effect of the domain geometry on the solutions to...
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Research ArticleEffect of the Domain Geometry on the Solutions to FractionalBrezis-Nirenberg Problem
Qiaoyu Tian and Yonglin Xu
School of Mathematics and Computer Northwest Minzu University Lanzhou Gansu 730000 China
Correspondence should be addressed to Qiaoyu Tian tianqiaoyu2004163com
Received 30 April 2019 Revised 7 July 2019 Accepted 7 July 2019 Published 18 July 2019
Academic Editor Yuri Latushkin
Copyright copy 2019 Qiaoyu Tian and Yonglin XuThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
In this paper we consider the Brezis-Nirenberg problem for the nonlocal fractional elliptic equationA120572119906(119909) = 119873(119873 minus 2120572)119906(119909)119901 +120576119906(119909) 119909 isin Ω 119906(119909) gt 0 119909 isin Ω 119906(119909) = 0 119909 isin 120597Ω where 0 lt 120572 lt 1 is fixed 119901 = (119873+2120572)(119873minus2120572) 120576 is a small parameter andΩ is abounded smooth domain ofR119873(119873 ge 4120572)A120572 denotes the fractional Laplace operator defined through the spectral decompositionUnder some geometry hypothesis on the domain Ω we show that all solutions to this problem are least energy solutions
1 Introduction and Main Results
In the famous paper of Brezis and Nirenberg [1] they studiedthe following nonlinear critical elliptic partial differentialequation
minusΔ119906 (119909) = 119873 (119873 minus 2) 119906 (119909)(119873+2)(119873minus2) + 120576119906 (119909) 119909 isin Ω
119906 (119909) gt 0 119909 isin Ω119906 (119909) = 0 119909 isin 120597Ω
(1)
where Ω is a bounded domain in R119873(119873 ge 3) They provedthat problem (1) has a positive nontrivial solution provided119873 ge 4 and 120576 isin (0 1205821) where 1205821 is the first eigenvalueof minusΔ in Ω This result was extended by Capozzi et al [2]for every parameter 120576 Rey [3] and Han [4] established theasymptotic behavior of positive solutions to problem (1) bydifferent methods independently
In this paper we study the following nonlocal Brezis-Nirenberg problem
A120572119906 (119909) = 119873 (119873 minus 2120572) 119906 (119909)119901 + 120576119906 (119909) 119909 isin Ω119906 (119909) gt 0 119909 isin Ω119906 (119909) = 0 119909 isin 120597Ω
(2)
where 0 lt 120572 lt 1 and 4120572 lt 119873 119901 = (119873 + 2120572)(119873 minus 2120572) Ωis a bounded smooth domain of R119873 and A120572 is the spectralfractional Laplacian defined in terms of the spectra of the minusΔinΩ for more details see Section 2
The qualitative properties of solutions to problem (2)such as existence nonexistence andmultiplicity results werewidely studied see [5ndash11] and references therein It is well-known that [5] problem (2) has at least one positive solutionfor each small 120576 gt 0 On the other hand when Ω is a11986211 domain by the Pohozaev identity [12 13] we know thatproblem (2) does not have any solutions in a star-shapeddomain when 120576 = 0 Consequently the solutions to problem(2) blow up at some points as 120576 997888rarr 0 Therefore as 120576 997888rarr 0there exist subsequences 120576119894 997888rarr 0 119906119894 119909119894 and a point 1199090 isin Ωsuch that 119909119894 997888rarr 1199090 and 119906119894(119909119894) fl 119906119894(119909119894)119871infin 997888rarr infin as119894 997888rarr infin In the following we mainly consider the solution119906119894
Based on fractional harmonic extension formula of Caf-farelli and Silvestre [14] (see Cabre and Tan [15] also) Choiet al [9] studied the asymptotic behavior of least energysolutions 119906 to problem (2) that is 119906 satisfies
