Effect of the Domain Geometry on the Solutions to Fractional Brezis-Nirenberg Problem

In this paper

Based on fractional harmonic extension formula of Caffarelli and Silvestre [14] (see Cabré and Tan [15] also), Choi et al. [9] studied the asymptotic behavior of least energy solutions  to problem (2); that is,  satisfies lim where S , is the best constant in fractional Sobolev inequality.For some other related results, see [6,8,16] and references therein.
The main goal of this paper is to show that, under some hypothesis on the domain Ω, all solutions to problem (2) automatically satisfy (3); that is, all solutions are least energy solutions.
Our main result is the following.

Theorem 1.
Let Ω be a smooth bounded domain of R  ( > 4), which is symmetric with respect to the coordinate hyperplanes {  = 0} and convex in the   −directions for  = 1; 2 ⋅ ⋅ ⋅ ; .Then all solutions to problem ( 2) are least energy solutions.
Remark 2. According to Theorem 1.3 in [9], we know that there exist a point  0 ∈ Ω and a family of solutions to (2), which blow up and concentrate at the point  0 as  → 0.
Without loss of generality we assume that  0 = 0 ∈ Ω in this paper.
Remark 3.This results are motivated by the work of Cerqueti and Grossi [17] about the classical Brezis-Nirenberg problem In [17], they obtained the asymptotical behavior of any solution to the above equation in a neighborhood of the origin.Furthermore, uniqueness and nondegeneracy result for the solutions also obtained.
The paper is organized as follows.Section 2 contains some notations and definitions.Section 3 is concerned with the proof of Theorem 1.

Useful Definitions
First of all, in this section we recall some basic properties of the spectral fractional Laplacian.
In this paper, the letter  will denote a positive constant, not necessarily the same everywhere.Let Ω be a smooth bounded domain of R  .  > 0,  = 1, 2 ⋅ ⋅ ⋅ , are the eigenvalues of the Dirichlet Laplacian on Ω, and   are the corresponding normalized eigenfunctions; namely, Define the fractional Laplacian A  :   0 (Ω) →  − 0 (Ω) as where fractional Sobolev space   0 (Ω)(0 <  < 1) is defined as It is interesting to note that another very popular "integral" fractional Laplacian is defined as up to a normalization constant which will be omitted for brevity.For differences between the spectral fractional Laplacian ( 6) and the fractional Laplacian (8), see [18][19][20].
In this paper, we mainly consider some properties of isolated blow-up point.Definition 4. Suppose that   is a solution to problem (2).The point  ∈ Ω is called a blow-up point of {  }, if there exists a sequence of point   ∈ Ω, such that   →  and   (  ) → ∞.

Proof of Theorem 1
In this section, we give the proof of Theorem 1, which will be divided into three lemmas.Firstly, according to Remark 2, we know that  = 0 is the blow-up point; in view of Definition 4, there exists a sequence of point   ∈ Ω and {  }, such that   → 0 and   (  ) → ∞.In the following, the index  is omitted for the sake of simplicity.The following lemma shows that  = 0 is an isolated blow-up point.Lemma 6.Let () be a solution to problem (2).Then there exists a constant  = () such that Proof.Suppose, on the contrary, there exists a   ∈ Ω, such that According to Lemma 3.1 in [9], there is a constant  > 0 such that sup ∈O(Ω,) () ≤ , where This fact implies that (  , ) ⊂ Ω. Define By Lemma 3.3 in [9], we know that, up to a subsequence, V  () converges to the function V uniformly on any compact set, where Obviously, for  > 0 and  0 is the extreme point of V. Then there exists a sequence of points   ∈ ( 0 , ), such that ∇V  (  ) = 0. Taking into account Remark 2, we find This fact, together with (11), implies that which contradicts the fact that   ∈ ( 0 , ) ⊂ Ω.This proves the validity of this lemma.
Proof.Analysis similar to that in the proof of Proposition 4.9 in [21] shows that there exists a positive constant  such that, for any This fact implies that there exist  > 0 and  > 0 such that for any || ≤ Particularly, Now we argue by contradiction to show (17) By a similar argument as Proposition 4.4 in [21], we derive that   () → () in  2  (R  ).Therefore, by the dominated convergence theorem, we find  +  |Ω| (−1)/(+1) . (28) This fact, combined with ( 22), shows that (3) holds.This completes the proof of Theorem 1.