The Existence of Solution of a Critical Fractional Equation

In this paper, we study the existence of solution of a critical fractional equation; we will use a variational approach to find the solution. Firstly, we will find a suitable functional to our problem; next, by using the classical concept and properties of the genus, we construct a mini-max class of critical points.


Introduction
In this paper, we focus our attention on the following problem: where Ω is a bounded domain in n R , (1. 2) The aim of this paper is to study the existence of solutions, we will see that if 1 2 p < < , then by concentration-compactness principle, together with mini- max arguments, we can prove the existence of solutions for (1.1).We now summarize the main result of the paper.Theorem 1.1.Let 1 2 p < < , ( ) ( ) ( ) ( ) ( ) ( ) . Moreover, ( ) 0 V x > is bounded on Ω .Then 1) For any 0 λ > , there exists 0 β > , then for any 0 β β < < , (1.1) has a consequence of weak solutions { } n u .
We denote by ( ) s n H R the usual fractional Sobolev space endowed with the so-called Gagliardo norm Then we defined endowed with the norm we refer to [1] for a general definition of ( ) X Ω and its properties.Observe that by [[2], Proposition 3.6] we have the following identity In this work, the Sobolev constant is given by (can be seen in [ [3], theorem where ( )

Statements of the Result
We will use a variational approach to find a solution of (1.1).Firstly, we will associate a suitable functional to our problem, the Euler-Lagrange functional related to problem (1) is given by ( ) To proof that J satisfy the Palais Smale condition at level c, we need the following lemma.
Lemma 2.1 [4] Letting φ be a regular function that satisfies that for some Ω → be a bilinear form defined by then, for every ( ) where B is defined in (2.4).
Lemma 2.4.[5] (Minimax principle) Assume that ( ) If the following conditions holds: 1) c is a finite real number; 2) there exists an 0 ε > , such that  is invariant with respect to the family of mappings; that is, for any T ∈ , there holds ( ) Then, E possesses a ( ) c PS sequence at level c define as (6.1.1);Furthermore, if E satisfies the ( ) c PS condition (or the ( ) c PS condition at level c), then c is a critical value of E.

Proof of Theorem 1.1
Firstly, recalling that J is said to satisfy the Palais Smale condition at level c if any sequence { } ( ) , where ( ) Which also yields the boundedness of ( ) c PS sequence { } n u .Lemma 3.2.Assume that 0 c < .Then 1) For any 0 λ > , there exists 0 0 β > , such that for any 0 0 β β < < , then J satisfies ( ) c PS .2) For any 0 β > there exists 0 0 λ > such that for any Following [6] it is easy to prove that ( ) X Ω could also be the ( ) x ∈ ⊂Ω and two sequences of nonnegative real numbers a nonincreasing cut-off function satisfying taking the derivative of (1.6), for any Then, taking by (3.10), we have x y u x u y u x x y x y Since φ is regular function with compact support, it is easy to see that it satis- fies the hypothesis of Lemma 2.1, by Lemma 2.2 and Lemma 2.3 applied to the sequence { } n u , it follows that the left hand side of (3.13) goes to zero.We obtain that Clearly, if according to the embedded theorem, we have However, if 0 β > is given, we can choose 0 0 λ > so small for every 0 0 λ λ < < that last term on the right-hand side above is greater than 0 which is contradiction when β is the same as λ greater than 0. We see that As consequence, we obtain that ( ) \ 0 : is closed in and symmetric with respect to the orign If there is no mapping φ as above for any m N ∈ , there ( ) = +∞ .we re- fer to [8]for the properties of the genus.
Proposition 3.3.[8] Let , A B ⊂ Α , 1) If there exists an odd map ( ) 4) If S is a sphere centered at the origin in m R , then ( ) According Holder inequality, we get that We define the function Then it is easy to see that given 0 β > , there exists 1 0 λ > so small that for every 1 0 λ λ < < , there exists Analogously, for given 0 λ > , we can choose 1 0 β > with the property that 0 1 , T T as above for each 1 0 β β < < .Clearly, ( ) ( ) As in [9], Let [ ] make the following truncation of the function J: 2) For any 0 λ > , there exists such that if 0 β β < < and 0 c < then J satisfies ( ) c PS .
H. Chen Journal of Applied Mathematics and Physics From the assumptions of ( ) V x , it is easy to see for every Therefore for given λ and β .we can choose PS satisfied, it follows from a standard argument (see [11]) that all m C are critical values of J .

X
Ω .# Next, by using the classical concept and properties of the genus, we construct a min-max class of the critical point.Journal of Applied Mathematics and Physics For a Banach space X, We define the set

C
∞ Ω with the respect to norm

(
< .Since all the norms are equivalent, we get

(
Γ and J is bounded from below.Proposition 3.6.Let , λ β be as in Lemma 3.5 (2) and (3).Then all m c giv- en by(3.25) are critical values of J and 0 Moreover, since ( ) c