much smaller set than X and so it is easier to study J
λ on N
λ.OnN
λ, we have
(9)
J
λ
(
u
)
=
(
1
p
-
1
α
+
1
)
∫
‍
(
a
|
∇
u
|
p
-
λ
|
u
|
p
)
d
x
=
(
1
p
-
1
α
+
1
)
∫
‍
(
b
|
u
|
α
+
1
)
d
x
.
The Nehari manifold is closely linked to the behavior of functions of the form ϕ
u
(
t
)
:
t
→
J
λ
(
t
u
)
(
t
>
0
). Such maps are known as fibering maps and were introduced by Drabek and Pohozaev [22] and are also discussed in Brown and Zhang [7]. If u
∈
X, we have
(10)
ϕ
u
(
t
)
=
t
p
p
∫
‍
a
|
∇
u
|
p
-
λ
|
u
|
p
d
x
-
t
α
+
1
α
+
1
∫
‍
b
|
u
|
α
+
1
d
x
,
(11)
ϕ
u
′
(
t
)
=
t
p
-
1
∫
‍
a
|
∇
u
|
p
-
λ
|
u
|
p
d
x
-
t
α
∫
‍
b
|
u
|
α
+
1
d
x
,
(12)
ϕ
u
′′
(
t
)
=
(
p
-
1
)
t
p
-
2
∫
‍
a
|
∇
u
|
p
-
λ
|
u
|
p
d
x
-
α
t
α
-
1
∫
‍
b
|
u
|
α
+
1
d
x
.
We can see that u
∈
N
λ if and only if ϕ
u
′
(
1
)
=
0 and more generally that ϕ
u
′
(
t
)
=
0 if and only if t
u
∈
N
λ; that is, elements in N
λ correspond to stationary points of fibering maps. It follows from (11) and (12) that if ϕ
u
′
(
t
)
=
0, then ϕ
u
′′
(
t
)
=
t
α
-
1
[
(
p
-
1
)
-
α
]
∫
‍
b
|
u
|
α
+
1
d
x. Thus, it is natural to subdivide N
λ into three parts corresponding to local minima, local maxima, and points of inflection. Consider
(13)
N
λ
+
=
{
u
∈
N
λ
:
∫
‍
b
|
u
|
α
+
1
d
x
>
0
}
,
N
λ
-
=
{
u
∈
N
λ
:
∫
‍
b
|
u
|
α
+
1
d
x
<
0
}
,
N
λ
0
=
{
u
∈
N
λ
:
∫
‍
b
|
u
|
α
+
1
d
x
=
0
}
.
So N
λ
+, N
λ
-, and N
λ
0 correspond to minima, maxima, and points of inflection, respectively.Letu
∈
X, then(i)
if∫
‍
a
|
∇
u
|
p
-
λ
|
u
|
p
d
x and ∫
‍
b
|
u
|
α
+
1
d
x have the same sign, then ϕ
u
(
t
) has a unique turning point at
(14)
t
(
u
)
=
[
∫
‍
b
|
u
|
α
+
1
d
x
∫
‍
(
a
|
∇
u
|
p
-
λ
|
u
|
p
)
d
x
]
1
/
(
(
p
-
1
)
-
α
)
.
This turning point is a local minimum (maximum) so that t
(
u
)
u
∈
N
λ
+
(
N
λ
-
) if and only if ∫
‍
b
|
u
|
α
+
1
d
x
>
0
(
<
0
).
(ii)
If∫
‍
a
|
∇
u
|
p
-
λ
|
u
|
p
d
x and ∫
‍
b
|
u
|
α
+
1
d
x have different signs, then ϕ
u
(
t
) has no turning points and so no multiples of u lie in N
λ.Thus, if we define(15)
L
+
(
λ
)
=
{
u
∈
X
:
∥
u
∥
=
1
,
∫
‍
a
|
∇
u
|
p
-
λ
|
u
|
p
d
x
>
0
}
,
B
+
(
λ
)
=
{
u
∈
X
:
∥
u
∥
=
1
,
∫
‍
b
|
u
|
α
+
1
d
x
>
0
}
.
Analogously, we can define L
-
(
λ
), L
0
(
λ
), B
-
(
λ
), and B
0
(
λ
) by replacing “>
0” by “<
0” or “=
0.” As appropriate, we have the following:(i)
ifu
∈
L
+
(
λ
)
∩
B
+
(
λ
), then t
→
ϕ
u
(
t
) has a local minimum at t
=
t
(
u
) and t
(
u
)
u
∈
N
λ
+,
(ii)
ifu
∈
L
-
(
λ
)
∩
B
-
(
λ
), then t
→
ϕ
u
(
t
) has a local maximum at t
=
t
(
u
) and t
(
u
)
u
∈
N
λ
-,
(iii)
ifu
∈
L
+
(
λ
)
∩
B
-
(
λ
), then t
→
ϕ
u
(
t
) is strictly increasing and no multiple of u lies in N
λ,
(iv)
ifu
∈
L
-
(
λ
)
∩
B
+
(
λ
), then t
→
ϕ
u
(
t
) is strictly decreasing and no multiple of u lies in N
λ.Next, we will prove the existence of solutions of (1) by investigating the existence of minimizers on N
λ. The following lemma proved that minimizers on N
λ are “usually” critical points for J
λ.Lemma 2.
Suppose thatu
0
∈
N
λ is a local maximum or minimum point for J
λ on N
λ, u
0
∉
N
λ
0; then J
λ
′
(
u
0
)
=
0 in X
-
1
(
Ω
).Proof.