lim120576997888rarr0
intΩ
10038161003816100381610038161003816A12120572 119906100381610038161003816100381610038162 119889119909(intΩ|119906|119901+1 119889119909)2(119901+1) = S119873120572 (3)
HindawiJournal of Function SpacesVolume 2019 Article ID 1093804 4 pageshttpsdoiorg10115520191093804
2 Journal of Function Spaces
whereS119873120572 is the best constant in fractional Sobolev inequal-ity For some other related results see [6 8 16] and referencestherein
The main goal of this paper is to show that under somehypothesis on the domain Ω all solutions to problem (2)automatically satisfy (3) that is all solutions are least energysolutions
Our main result is the following
Theorem 1 Let Ω be a smooth bounded domain of R119873(119873 gt4120572) which is symmetric with respect to the coordinate hyper-planes 119909119896 = 0 and convex in the 119909119896minusdirections for 119896 =1 2 sdot sdot sdot 119873 Then all solutions to problem (2) are least energysolutions
Remark 2 According to Theorem 13 in [9] we know thatthere exist a point 1199090 isin Ω and a family of solutions to (2)which blow up and concentrate at the point 1199090 as 120576 997888rarr 0Without loss of generality we assume that 1199090 = 0 isin Ω in thispaper
Remark 3 This results are motivated by the work of Cerquetiand Grossi [17] about the classical Brezis-Nirenberg problem
minusΔ119906 (119909) = 119873 (119873 minus 2) 119906 (119909)(119873+2)(119873minus2) + 120576119906 (119909) 119909 isin Ω
119906 (119909) gt 0 119909 isin Ω119906 (119909) = 0 119909 isin 120597Ω
(4)
In [17] they obtained the asymptotical behavior of anysolution to the above equation in a neighborhood of theorigin Furthermore uniqueness and nondegeneracy resultfor the solutions also obtained
Thepaper is organized as follows Section 2 contains somenotations and definitions Section 3 is concerned with theproof of Theorem 1
2 Useful Definitions
First of all in this section we recall some basic properties ofthe spectral fractional Laplacian
In this paper the letter 119862 will denote a positive constantnot necessarily the same everywhere Let Ω be a smoothbounded domain of R119873 120582119896 gt 0 119896 = 1 2 sdot sdot sdot are theeigenvalues of the Dirichlet Laplacian on Ω and 120601119896 are thecorresponding normalized eigenfunctions namely
Δ120601119896 (119909) = 120582119896120601119896 (119909) 119909 isin Ω120601119896 (119909) = 0 119909 isin 120597Ω (5)
Define the fractional LaplacianA120572 1198671205720 (Ω) 997888rarr 119867minus1205720 (Ω) asA120572(infinsum119896=1
119886119896120601119896) = infinsum119896=1
119886119896120582120572119896120601119896 (6)
where fractional Sobolev space1198671205720 (Ω)(0 lt 120572 lt 1) is definedas
1198671205720 (Ω) = 119891 =infinsum119896=1
120582119896120601119896 isin 1198712 (Ω) infinsum119896=1
12058221198961206012119896 lt infin (7)
It is interesting to note that another very popular ldquointe-gralrdquo fractional Laplacian is defined as
(minusΔ)119904 119906 (119909) = 119875119881 intR119873
119906 (119909) minus 119906 (119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899+2119904 119889119910
= lim120576997888rarr0
intR119873119861120576(119909)
119906 (119909) minus 119906 (119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899+2119904 119889119910(8)
up to a normalization constant which will be omitted forbrevity For differences between the spectral fractional Lapla-cian (6) and the fractional Laplacian (8) see [18ndash20]
In this paper we mainly consider some properties ofisolated blow-up point
