Existence and Uniqueness of Solution to Semilinear Fractional Elliptic Equation

In this work, we study the following problem. ( ) ( ) ( ) , , 0, \ , s N u g u f x x u x  −∆ + = ∈Ω   = ∈ Ω  , where ( ) −∆ is the fractional Laplacian and Ω is a bounded domain in N with Lipschitz boundary. : g → is an increasing locally Lipschitz continuous function. and ( ) m f L ∈ Ω , 2 2 N m N s ≥ + . We use Stampacchia’s theorem to study existence of the solution u, and we prove the uniqueness of u by contradiction.


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 where ( ) S. J. Liu where ( ) ( ) It is worthy to point out that we can refer [3].

0,1 s ∈
, we can also define the fractional Laplacian ( ) s −∆ as the operator given by the Fourier multiplier 2s ξ ,that is, for where we denote by ( ) We introduce the Sobolev space : and the space ( ) endowed with the norm where . This space allows us to deal with the problems proposed in a bounded domain Ω , as we need.

Preliminaries
In this section, we give some basic results of fractional Sobolev space ( ) H Ω that will be used in the next section.Definition 2.1 We say that ( ) where is the constant defined in (1.3).
Proof.Fixed y we change coordinates z x y = − and apply Plancherel.
Recalling that The integral in brackets is of the form where ( ) is the Bessel function of the first kind of order ( ) can refer [6].

.20)
In conclusion the second term of the right-hand side of (2.18) is less than Lemma 2.6.(Hölder inequality) [11] Let p and q are dual indicators, stisfies the defined function belongs to ( ) L Ω , and we have

.24)
If and only if there is a real constant m that makes the following formula hold e im fg f g = . (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 con- ∈ Ω , and when ∈ Ω , and when ( ) 0 g x ≠ , we have ( )

Proof of Theorem 1.1
Theorem 3.1.Let : g → » » be an increasing function, and g is Lipschitz continuous, that is, there exists a positive constant µ such that for any , s t ∈ » we have ( ) ( ) Proof.We define the following form on ( ) ( )

N s D u x u y w x w y a u w x y g u w x x y
Using Hölder inequality and (3.1) we have

N s N s D u x u y w x w y a u w x y u g w x x y
x y that is, a is well defined.By the definition of a, we know that a is continuous and linear in the second variable.  ( ) the fractional Laplacian and Ω is a bounded domain in N » with Lipschitz boundary.: g → » » is an increasing locally Lipschitz continuous function.and is a bounded domain with Lipschitz boun- dary.( ) s −∆ as the fractional Laplacian, which defined as(

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Journal of Applied Mathematics and Physics for any last inequality following from Hölder inequality and (3.1), by lemma 2.2 ( ) ( . in Ω .