Ifu
0 is a local minimizer point for J
λ on N
λ, then u
0 is a solution of the optimization problem:
(16)
Minimize
J
λ
(
u
)
subject
to
r
(
u
)
=
〈
J
λ
′
(
u
)
,
u
〉
=
0
.
Hence, by the theory of Lagrange multipliers, there existsμ
∈
R such that
(17)
J
λ
′
(
u
0
)
=
μ
r
λ
′
(
u
0
)
,
in
X
-
1
(
Ω
)
.
Thus,
(18)
〈
J
λ
′
(
u
0
)
,
u
0
〉
X
=
μ
〈
r
λ
′
(
u
0
)
,
u
0
〉
X
.
Since u
0
∈
N
λ, 〈
J
λ
′
(
u
0
)
,
u
0
〉
=
0 and so
(19)
∫
‍
(
a
|
∇
u
0
|
p
-
λ
|
u
0
|
p
)
d
x
=
∫
‍
b
|
u
0
|
α
+
1
d
x
.
Hence,
(20)
〈
r
λ
′
(
u
0
)
,
u
0
〉
X
=
p
∫
‍
(
a
|
∇
u
0
|
p
-
λ
|
u
0
|
p
)
d
x
-
(
α
+
1
)
∫
‍
b
|
u
0
|
α
+
1
d
x
=
[
(
p
-
1
)
-
α
]
∫
‍
(
a
|
∇
u
0
|
p
-
λ
|
u
0
|
p
)
d
x
.
Thus, if u
0
∉
N
λ
0, 〈
r
λ
′
(
u
0
)
,
u
0
〉
X
≠
0  and so by (18) μ
=
0. This completes the proof.


## 3. Properties of the Nehari Manifold

In this section, we will discuss the vital role played by the conditionL
-
(
λ
)
⊆
B
-
(
λ
) in determining the nature of the Nehari manifold. When λ
<
λ
1, ∫
‍
(
a
|
∇
u
|
p
-
λ
|
u
|
p
)
d
x
>
0, for all u
∈
X, and so L
+
(
λ
)
=
{
u
∈
X
:
∥
u
∥
=
1
} and L
-
(
λ
)
,
L
0
(
λ
)
=
∅. When λ
=
λ
1, we have L
-
(
λ
)
=
∅ and L
0
(
λ
)
=
{
ϕ
1
} and when λ is greater than λ
1, L
-
(
λ
) becomes nonempty and gets bigger as λ increases.Theorem 3.
Suppose there existsλ
~ such that, for all λ
<
λ
~, L
-
(
λ
)
⊆
B
-
(
λ
). Then, for ∀
λ
<
λ
~,(i)
L
0
(
λ
)
⊆
B
-
(
λ
) and so L
0
(
λ
)
∩
B
0
(
λ
)
=
∅,
(ii)
N
λ
+ is bounded,
(iii)
0
∉
N
λ
-
¯ and N
λ
- is closed,
(iv)
N
λ
+
¯
∩
N
λ
-
=
∅.Proof.
(i) Suppose that the result is false. Then there existsu
∈
L
0
(
λ
) such that u
∉
B
-
(
λ
); if λ
<
μ
<
λ
~, then u
∈
L
-
(
μ
), and so L
-
(
μ
)
⊈
B
-
(
λ
) which is a contradiction.
(ii) Suppose thatN
λ
+ is unbounded. Then there exists {
u
n
}
⊆
N
λ
+ such that ∥
u
n
∥
→
∞ as n
→
∞. Let v
n
=
u
n
/
∥
u
n
∥; without loss of generality, we may assume that v
n
⇀
v
0 in X and so v
n
→
v
0 in L
p
(
Ω
) and in L
α
+
1
(
Ω
). Since u
n
∈
N
λ
+, ∫
‍
(
b
|
v
n
|
α
+
1
)
d
x
>
0 and so ∫
‍
(
b
|
v
0
|
α
+
1
)
d
x
≥
0. Since u
n
∈
N
λ
+
⊆
N
λ, so, by (8), we have
(21)
∫
‍
(
a
|
∇
u
n
|
p
-
λ
|
u
n
|
p
)
d
x
=
∫
‍
b
|
u
n
|
α
+
1
d
x
,and divide by ∥
u
n
∥
p, so
(22)
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
=
∫
‍
(
b
|
v
n
|
α
+
1
∥
u
n
∥
(
α
+
1
)
-
p
)
d
x
⟶
0
.
Suppose v
n
↛
v
0 in X. Then ∫
‍
(
a
|
∇
v
0
|
p
)
d
x
<
lim
⁡
_
n
→
∞
∫
‍
(
a
|
∇
v
n
|
p
)
d
x, and so ∫
‍
(
a
|
∇
v
0
|
p
-
λ
|
v
0
|
p
)
d
x
<
lim
⁡
n
→
∞
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
=
0. Thus, v
0
/
∥
v
0
∥
∈
L
-
(
λ
)
⊆
B
-
(
λ
) which is impossible as ∫
‍
(
b
|
v
0
|
α
+
1
)
d
x
≥
0. Hence, v
n
→
v
0 in X. Thus, ∥
v
0
∥
=
1 and
(23)
∫
‍
(
a
|
∇
v
0
|
p
-
λ
|
v
0
|
p
)
d
x
=
lim
⁡
n
→
∞
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
=
0
.
Thus, v
0
∈
L
0
(
λ
)
⊆
B
-
(
λ
) which is impossible as ∫
‍
(
b
|
v
0
|
α
+
1
)
d
x
≥
0. Therefore, N
λ
+ is bounded.