Definition 4 Suppose that 119906119894 is a solution to problem (2)Thepoint 119909 isin Ω is called a blow-up point of 119906119894 if there exists asequence of point 119909119894 isin Ω such that 119909119894 997888rarr 119909 and 119906119894(119909119894) 997888rarrinfin
The concept of an isolated blow-up point was firstintroduced by Schoen for more details of the definition ofisolated blow-up point see [17 21]
Definition 5 Let 119906119894 be a solution to problem (2) A point 119909 isinΩ is an isolated blow-up point of 119906119894 if there exist 0 lt 119903 lt119889119894119904119905(119909 120597Ω) 119862 gt 0 and a sequence 119909119894 997888rarr 119909 such that 119909119894is a local maximum point of 119906119894 119906119894(119909119894) 997888rarr infin and for any119909 isin 119861119903(119909119894)
119906119894 (119909) le 119862 1003816100381610038161003816119909 minus 1199091198941003816100381610038161003816minus2120572(119901minus1) (9)
3 Proof of Theorem 1
In this section we give the proof of Theorem 1 which will bedivided into three lemmas
Firstly according to Remark 2 we know that 119909 = 0is the blow-up point in view of Definition 4 there exists asequence of point 119909119894 isin Ω and 119906119894 such that 119909119894 997888rarr 0 and119906119894(119909119894) 997888rarr infin In the following the index 119894 is omitted for thesake of simplicity The following lemma shows that 119909 = 0 isan isolated blow-up point
Lemma 6 Let 119906(119909) be a solution to problem (2) Then thereexists a constant 119862 = 119862(119873) such that
119906 (119909) le 119862|119909|(119873minus2120572)2 119909 isin Ω (10)
Proof Suppose on the contrary there exists a 119909119895 isin Ω suchthat
119872119895 fl 119906 (119909119895) 1003816100381610038161003816100381611990911989510038161003816100381610038161003816(119873minus2120572)2 = sup119909isinΩ
119906 (119909) |119909|(119873minus2120572)2997888rarr infin
(11)
Journal of Function Spaces 3
According to Lemma 31 in [9] there is a constant119862 gt 0 suchthat sup119909isinO(Ω119903)119906(119909) le 119862 where
O (Ω 119903) = 119909 isin Ω 119889119894119904119905 (119909 120597Ω) lt 119903 119903 gt 0 (12)
This fact implies that 119861(119909119895 119903) sub Ω DefineV119895 (119909) = 119906 (119909119895)minus1 119906 (119906 (119909119895)minus2(119873minus2120572) 119909 + 119909119895)
119909 isin 119906 (119909119895)2(119873minus2120572) (Ω minus 119909119895) (13)
By Lemma 33 in [9] we know that up to a subsequence V119895(119909)converges to the function V uniformly on any compact setwhere
V (119909) = ( 1205821 + 1205822 1003816100381610038161003816119909 minus 119909010038161003816100381610038162)
(119873minus2120572)2
(14)
Obviously V(0) = 1 then for fixed 120582 ge 1 120582 = 1 + 1205822|1199090|2which implies that |1199090| le 1 Note that V119894 997888rarr V in 1198621(119861(1199090 119903))for 119903 gt 0 and 1199090 is the extreme point of V Then there exists asequence of points119909119894 isin 119861(1199090 119903) such thatnablaV119894(119909119894) = 0 Takinginto account Remark 2 we find
119906 (119909119894)minus2(119873minus2120572) 119909 + 119909119894 = 0 (15)
This fact together with (11) implies that10038161003816100381610038161199091198941003816100381610038161003816 = 119906 (119909119894)2(119873minus2120572) 10038161003816100381610038161199091198941003816100381610038161003816 997888rarr infin (16)
which contradicts the fact that 119909119894 isin 119861(1199090 119903) sub Ω This provesthe validity of this lemma
Lemma 7 Let 119906(119909) be a solution of problem (2) Then thereexist two positive constants 119862 and 120575 such that119906 (119909) le 119862119906 (119909)119871infin 120575119873minus2120572 119909 isin 119863 fl Ω cap |119909| gt 120575 (17)
