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

Authors: Qiaoyu Tian; Yonglin Xu
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, we consider the Brezis-Nirenberg problem for the nonlocal fractional elliptic equationAαux=NN-2αuxp+εux, x∈Ω, ux>0, x∈Ω, u(x)=0, x∈∂Ω, where 0<α<1 is fixed, p=N+2α/N-2α, ε is a small parameter, and Ω is a bounded smooth domain of RN(N≥4α). Aα denotes the fractional Laplace operator defined through the spectral decomposition. Under some geometry hypothesis on the domain Ω, we show that all solutions to this problem are least energy solutions.


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## Body

## 1. Introduction and Main Results

In the famous paper of Brezis and Nirenberg [1], they studied the following nonlinear critical elliptic partial differential equation:(1)-Δux=NN-2uxN+2/N-2+εux,x∈Ω,ux>0,x∈Ω,ux=0,x∈∂Ω,where Ω is a bounded domain in RN(N≥3). They proved that problem (1) has a positive nontrivial solution provided N≥4 and ε∈(0,λ1), where λ1 is the first eigenvalue of -Δ in Ω. This result was extended by Capozzi et al. [2] for every parameter ε. Rey [3] and Han [4] established the asymptotic behavior of positive solutions to problem (1) by different methods independently.In this paper, we study the following nonlocal Brezis-Nirenberg problem:(2)Aαux=NN-2αuxp+εux,x∈Ω,ux>0,x∈Ω,ux=0,x∈∂Ω,where 0<α<1 and 4α<N, p=N+2α/N-2α, Ω is a bounded smooth domain of RN, and Aα is the spectral fractional Laplacian defined in terms of the spectra of the -Δ in Ω; for more details see Section 2.The qualitative properties of solutions to problem (2), such as existence, nonexistence, and multiplicity results, were widely studied; see [5–11] and references therein. It is well-known that [5] problem (2) has at least one positive solution for each small ε>0. On the other hand, when Ω is a C1,1 domain, by the Pohozaev identity [12, 13], we know that problem (2) does not have any solutions in a star-shaped domain when ε=0. Consequently, the solutions to problem (2) blow up at some points as ε→0. Therefore, as ε→0, there exist subsequences εi→0, ui, xi and a point x0∈Ω, such that xi→x0 and ui(xi)≔uixiL∞→∞ as i→∞. In the following, we mainly consider the solution ui.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 u to problem (2); that is, u satisfies(3)limε→0⁡∫ΩAα1/2u2dx∫Ωup+1dx2/p+1=SN,α,where SN,α 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 domainΩ, all solutions to problem (2) automatically satisfy (3); that is, all solutions are least energy solutions.Our main result is the following.Theorem 1.
LetΩ be a smooth bounded domain of RN(N>4α), which is symmetric with respect to the coordinate hyperplanes {xk=0} and convex in the xk-directions for k=1;2⋯;N. 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 x0∈Ω and a family of solutions to (2), which blow up and concentrate at the point x0 as ε→0. Without loss of generality we assume that x0=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 (4)-Δux=NN-2uxN+2/N-2+εux,x∈Ω,ux>0,x∈Ω,ux=0,x∈∂Ω.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. Section2 contains some notations and definitions. Section 3 is concerned with the proof of Theorem 1.


## 2. Useful Definitions

First of all, in this section we recall some basic properties of the spectral fractional Laplacian.In this paper, the letterC will denote a positive constant, not necessarily the same everywhere. Let Ω be a smooth bounded domain of RN. λk>0,k=1,2⋯, are the eigenvalues of the Dirichlet Laplacian on Ω, and ϕk are the corresponding normalized eigenfunctions; namely, (5)Δϕkx=λkϕkx,x∈Ω,ϕkx=0,x∈∂Ω.Define the fractional Laplacian Aα:H0α(Ω)→H0-α(Ω) as(6)Aα∑k=1∞akϕk=∑k=1∞akλkαϕk,where fractional Sobolev space H0α(Ω)(0<α<1) is defined as (7)H0αΩ=f=∑k=1∞λkϕk∈L2Ω:∑k=1∞λk2ϕk2<∞.It is interesting to note that another very popular “integral” fractional Laplacian is defined as(8)-Δsux=P.V.∫RNux-uyx-yn+2sdy=limε→0∫RN∖Bεxux-uyx-yn+2sdy,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 [18–20].In this paper, we mainly consider some properties of isolated blow-up point.Definition 4.
Suppose thatui is a solution to problem (2). The point x¯∈Ω is called a blow-up point of {ui}, if there exists a sequence of point xi∈Ω, such that xi→x¯ and ui(xi)→∞.The concept of an isolated blow-up point was first introduced by Schoen; for more details of the definition of isolated blow-up point, see [17, 21].Definition 5.
Letui be a solution to problem (2). A point x¯∈Ω is an isolated blow-up point of {ui} if there exist 0<r¯<dist(x¯,∂Ω),C>0, and a sequence xi→x¯, such that xi is a local maximum point of ui, ui(xi)→∞, and for any x∈Br¯(xi)(9)uix≤Cx-xi-2α/p-1.