(iii) Suppose0
∈
N
λ
-
¯. Then there exists {
u
n
}
⊆
N
λ
- such that lim
⁡
n
→
∞
u
n
=
0; let v
n
=
u
n
/
∥
u
n
∥ and then we may assume that v
n
⇀
v
0 in X and so v
n
→
v
0 in L
p
(
Ω
) and in L
α
+
1
(
Ω
). Since u
n
∈
N
λ
-, we have
(24)
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
=
1
∥
u
n
∥
(
p
-
1
)
-
α
∫
‍
(
b
|
v
n
|
α
+
1
)
d
x
≤
0
;
since the {
v
n
} is bounded, it follows that lim
⁡
n
→
∞
∫
‍
(
b
|
v
n
|
α
+
1
)
d
x
=
0 and so ∫
‍
(
b
|
v
0
|
α
+
1
)
d
x
=
0. Suppose v
n
→
v
0 in X; then ∥
v
0
∥
=
1 and so v
0
∈
B
0
(
λ
). Moreover
(25)
∫
‍
(
a
|
∇
v
0
|
p
-
λ
|
v
0
|
p
)
d
x
=
lim
⁡
n
→
∞
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
≤
0
,
and so v
0
∈
L
0
(
λ
) or L
-
(
λ
). Hence, v
0
∈
B
-
(
λ
) and this is impossible as ∫
‍
(
b
|
v
0
|
α
+
1
)
d
x
=
0. Thus, we must have that v
n
↛
v
0 in X; then
(26)
∫
‍
(
a
|
∇
v
0
|
p
-
λ
|
v
0
|
p
)
d
x
<
lim
⁡
n
→
∞
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
≤
0
.
Hence v
0
/
∥
v
0
∥
∈
L
-
(
λ
)
∩
B
0
(
λ
) which is impossible and so 0
∉
N
λ
-
¯. We now prove that N
λ
- is closed. Suppose {
u
n
}
⊆
N
λ
- and u
n
→
u in X. Then u
∈
N
λ
-
¯ and so u
≢
0. Moreover,
(27)
∫
‍
(
a
|
∇
u
|
p
-
λ
|
u
|
p
)
d
x
=
∫
‍
b
|
u
|
α
+
1
d
x
≤
0
.
If both integrals equal 0, then u
/
∥
u
∥
∈
L
0
(
λ
)
∩
B
0
(
λ
) which contradicts (i). Hence both integrals must be negative and so u
∈
N
λ
-. Thus N
λ
- is closed.
(iv) Letu
∈
N
λ
+
¯
∩
N
λ
-, as u
∈
N
λ
-, u
≢
0. Moreover, it is clear that
(28)
∫
‍
(
a
|
∇
u
|
p
-
λ
|
u
|
p
)
d
x
=
∫
‍
b
|
u
|
α
+
1
d
x
=
0
,
and so u
/
∥
u
∥
∈
L
0
(
λ
)
∩
B
0
(
λ
) which is impossible. Thus, N
λ
+
¯
∩
N
λ
-
=
∅.The following theorem presentsJ
λ
(
u
)
>
0 on N
λ
- and the behavior of J
λ
(
u
) on N
λ
+.Theorem 4.
Suppose the same hypotheses are satisfied as in Theorem3; then(i)
J
λ is bounded below on N
λ
+,
(ii)
inf
⁡
u
∈
N
λ
-
J
λ
(
u
)
>
0 provided N
λ
- is nonempty.Proof.
The proof of (i) is an immediate consequence of the boundedness ofN
λ
+.
(ii) Supposeinf
⁡
u
∈
N
λ
-
J
λ
(
u
)
=
0. Then there exists {
u
n
}
⊆
N
λ
- such that lim
⁡
n
→
∞
J
λ
(
u
n
)
=
0. And it is clear from (9) that ∫
‍
(
a
|
∇
u
n
|
p
-
λ
|
u
n
|
p
)
d
x
→
0 and ∫
‍
b
|
u
n
|
α
+
1
d
x
→
0 as n
→
∞. Let v
n
=
u
n
/
∥
u
n
∥; since 0
∉
N
λ
-
¯, {
∥
u
n
∥
} is bounded away from 0. Hence lim
⁡
n
→
∞
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
=
0, and lim
⁡
n
→
∞
∫
‍
b
|
v
n
|
α
+
1
d
x
=
0; we may assume that v
n
⇀
v
0 in X and v
n
→
v
0 in L
p
(
Ω
) and in L
α
+
1
(
Ω
). Then ∫
‍
b
|
v
0
|
α
+
1
d
x
=
0. If v
n
→
v
0 in X, we have ∥
v
0
∥
=
1 and ∫
‍
(
a
|
∇
v
0
|
p
-
λ
|
v
0
|
p
)
d
x
=
0; that is, v
0
∈
L
0
(
λ
), whereas if v
n
↛
v
0 in X, ∫
‍
(
a
|
∇
v
0
|
p
-
λ
|
v
0
|
p
)
d
x
<
0; that is, v
0
/
∥
v
0
∥
∈
L
-
(
λ
). In both cases, however, we must also have v
0
/
∥
v
0
∥
∈
B
0
(
λ
) and this is a contradiction. Thus, inf
⁡
u
∈
N
λ
-
J
λ
(
u
)
>
0.


## 4. The Existence of Weak Solution

We now show that there exists a minimizer onN
λ
+
(
N
λ
-
) which is a critical point of J
λ
(
u
) and so a nontrivial positive solution of (1).Theorem 5.
SupposeL
-
(
λ
)
⊆
B
-
(
λ
) for all λ
<
λ
~; then, for ∀
λ
<
λ
~,
(i) there exists a minimizer point forJ
λ on N
λ
+,
(ii) there exists a minimizer point forJ
λ on N
λ
- provided that L
-
(
λ
) is nonempty.Proof.