Remark 8 It can be easily seen that lim119894997888rarrinfin119906119894(119909) = 0uniformly for 119909 isin Ω cap |119909| gt 120575Proof Analysis similar to that in the proof of Proposition 49in [21] shows that there exists a positive constant 119862 such thatfor any |119909 minus 119909119894| le 1
119906119894 (119909) le 119862119906minus1119894 (119909119894) 1003816100381610038161003816119909 minus 11990911989410038161003816100381610038162120572minus119873 (18)
This fact implies that there exist 120575 gt 0 and 119862 gt 0 such thatfor any |119909| le 120575
119906119894 (119909) le 119862 10038171003817100381710038171199061198941003817100381710038171003817minus1 |119909|2120572minus119873 (19)
Particularly
119906119894 (119909) le 119862 10038171003817100381710038171199061198941003817100381710038171003817minus1 1205752120572minus119873 |119909| = 120575 (20)
Now we argue by contradiction to show (17) holds Supposethat there exists a point 119909119894 isin 119863 say the maximum point of 119906119894in119863 such that
119906119894 (119909) gt 119862 1003817100381710038171003817119906119894 (119909119894)1003817100381710038171003817minus1 1205752120572minus119873 gt 0 (21)
It can be easily seen that 119909119894 isin 119863 Thus nabla119906119894(119909119894) = 0Consequently 119909119894 = 0 this contradicts the fact that 119909119894 isin119863
Remark 9 It is worth pointing out that in [21] Jin et alconsider the integral fractional Laplacian but the argumentsof Propositions 44 and 49 in [21] still hold for spectralfractional Laplacian
Lemma 10 Let 119906119894(119909) be a solution to problem (2) Then
lim119894997888rarrinfin
intΩ119906119894 (119909)119901+1 119889119909 = int
R119873119880 (119909)119901+1 119889119909 (22)
Proof Decompose Ω as Ω = Ω1 cup Ω2 where Ω1 = 119909 isin Ω |119909| le 120575 and Ω2 = 119909 isin Ω |119909| gt 120575 120575 is the constant whichappears in Lemma 7 we get
intΩ119906119894 (119909)119901+1 119889119909 = int
Ω1
119906119894 (119909)119901+1 119889119909 + intΩ2
119906119894 (119909)119901+1 119889119909 (23)
By (17) it is obvious that intΩ2119906119894(119909)119901+1119889119909 997888rarr 0 On the other
hand for any |119910| lt 120575119906119894(0)(1minus119901)2120572 define119906119894 (119910) = 119906119894 (0)minus1 119906119894 (119906119894 (0)(1minus119901)2120572 119910) (24)
By a similar argument as Proposition 44 in [21] we derivethat 119906119894(119910) 997888rarr 119880(119910) in1198622119897119900119888(R119873)Therefore by the dominatedconvergence theorem we find
intΩ1
119906119894 (119909)119901+1 119889119909 = int|119910|lt119906119894
(119901minus1)2120572119906119894 (119910)119901+1 119889119910
997888rarr intR119873119880 (119909)119901+1 119889119909
(25)
This finishes the proof of (22)
Proof ofTheorem 1 Since 119906119894(119909) is the solution to problem (2)therefore
intΩ
10038161003816100381610038161003816A12120572 119906100381610038161003816100381610038162 119889119909 = intΩA12120572 119906 sdotA12120572 119906 119889119909
= 119873 (119873 minus 2120572)intΩ119906119901+1119889119909
+ 120576intΩ1199062119889119909
(26)
According to the Holder inequality we find
intΩ1199062119889119909 le |Ω|(119901minus1)(119901+1) (int
Ω119906119901+1119889119909)2(119901+1) (27)
Thus taking into account (26) and (27) we obtain
intΩ
10038161003816100381610038161003816A12120572 119906100381610038161003816100381610038162 119889119909(intΩ119906119901+1119889119909)2(119901+1)
le 119873 (119873 minus 2120572) (intΩ119906119901+1119889119909)(119901minus1)(119901+1)