## 3. Proof of Theorem1

In this section, we give the proof of Theorem1, which will be divided into three lemmas.Firstly, according to Remark2, we know that x=0 is the blow-up point; in view of Definition 4, there exists a sequence of point xi∈Ω and {ui}, such that xi→0 and ui(xi)→∞. In the following, the index i is omitted for the sake of simplicity. The following lemma shows that x=0 is an isolated blow-up point.Lemma 6.
Letu(x) be a solution to problem (2). Then there exists a constant C=C(N) such that(10)ux≤CxN-2α/2,x∈Ω.Proof.
Suppose, on the contrary, there exists axj∈Ω, such that(11)Mj≔uxjxjN-2α/2=supx∈Ω⁡uxxN-2α/2→∞.According to Lemma 3.1 in [9], there is a constant C>0 such that supx∈O(Ω,r)u(x)≤C, where (12)OΩ,r=x∈Ω:distx,∂Ω<r,r>0.This fact implies that B(xj,r)⊂Ω. Define (13)vjx=uxj-1uuxj-2/N-2αx+xj,x∈uxj2/N-2αΩ-xj.By Lemma 3.3 in [9], we know that, up to a subsequence, vj(x) converges to the function v uniformly on any compact set, where (14)vx=λ1+λ2x-x02N-2α/2. Obviously, v(0)=1; then for fixed λ≥1, λ=1+λ2|x0|2, which implies that |x0|≤1. Note that vi→v in C1(B(x0,r)) for r>0 and x0 is the extreme point of v. Then there exists a sequence of points xi∈B(x0,r), such that ∇vi(xi)=0. Taking into account Remark 2, we find (15)uxi-2/N-2αx+xi=0. This fact, together with (11), implies that (16)xi=uxi2/N-2αxi→∞, which contradicts the fact that xi∈B(x0,r)⊂Ω. This proves the validity of this lemma.Lemma 7.
Letu(x) be a solution of problem (2). Then there exist two positive constants C and δ, such that(17)ux≤CuxL∞δN-2α,x∈D≔Ω∩x>δ.Remark 8.
It can be easily seen thatlimi→∞ui(x)=0 uniformly for x∈Ω∩{|x|>δ}.Proof.
Analysis similar to that in the proof of Proposition 4.9 in [21] shows that there exists a positive constant C such that, for any |x-xi|≤1, (18)uix≤Cui-1xix-xi2α-N.This fact implies that there exist δ>0 and C>0 such that for any |x|≤δ(19)uix≤Cui-1x2α-N.Particularly, (20)uix≤Cui-1δ2α-N,x=δ.Now we argue by contradiction to show (17) holds. Suppose that there exists a point xi∈D¯, say the maximum point of ui in D¯, such that (21)uix>Cuixi-1δ2α-N>0.It can be easily seen that xi∈D. Thus ∇ui(xi)=0. Consequently, xi=0; this contradicts the fact that xi∈D.Remark 9.
It is worth pointing out that, in [21], Jin et al. consider the integral fractional Laplacian, but the arguments of Propositions 4.4 and 4.9 in [21] still hold for spectral fractional Laplacian.Lemma 10.
Letui(x) be a solution to problem (2). Then(22)limi→∞⁡∫Ωuixp+1dx=∫RNUxp+1dx.Proof.
DecomposeΩ as Ω=Ω1∪Ω2, where Ω1={x∈Ω:|x|≤δ} and Ω2={x∈Ω:|x|>δ}; δ is the constant which appears in Lemma 7; we get (23)∫Ωuixp+1dx=∫Ω1uixp+1dx+∫Ω2uixp+1dx.By (17), it is obvious that ∫Ω2ui(x)p+1dx→0. On the other hand, for any |y|<δui(0)1-p/2α, define (24)u¯iy=ui0-1uiui01-p/2αy.By a similar argument as Proposition 4.4 in [21], we derive that u¯i(y)→U(y) in Cloc2(RN). Therefore, by the dominated convergence theorem, we find (25)∫Ω1uixp+1dx=∫y<uip-1/2αu¯iyp+1dy→∫RNUxp+1dx.This finishes the proof of (22).Proof of Theorem1.
Sinceui(x) is the solution to problem (2), therefore(26)∫ΩAα1/2u2dx=∫ΩAα1/2u·Aα1/2udx=NN-2α∫Ωup+1dx+ε∫Ωu2dx.According to the Hölder inequality, we find(27)∫Ωu2dx≤Ωp-1/p+1∫Ωup+1dx2/p+1.Thus, taking into account (26) and (27), we obtain(28)∫ΩAα1/2u2dx∫Ωup+1dx2/p+1≤NN-2α∫Ωup+1dxp-1/p+1+εΩp-1/p+1.This fact, combined with (22), shows that (3) holds. This completes the proof of Theorem 1.



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*Source: 1093804-2019-07-18.xml*