(i) By Theorem4, J
λ is bounded below on N
λ
+. Let {
u
n
}
⊆
N
λ
+ be a minimizing sequence; that is,
(29)
lim
⁡
n
→
∞
J
λ
(
u
n
)
=
inf
⁡
u
∈
N
λ
+
J
λ
(
u
)
<
0
.
Since N
λ
+ is bounded, we may assume that u
n
⇀
u
0 in X and u
n
→
u
0 in L
P
(
Ω
) and in L
α
+
1
(
Ω
). Since J
λ
(
u
n
)
=
(
1
/
p
-
1
/
(
α
+
1
)
)
∫
‍
b
|
u
n
|
α
+
1
d
x, it follows that
(30)
∫
‍
b
|
u
0
|
α
+
1
d
x
=
lim
⁡
n
→
∞
∫
‍
b
|
u
n
|
α
+
1
d
x
>
0
,
and so u
0
/
∥
u
0
∥
∈
B
+
(
λ
). Hence, by Theorem 3, u
0
/
∥
u
0
∥
∈
L
+
(
λ
) and so the fibering map ϕ
u
0 has a unique minimum at t
(
u
0
), such that t
(
u
0
)
u
0
∈
N
λ
+. Suppose u
n
↛
u
0 in X; then
(31)
∫
‍
(
a
|
∇
u
0
|
p
-
λ
|
u
0
|
p
)
d
x
<
lim
⁡
n
→
∞
∫
‍
(
a
|
∇
u
n
|
p
-
λ
|
u
n
|
p
)
d
x
=
lim
⁡
n
→
∞
∫
‍
b
|
u
n
|
α
+
1
d
x
=
∫
‍
b
|
u
0
|
α
+
1
d
x
,
and so t
(
u
0
)
>
1. Hence
(32)
J
λ
(
t
(
u
0
)
u
0
)
<
J
λ
(
u
0
)
<
lim
⁡
n
→
∞
J
λ
(
u
n
)
=
inf
⁡
u
∈
N
λ
+
J
λ
(
u
)
,
which is impossible.
Henceu
n
→
u
0 in X and so u
0
∈
N
λ. It now follows easily that u
0 is a minimizer point for J
λ on N
λ
+, since J
λ
(
u
)
=
J
λ
(
|
u
|
) and we may assume that u
0 is a nonnegative in Ω, since J
λ
(
u
0
)
<
0, u
0 is a local minimum point for J
λ on N
λ. It follows from Lemma 2 that u
0 is a critical point of J
λ and so is a weak solution of (1).
(ii) Let{
u
n
}
⊆
N
λ
- be a minimizing sequence. Then by Theorem 4, we must have lim
⁡
n
→
∞
J
λ
(
u
n
)
=
inf
⁡
u
∈
N
λ
-
J
λ
(
u
)
>
0.
Suppose that{
u
n
} is unbounded; we may suppose that ∥
u
n
∥
→
∞ as n
→
∞. Let v
n
=
u
n
/
∥
u
n
∥; since {
J
λ
(
u
n
)
} is bounded, it follows that {
∫
‍
(
a
|
∇
u
n
|
p
-
λ
|
u
n
|
p
)
d
x
} and {
∫
‍
(
b
|
u
n
|
α
+
1
)
d
x
} are bounded and so
(33)
lim
⁡
n
→
∞
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
=
lim
⁡
n
→
∞
∫
‍
(
b
|
v
n
|
α
+
1
)
d
x
=
0
;
since {
v
n
} is bounded, we may assume that v
n
⇀
v
0 in X and v
n
→
v
0 in L
p
(
Ω
) and in L
α
+
1
(
Ω
), so that ∫
‍
(
b
|
v
0
|
α
+
1
)
d
x
=
0. If v
n
→
v
0 in X, it is easy to see that v
0
∈
L
0
(
λ
)
∩
B
0
(
λ
) which is impossible because of Theorem 3(i). Hence, v
n
↛
v
0 in X and so
(34)
∫
‍
(
a
|
∇
v
0
|
p
-
λ
|
v
0
|
p
)
d
x
<
lim
⁡
n
→
∞
∫
‍
(
a
|
∇
v
n
|
p
-
λ
|
v
n
|
p
)
d
x
=
0
;
hence, v
0
≠
0 and v
0
/
∥
v
0
∥
∈
L
-
(
λ
)
∩
B
0
(
λ
), which is again impossible. Thus, {
u
n
} is bounded and so we may assume that u
n
⇀
u
0 in X and u
n
→
u
0 in L
p
(
Ω
) and in L
α
+
1
(
Ω
). Suppose u
n
↛
u
0 in X. Then
(35)
∫
‍
(
b
|
u
0
|
α
+
1
)
d
x
=
lim
⁡
n
→
∞
∫
‍
(
b
|
u
n
|
α
+
1
)
d
x
=
(
1
p
-
1
α
+
1
)
-
1
lim
⁡
n
→
∞
J
λ
(
u
n
)
<
0
,
∫
‍
(
a
|
∇
u
0
|
p
-
λ
|
u
0
|
p
)
d
x
<
lim
⁡
n
→
∞
∫
‍
(
a
|
∇
u
n
|
p
-
λ
|
u
n
|
p
)
d
x
=
lim
⁡
n
→
∞
∫
‍
(
b
|
u
n
|
α
+
1
)
d
x
=
∫
‍
(
b
|
u
0
|
α
+
1
)
d
x
.
Hence u
0
/
∥
u
0
∥
∈
L
-
(
λ
)
∩
B
-
(
λ
), and so t
(
u
0
)
u
0
∈
N
λ
-, where
(36)
t
(
u
0
)
=
[
∫
‍
(
b
|
u
0
|
α
+
1
)
d
x
∫
‍
(
a
|
∇
u
0
|
p
-
λ
|
u
0
|
p
)
d
x
]
1
/
(
(
p
-
1
)
-
α
)
<
1
.