+ 120576 |Ω|(119901minus1)(119901+1)
(28)
This fact combined with (22) shows that (3) holds Thiscompletes the proof of Theorem 1
4 Journal of Function Spaces
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All the authors equally contributed in this work
Acknowledgments
This work is partially supported by the National Sci-ence Foundation of China (No 11761059) General Top-ics of National Education and Scientific Research (NoZXYB18019) Fundamental Research Funds for the CentralUniversities (No 31920170001) and Key Subject of GansuProvince
References
[1] H Brezis and L Nirenberg ldquoPositive solutions of nonlinearelliptic equations involving critical Sobolev exponentsrdquo Com-munications on Pure and Applied Mathematics vol 36 no 4pp 437ndash477 1983
[2] A Capozzi D Fortunato and G Palmieri ldquoAn existenceresult for nonlinear elliptic problems involving critical Sobolevexponentrdquo Annales de lrsquoInstitut Henri Poincare (C) Analyse NonLineaire vol 2 no 6 pp 463ndash470 1985
[3] O Rey ldquoThe role of the Greenrsquos function in a nonlinear ellipticequation involving the critical Sobolev exponentrdquo Journal ofFunctional Analysis vol 89 no 1 pp 1ndash52 1990
[4] Z Han ldquoAsymptotic approach to singular solutions for non-linear elliptic equations involving critical Sobolev exponentrdquoAnnales de lrsquoInstitut Henri Poincare (C) Analyse Non Lineairevol 8 no 2 pp 159ndash174 1991
[5] B Barrios E Colorado A de Pablo and U Sanchez ldquoOn somecritical problems for the fractional Laplacian operatorrdquo Journalof Differential Equations vol 252 no 11 pp 6133ndash6162 2012
[6] M Bhakta and D Mukherjee ldquoNonlocal scalar field equationsQualitative properties asymptotic profiles and local uniquenessof solutionsrdquo Journal of Differential Equations vol 266 no 11pp 6985ndash7037 2019
[7] C Bucur and M Medina ldquoA fractional elliptic problem in RNwith critical growth and convex nonlinearitiesrdquo ManuscriptaMathematica vol 158 no 3-4 pp 371ndash400 2019
[8] W Choi and S Kim ldquoClassification of finite energy solutionsto the fractional LanendashEmdenndashFowler equations with slightlysubcritical exponentsrdquoAnnali di Matematica Pura ed Applicatavol 196 no 1 pp 269ndash308 2017
[9] W Choi S Kim and K Lee ldquoAsymptotic behavior of solutionsfor nonlinear elliptic problems with the fractional LaplacianrdquoJournal of Functional Analysis vol 266 no 11 pp 6531ndash65982014
[10] S Huang and Q Tian ldquoMarcinkiewicz estimates for solution tofractional elliptic Laplacian equationrdquo Computers ampMathemat-ics with Applications 2019
[11] R Servadei and E Valdinoci ldquoThe Brezis-Nirenberg resultfor the fractional Laplacianrdquo Transactions of the AmericanMathematical Society vol 367 no 1 pp 67ndash102 2015
[12] C Brandle E Colorado A de Pablo and U Sanchez ldquoAconcavemdashconvex elliptic problem involving the fractionalLaplacianrdquo Proceedings of the Royal Society of Edinburgh Sec-tion A Mathematics vol 143 no 1 pp 39ndash71 2013
[13] S Yan J Yang and X Yu ldquoEquations involving fractionallaplacian operator compactness and applicationrdquo Journal ofFunctional Analysis vol 269 no 1 pp 47ndash79 2015
[14] L A Caffarelli and L Silvestre ldquoAn extension problem related tothe fractional LaplacianrdquoCommunications in PartialDifferentialEquations vol 32 no 7ndash9 pp 1245ndash1260 2007
[15] X Cabre and J Tan ldquoPositive solutions of nonlinear problemsinvolving the square root of the Laplacianrdquo Advances in Mathe-matics vol 224 no 5 pp 2052ndash2093 2010
