Existence and Uniqueness of Solution to Semilinear Fractional Elliptic Equation ()
1. Introduction
In recent years, many people pay attention to the fractional Laplacian. One of the reasons for this comes from the fact that this operator naturally arises in several phenomena like flames propagation and geophysical fluid dynamics, or in mathematical finance. About the Fractional Sobolev space we can refer [1] [2] . In this work, we consider the problem
(1.1)
where
,
,
is a bounded domain with Lipschitz boundary.
as the fractional Laplacian, which defined as
, (1.2)
where
(1.3)
It is worthy to point out that
, (1.4)
we can refer [3] .
For
, we can also define the fractional Laplacian
as the operator given by the Fourier multiplier
,that is, for
, (1.5)
where we denote by
the class of all Schwartz functions in
.
We introduce the Sobolev space
, (1.6)
and the space
, (1.7)
endowed with the norm
, (1.8)
where
,
. This space allows us to deal with the problems proposed in a bounded domain
, as we need. The pair
yields a Hilbert space [4] . Moreover, it can be seen that
(1.9)
is a continuous operator.
Theorem 1.1. Let
be an increasing locally Lipschitz continuous function. Let
. Then (1.1) have a unique solution
. Moreover,
.
2. Preliminaries
In this section, we give some basic results of fractional Sobolev space
that will be used in the next section.
Definition 2.1 We say that
is a weak solution to (1.1) if we have
, (2.10)
for any
.
Lemma 2.1. [5] Let
and
.Then for all
we have
, (2.11)
where
is the constant defined in (1.3).
Proof. Fixed y we change coordinates
and apply Plancherel.
Recalling that
we obtain
(2.12)
The integral in brackets is of the form
, with
(2.13)
where
is the Bessel function of the first kind of order
, we can refer [6] .
Recall that
. The formula (1.3) for
now follows from
, (2.14)
for
, we can see [5] .
Lemma 2.2. [7] For
,
, there exists a positive constant
, for any
,we have
, (2.15)
where
is called fractional critical Sobolev exponent. In particular, if
then
. (2.16)
Lemma 2.3. (Egorov’s theorem) [8] Let
be a sequence of functions and f be a function defined on E, with
. Assume that
a.e. in E. Then for every
there exists a measurable subset A of E such that
and
uniformly on A,as
.
Lemma 2.4. (Vitali) [9] Let
be a sequence of functions and f be a function in
. Assume that
1)
a.e. in
;
2) if E is a measurable subset of
, and we have
, (2.17)
uniformly with respect n. Where
means measure representing E. Then
in
.
Proof. Fixed
, let
be a measurable set, we have
. (2.18)
Using assumption (2), we know that there exists
such that, if
, then for any
we have
. (2.19)
Since
there exists
such that if
, then
. (2.20)
In conclusion the second term of the right-hand side of (2.18) is less than
. Let us study the first one. We set
, and use Egorov’s theorem, there exist
and a measurable set
such that
, and
, (2.21)
for any
.Choosing
in(2.18), we get the result.
Lemma 2.5. (Stampacchia) [10] Let H be a Hilbert space,
is a continuous and linear form in the second variable such that
1) for
,any
, we have
, (2.22)
2) for a positive constant C, any
we have
. (2.23)
Lemma 2.6. (Hölder inequality) [11] Let p and q are dual indicators, stisfies
,
where
, if
, and
, then the product of
the defined function belongs to
, and we have
. (2.24)
If and only if there is a real constant m that makes the following formula hold
. (2.25)
The first unequal sign of (2.24) is established. If f not constant equals 0,then the second unequal sign of (2.24) is established, if and only if there exists a constant
, such that
1) if
, then
.
2) if
, then
, and when
, we have
.
3) if
, then
, and when
, we have
.
3. Proof of Theorem 1.1
Theorem 3.1. Let
be an increasing function, and g is Lipschitz continuous, that is, there exists a positive constant
such that for any
we have
, (3.1)
Let
. Then (1.1) exists a unique solution
.
Proof. We define the following form on
:
. (3.2)
Using Hölder inequality and (3.1) we have
, (3.3)
that is, a is well defined. By the definition of a, we know that a is continuous and linear in the second variable. If
in
, then
, (3.4)
. (3.5)
Since
(3.6)
the last inequality following from Hölder inequality and (3.1), by lemma 2.2
. (3.7)
Since
(3.8)
by (3.1)
, (3.9)
then
. (3.10)
We know that a satisfies lemma 2.4 from (3.2) and (3.10), the result follows from lemma 2.4.
We define the following function, for
:
(3.11)
Proof of theorem 1.1: First, we proof the existence of a solution by approximation. Let
, By theorem 3.1 we know that there exists
be the solution to problems
(3.12)
We use
as a test function in (3.12), we get
. (3.13)
Then use Hölder inequality on the right-hand side implies
(3.14)
Because g is increasing, then
. This means
is uniformly bounded. We can deduce there exists
weakly in
and a.e., since
, by (3.13) there exists a positive constant C such that
, (3.14)
for every n.
Now we prove
in
. Since g is continuous in
then it is clear that
a.e. in
. If E is a subset of
, for
have
(3.15)
combining (3.14), for
we have
(3.16)
Using lemma 2.4, we know that
in
. Then for any
we from
(3.17)
get
. (3.18)
Finally we prove the solution of problem (1.1) is unique. We assume
and
are two solutions,
, we take
as a test function
(3.19)
(3.20)
We can deduce from (3.19) and (3.20)
(3.21)
This means
. (3.22)
By the monotonicity of g we know
. (3.23)
Combining (3.22) and (3.23) we know
a.e. in
.