Moreover, t
(
u
0
)
u
n
⇀
t
(
u
0
)
u
0 but t
(
u
0
)
u
n
↛
t
(
u
0
)
u
0 and so
(37)
J
λ
(
t
(
u
0
)
u
0
)
<
lim
⁡
_
n
→
∞
⁡
J
λ
(
t
(
u
0
)
u
n
)
;
since the map t
→
J
λ
(
t
u
n
) attains its maximum at t
=
1,
(38)
lim
⁡
_
n
→
∞
⁡
J
λ
(
t
(
u
0
)
u
n
)
≤
lim
⁡
n
→
∞
J
λ
(
u
n
)
=
inf
⁡
u
∈
N
λ
-
J
λ
(
u
)
.
Hence, J
λ
(
t
(
u
0
)
u
0
)
<
inf
⁡
u
∈
N
λ
-
J
λ
(
u
) which is impossible. Thus, u
n
→
u
0 in X and it follows that u
0 is a minimizer point for J
λ
(
u
) on N
λ
-. Since J
λ
(
u
)
=
J
λ
(
|
u
|
), we may assume that u
0 is a nonnegative in Ω; since N
λ
+ is closed, u
0 is a local minimum point for J
λ on N
λ. It follows from Lemma 2 that u
0 is a critical point of J
λ and so is a weak solution of (1).


## 5. The Corresponding Time-Fractional Equation

In this section, we switch our viewpoint to the fractional order equation(
E
λ
,
b
,
t
) in Sobolev space H
0
1
(
Ω
).To discuss the existence of the positive solution for the equation(
E
λ
,
b
,
t
), we present some basic notations, definitions, and preliminary results which will be used throughout this section.Definition 6 (see [23]).
The Caputo fractional derivative of orderα of a function f
(
t
), t
>
0, is defined as
(39)
D
α
f
(
t
)
=
1
Γ
(
1
-
{
α
}
)
∫
0
t
‍
1
(
t
-
s
)
{
α
}
f
(
[
α
]
+
1
)
d
s
,
where {
α
} and [
α
] denote the fractional and the integer part of the real number α, respectively, and Γ
(
·
) is the Gamma function.Definition 7 (see [23]).
The Riemann-Liouville fractional integral of orderα of a function f
(
t
), t
>
0, is defined as
(40)
I
0
+
α
f
(
t
)
=
1
Γ
(
α
)
∫
0
t
‍
(
t
-
s
)
α
-
1
f
(
s
)
d
s
,
provided that the right side is pointwise defined on (
0
,
∞
).Lemma 8 (see [23]).
Assumey
∈
C
[
0
,
T
], 0
<
T
<
1, and 0
<
α
≤
1; then the problem
(41)
D
α
u
(
t
)
=
y
(
t
)
,
t
∈
[
0
,
T
]
,
has the unique solution
(42)
u
(
t
)
=
u
(
0
)
+
u
′
(
0
)
t
+
1
Γ
(
α
)
∫
0
t
‍
(
t
-
s
)
α
-
1
y
(
s
)
d
s
.Now, we establish some results of the existence of weak solution for(
E
λ
,
b
,
t
).By Lemma8, we may reduce (
E
λ
,
b
,
t
) to an equivalent integral equation as follows:
(
E
λ
,
b
,
integral
)
-
ϕ
(
x
)
-
ψ
(
x
)
t
+
u
(
x
,
t
)
=
1
Γ
(
β
)
∫
0
t
‍
(
t
-
s
)
β
-
1
×
(
div
⁡
(
a
(
x
)
|
∇
u
(
x
,
s
)
|
p
-
2
∇
u
(
x
,
s
)
)
+
λ
|
u
(
x
,
s
)
|
p
-
2
u
(
x
,
s
)
+
b
(
x
)
|
u
(
x
,
s
)
|
α
-
1
u
(
x
,
s
)
)
d
s
,
in
Ω
T
,
u
(
x
,
t
)
=
0
,
on
⁡
∂
Ω
T
.Now we define(
E
λ
,
b
,
fixed
)
Φ
(
u
)
=
ϕ
(
x
)
+
ψ
(
x
)
t
+
1
Γ
(
β
)
∫
0
t
‍
(
t
-
s
)
β
-
1
×
(
div
⁡
(
a
(
x
)
|
∇
u
(
x
,
s
)
|
p
-
2
∇
u
(
x
,
s
)
)
+
λ
|
u
(
x
,
s
)
|
p
-
2
u
(
x
,
s
)
+
b
(
x
)
|
u
(
x
,
s
)
|
α
-
1
u
(
x
,
s
)
)
d
s
,
in
Ω
T
,
u
(
x
,
t
)
=
0
,
on
⁡
∂
Ω
T
.Definition 9.
One callsu
∈
C
(
[
0
,
T
]
;
H
0
1
(
Ω
)
), 0
<
T
<
1 to be a weak solution of the fractional order equation (
E
λ
,
b
,
t
), if ∫
Ω
‍
(
u
-
Φ
(
u
)
)
v
d
x
=
0, ∀
t
∈
[
0
,
T
] for every v
∈
H
0
1
(
Ω
).Lemma 10.
Let∥
a
(
x
)
∥
L
∞
(
Ω
) and ∥
b
(
x
)
∥
L
∞
(
Ω
) be bounded; then the operator Φ
(
u
)
:
H
0
1
(
Ω
)
→
H
1
(
Ω
) is completely continuous.Proof.
Put(43)
F
(
u
)
=
div
⁡
(
a
(
x
)
|
∇
u
(
x
,
s
)
|
p
-
2
∇
u
(
x
,
s
)
)
+
λ
|
u
(
x
,
s
)
|
p
-
2
u
(
x
,
s
)
+
b
(
x
)
|
u
(
x
,
s
)
|
α
-
1
u
(
x
,
s
)
.
We can rewrite
(44)
Φ
(
u
)
=
ϕ
(
x
)
+
ψ
(
x
)
t
+
1
Γ
(
β
)
∫
0
t
‍
(
t
-
s
)
β
-
1
F
(
u
)
d
s
.
For each v
∈
H
0
1
(
Ω
) and ∥
v
∥
H
0
1
(
Ω
)
=
1 integration by parts, we can get
(45)
|
〈
F
(
u
)
,
v
〉
|
=
|
∫
‍
(
a
(
x
)
|
∇
u
|
p
-
1
∇
v
+
λ
|
u
|
p
-
2
u
v
+
b
(
x
)
|
u
|
α
-
1
u
v
)
d
x
∫
‍
(
a
(
x
)
|
∇
u
|
p
-
1
∇
v
)
|
.
By Lemma 2, we know r
(
u
)
=
〈
J
λ
′
(
u
)
,
u
〉
=
0; that is,
(46)
∫
‍
(
a
|
∇
u
|
p
)
d
x
=
∫
‍
λ
|
u
|
p
d
x
+
∫
‍
b
|
u
|
α
+
1
d
x
.
Since ∥
u
∥
L
p
(
Ω
)
≤
S
p
∥
u
∥, so
(47)
∫
‍
λ
|
u
|
p
d
x
≤
|
λ
|
∫
‍
|
u
|
p
d
x
=
|
λ
|
∥
u
∥
L
p
(
Ω
)
p
≤
|
λ
|
S
p
p
∥
u
∥
p
.
And using the same proof as above, we can get ∫
‍
b
|
u
|
α
+
1
d
x
≤
∥
b
∥
L
∞
(
Ω
)
S
α
+
1
α
+
1
∥
u
∥
α
+
1. Thus, we deduce
(48)
∥
u
∥
p
≤
|
λ
|
S
p
p
∥
u
∥
p
+
∥
b
∥
L
∞
(
Ω
)
S
α
+
1
α
+
1
∥
u
∥
α
+
1
.
So
(49)
∥
u
∥
≤
(
∥
b
∥
L
∞
(
Ω
)
S
α
+
1
α
+
1
1
-
|
λ
|
S
p
p
)
1
/
(
(
p
-
1
)
-
α
)
.
And since 2
<
p
≤
p
*
(
p
*
=
n
p
/
(
n
-
p
)
), and a
(
x
) is a positive sufficiently smooth function, there exists a positive constant C, such that ∥
a
(
x
)
∥
L
2
(
Ω
)
≥
∥
a
(
x
)
∥
L
1
(
Ω
)
≥
C. Hence
(50)
∫
Ω
‍
(
a
(
x
)
|
∇
u
|
p
)
d
x
≥
∫
Ω
‍
(
a
(
x
)
|
∇
u
|
2
)
d
x
=
∥
a
(
x
)
∥
L
1
(
Ω
)
∫
Ω
‍
|
∇
u
|
2
d
x
≥
C
∫
Ω
‍
|
∇
u
|
2
d
x
≥
C
∫
Ω
‍
|
u
|
2
d
x
,
∀
x
∈
Ω
,
and
a
.
e
.
time
0
≤
t
≤
T
.
We used Poincare’s inequality in the last inequality above. Thus, by Sobolev imbedding theorem [11], we have
(51)
W
0
1
,
p
(
a
(
x
)
,
Ω
)
↪
H
0
1
(
a
(
x
)
,
Ω
)
↪
H
0
1
(
Ω
)
↪
L
2
(
Ω
)
,
and thus,
(52)
∥
u
∥
H
0
1
(
Ω
)
≤
C
∥
u
∥
W
0
1
,
p
(
a
(
x
)
,
Ω
)
=
C
∥
u
∥
.
In the following, we denote ∥
u
∥
H
0
1
(
Ω
) and ∥
u
∥
H
-
1
(
Ω
) by ∥
u
∥
H
0
1 and ∥
u
∥
H
-
1, respectively. Hence, by Cauchy-Schwarz inequalities, Poincare inequalities, Hölder inequalities, Sobolev imbedding theorem, and (49), for 1
<
α
<
p
-
1, we can get
(53)
|
〈
F
(
u
)
,
v
〉
|
=
|
∫
(
a
(
x
)
|
∇
u
|
p
-
1
∇
v
+
λ
|
u
|
p
-
2
u
v
+
b
(
x
)
|
u
|
α
-
1
u
v
)
d
x
∫
(
a
(
x
)
|
∇
u
|
p
-
1
∇
v
)
|
≤
(
∫
‍
|
a
(
x
)
|
∇
u
|
p
-
1
|
2
d
x
)
1
/
2
(
∫
‍
|
∇
v
|
2
d
x
)
1
/
2
+
(
∫
‍
|
λ
|
u
|
p
-
2
u
|
2
d
x
)
1
/
2
(
∫
‍
|
v
|
2
d
x
)
1
/
2
+
(
∫
‍
|
b
(
x
)
|
u
|
α
-
1
u
|
2
d
x
)
1
/
2
(
∫
‍
|
v
|
2
d
x
)
1
/
2
≤
∥
a
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
|
∇
u
|
p
-
1
|
2
d
x
)
1
/
2
(
∫
‍
|
∇
v
|
2
d
x
)
1
/
2
+
|
λ
|
(
∫
‍
|
|
u
|
p
-
2
u
|
2
d
x
)
1
/
2
(
∫
‍
|
∇
v
|
2
d
x
)
1
/
2
+
∥
b
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
|
u
|
α
-
1
u
|
2
d
x
)
1
/
2
(
∫
‍
|
∇
v
|
2
d
x
)
1
/
2
≤
[
∥
a
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
|
∇
u
|
p
-
1
|
2
d
x
)
1
/
2
+
|
λ
|
(
∫
‍
|
|
u
|
p
-
2
u
|
2
d
x
)
1
/
2
+
∥
b
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
|
u
|
α
-
1
u
|
2
d
x
)
1
/
2
]
∥
v
∥
H
0
1
≤
∥
a
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
|
∇
u
|
p
-
1
|
2
d
x
)
1
/
2
+
|
λ
|
(
∫
‍
|
|
u
|
p
-
2
u
|
2
d
x
)
1
/
2
+
∥
b
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
|
u
|
α
-
1
u
|
2
d
x
)
1
/
2
≤
∥
a
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
∇
u
|
2
(
p
-
1
)
d
x
)
(
1
/
2
(
p
-
1
)
)
(
p
-
1
)
+
|
λ
|
(
∫
‍
|
u
|
2
(
p
-
1
)
d
x
)
(
1
/
2
(
p
-
1
)
)
(
p
-
1
)
+
∥
b
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
u
|
2
α
d
x
)
(
1
/
2
α
)
α
≤
∥
a
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
∇
u
|
2
(
p
-
1
)
d
x
)
(
1
/
2
(
p
-
1
)
)
(
p
-
1
)
+
|
λ
|
(
∫
‍
|
∇
u
|
2
(
p
-
1
)
d
x
)
(
1
/
2
(
p
-
1
)
)
(
p
-
1
)
+
∥
b
(
x
)
∥
L
∞
(
Ω
)
(
∫
‍
|
∇
u
|
2
α
d
x
)
(
1
/
2
α
)
α
=
∥
a
(
x
)
∥
L
∞
(
Ω
)
∥
∇
u
∥
L
2
(
p
-
1
)
(
Ω
)
p
-
1
+
|
λ
|
∥
∇
u
∥
L
2
(
p
-
1
)
(
Ω
)
p
-
1
+
∥
b
(
x
)
∥
L
∞
(
Ω
)
∥
∇
u
∥
L
2
(
α
)
(
Ω
)
α
≤
C
∥
u
∥
H
0
1
p
-
1
+
∥
b
(
x
)
∥
L
∞
(
Ω
)
∥
u
∥
H
0
1
α
≤
C
1
∥
u
∥
p
-
1
+
C
2
∥
b
(
x
)
∥
L
∞
(
Ω
)
∥
u
∥
α
≤
C
1
(
∥
b
(
x
)
∥
L
∞
(
Ω
)
S
α
+
1
α
+
1
1
-
|
λ
|
S
p
p
)
(
p
-
1
)
/
(
(
p
-
1
)
-
α
)
+
C
2
∥
b
(
x
)
∥
L
∞
(
Ω
)
(
∥
b
(
x
)
∥
L
∞
(
Ω
)
S
α
+
1
α
+
1
1
-
|
λ
|
S
p
p
)
α
/
(
(
p
-
1
)
-
α
)
=
M
.
Here, C
=
max
⁡
{
∥
a
(
x
)
∥
L
∞
(
Ω
)
,
|
λ
|
} and C
1, C
2 from the Sobolev imbedding theorem.
Thus, by Cauchy-Schwarz inequalities, we obtain(54)
∥
Φ
(
u
)
∥
H
-
1
=
sup
⁡
∥
v
∥
H
0
1
≤
1
|
〈
Φ
(
u
)
,
v
〉
|
=
sup
⁡
∥
v
∥
H
0
1
≤
1
|
∫
0
t
‍
(
t
-
s
)
β
-
1
〈
F
(
u
)
,
v
〉
d
s
〈
ϕ
(
x
)
,
v
〉
+
〈
ψ
(
x
)
,
v
〉
t
+
1
Γ
(
β
)
∫
0
t
‍
(
t
-
s
)
β
-
1
〈
F
(
u
)
,
v
〉
d
s
|
≤
|
〈
ϕ
(
x
)
,
v
〉
|
+
|
〈
ψ
(
x
)
,
v
〉
t
|
+
|
1
Γ
(
β
)
∫
0
t
‍
(
t
-
s
)
β
-
1
〈
F
(
u
)
,
v
〉
d
s
|
≤
∥
ϕ
(
x
)
∥
L
∞
(
Ω
)
∥
v
∥
H
0
1
+
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
∥
v
∥
H
0
1
T
+
|
〈
F
(
u
)
,
v
〉
|
|
1
Γ
(
β
)
∫
0
t
‍
(
t
-
s
)
β
-
1
d
s
|
≤
∥
ϕ
(
x
)
∥
L
∞
(
Ω
)
+
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
T
+
M
Γ
(
β
)
|
∫
0
t
‍
(
t
-
s
)
β
-
1
d
s
|
≤
∥
ϕ
(
x
)
∥
L
∞
(
Ω
)
+
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
T
+
M
β
Γ
(
β
)
t
β
≤
∥
ϕ
(
x
)
∥
L
∞
(
Ω
)
+
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
T
+
M
β
Γ
(
β
)
T
β
.
Hence, Φ
(
u
) is bounded.
On the other hand, givenϵ
>
0, setting
(55)
δ
=
{
(
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
+
M
Γ
(
β
)
)
-
1
ϵ
}
1
/
β
,
then, for every v
∈
H
0
1
(
Ω
), t
1
<
t
2
,
t
1
,
t
2
∈
[
0
,
T
], 0
<
T
<
1, and t
2
-
t
1
<
δ, one has ∥
Φ
u
(
t
2
)
-
Φ
u
(
t
1
)
∥
H
-
1
=
sup
⁡
∥
v
∥
H
0
1
≤
1
|
〈
Φ
u
(
t
2
)
-
Φ
u
(
t
1
)
,
v
〉
|
≤
ϵ. That is to say, Φ
(
u
) is equicontinuity. In fact,
(56)
∥
Φ
u
(
t
2
)
-
Φ
u
(
t
1
)
∥
H
-
1
=
sup
⁡
∥
v
∥
H
0
1
≤
1
|
〈
Φ
u
(
t
2
)
-
Φ
u
(
t
1
)
,
v
〉
|
=
sup
⁡
∥
v
∥
H
0
1
≤
1
|
∫
0
t
1
‍
(
t
1
-
s
)
β
-
1
〈
F
(
u
)
,
v
〉
d
s
〈
ψ
(
x
)
,
v
〉
(
t
2
-
t
1
)
+
1
Γ
(
β
)
∫
0
t
2
‍
(
t
2
-
s
)
β
-
1
〈
F
(
u
)
,
v
〉
d
s
-
1
Γ
(
β
)
∫
0
t
1
‍
(
t
1
-
s
)
β
-
1
〈
F
(
u
)
,
v
〉
d
s
|
≤
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
∥
v
∥
H
0
1
|
t
2
-
t
1
|
+
1
Γ
(
β
)
|
〈
F
(
u
)
,
v
〉
|
∫
0
t
2
‍
|
(
t
2
-
s
)
β
-
1
|
d
s
+
1
Γ
(
β
)
|
〈
F
(
u
)
,
v
〉
|
∫
0
t
1
‍
|
(
t
2
-
s
)
β
-
1
-
(
t
1
-
s
)
β
-
1
|
d
s
≤
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
|
t
2
-
t
1
|
+
M
β
Γ
(
β
)
t
2
β
-
M
β
Γ
(
β
)
t
1
β
=
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
|
t
2
-
t
1
|
+
M
β
Γ
(
β
)
(
t
2
β
-
t
1
β
)
.
In the following, we divide the proof into two cases.
Case 1. δ
≤
t
1
<
t
2
<
T
<
1; since 0
<
β
≤
1, we get
(57)
∥
Φ
u
(
t
2
)
-
Φ
u
(
t
1
)
∥
H
-
1
=
sup
⁡
∥
v
∥
H
0
1
≤
1
|
〈
Φ
u
(
t
2
)
-
Φ
u
(
t
1
)
,
v
〉
|
≤
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
|
t
2
-
t
1
|
+
M
β
Γ
(
β
)
(
t
2
β
-
t
1
β
)
=
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
|
t
2
-
t
1
|
+
M
β
Γ
(
β
)
β
t
β
-
1
(
t
2
-
t
1
)
≤
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
|
t
2
-
t
1
|
+
M
Γ
(
β
)
δ
1
-
β
|
t
2
-
t
1
|
=
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
δ
+
M
Γ
(
β
)
δ
β
≤
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
δ
β
+
M
Γ
(
β
)
δ
β
=
(
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
+
M
Γ
(
β
)
)
δ
β
≤
ϵ
;
here, t
1
<
t
<
t
2, and we apply the mean theorem t
2
β
-
t
1
β
=
β
t
β
-
1
(
t
2
-
t
1
).
Case 2. 0
≤
t
1
<
δ, t
2
<
β
1
/
β
δ,
(58)
∥
Φ
u
(
t
2
)
-
Φ
u
(
t
1
)
∥
H
-
1
=
sup
⁡
∥
v
∥
H
0
1
≤
1
|
〈
Φ
u
(
t
2
)
-
Φ
u
(
t
1
)
,
v
〉
|
≤
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
|
t
2
-
t
1
|
+
M
β
Γ
(
β
)
(
t
2
β
-
t
1
β
)
≤
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
δ
+
M
β
Γ
(
β
)
(
β
1
/
β
δ
)
β
≤
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
δ
β
+
M
Γ
(
β
)
δ
β
=
(
∥
ψ
(
x
)
∥
L
∞
(
Ω
)
+
M
Γ
(
β
)
)
δ
β
≤
ϵ
.
By the means of the Arzela-Ascoli theorem, we know thatΦ
(
u
)
:
H
0
1
(
Ω
)
→
H
1
(
Ω
) is completely continuous. This completes the proof.By Lemma10, we know that ∫
Ω
‍
(
u
-
Φ
(
u
)
)
v
d
x
=
0, ∀
t
∈
[
0
,
T
], 0
<
T
<
1, for every v
∈
H
0
1
(
Ω
). That is to say, the fractional order equation (
E
λ
,
b
,
t
) has a unique weak solution u
∈
C
(
[
0
,
T
]
;
H
0
1
(
Ω
)
).


## 6. Conclusion

In this paper, we discussed the existence of a class of semilinear equations in the case of1
<
α
<
p
-
1 in a weighted Sobolev space. By exploiting the relationship between the Nehari manifold and fibering maps, we first analyzed the properties of the Nehari manifold in the weighted Sobolev space, using these properties and combining the compact imbedding theorem and the behavior of Palais-Smale sequences on the Nehari manifold, we obtained the existence of the positive solutions for the problem (1). Finally, by using the Arzela-Ascoli fixed point theorem, the existence of weak solution for the time-fractional equation (
E
λ
,
b
,
t
) was obtained.



---

*Source: 231892-2014-07-02.xml*