[16] M Bhakta D Mukherjee and S Santra ldquoProfile of solutionsfor nonlocal equations with critical and supercritical nonlinear-itiesrdquo Communications in Contemporary Mathematics vol 21no 01 Article ID 1750099 2019
[17] K Cerqueti and M Grossi ldquoLocal estimates for a semilinearelliptic equation with Sobolev critical exponent and applicationto a uniqueness resultrdquo Nonlinear Differential Equations andApplications NoDEA vol 8 no 3 pp 251ndash283 2001
[18] N Abatangelo and E Valdinoci ldquoGetting acquainted with thefractional laplacian contemporary research in elliptic PDEs andrelated topicsrdquo vol 33 of Springer INdAM Ser Springer ChamSwitzerland 2019
[19] R Musina and A I Nazarov ldquoOn fractional laplaciansrdquo Com-munications in Partial Differential Equations vol 39 no 9 pp1780ndash1790 2014
[20] R Servadei and E Valdinoci ldquoOn the spectrum of two differentfractional operatorsrdquo Proceedings of the Royal Society of Edin-burgh Section AMathematics vol 144 no 4 pp 831ndash855 2014
[21] T Jin Y Li and J Xiong ldquoOn a fractional Nirenberg problempart I blow up analysis and compactness of solutionsrdquo Journalof the EuropeanMathematical Society vol 16 no 6 pp 1111ndash11712014
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2 Journal of Function Spaces
whereS119873120572 is the best constant in fractional Sobolev inequal-ity For some other related results see [6 8 16] and referencestherein
The main goal of this paper is to show that under somehypothesis on the domain Ω all solutions to problem (2)automatically satisfy (3) that is all solutions are least energysolutions
Our main result is the following
Theorem 1 Let Ω be a smooth bounded domain of R119873(119873 gt4120572) which is symmetric with respect to the coordinate hyper-planes 119909119896 = 0 and convex in the 119909119896minusdirections for 119896 =1 2 sdot sdot sdot 119873 Then all solutions to problem (2) are least energysolutions
Remark 2 According to Theorem 13 in [9] we know thatthere exist a point 1199090 isin Ω and a family of solutions to (2)which blow up and concentrate at the point 1199090 as 120576 997888rarr 0Without loss of generality we assume that 1199090 = 0 isin Ω in thispaper
Remark 3 This results are motivated by the work of Cerquetiand Grossi [17] about the classical Brezis-Nirenberg problem
minusΔ119906 (119909) = 119873 (119873 minus 2) 119906 (119909)(119873+2)(119873minus2) + 120576119906 (119909) 119909 isin Ω
119906 (119909) gt 0 119909 isin Ω119906 (119909) = 0 119909 isin 120597Ω
(4)
In [17] they obtained the asymptotical behavior of anysolution to the above equation in a neighborhood of theorigin Furthermore uniqueness and nondegeneracy resultfor the solutions also obtained
Thepaper is organized as follows Section 2 contains somenotations and definitions Section 3 is concerned with theproof of Theorem 1
2 Useful Definitions
First of all in this section we recall some basic properties ofthe spectral fractional Laplacian
In this paper the letter 119862 will denote a positive constantnot necessarily the same everywhere Let Ω be a smoothbounded domain of R119873 120582119896 gt 0 119896 = 1 2 sdot sdot sdot are theeigenvalues of the Dirichlet Laplacian on Ω and 120601119896 are thecorresponding normalized eigenfunctions namely
Δ120601119896 (119909) = 120582119896120601119896 (119909) 119909 isin Ω120601119896 (119909) = 0 119909 isin 120597Ω (5)
Define the fractional LaplacianA120572 1198671205720 (Ω) 997888rarr 119867minus1205720 (Ω) asA120572(infinsum119896=1
119886119896120601119896) = infinsum119896=1
119886119896120582120572119896120601119896 (6)
where fractional Sobolev space1198671205720 (Ω)(0 lt 120572 lt 1) is definedas
1198671205720 (Ω) = 119891 =infinsum119896=1
120582119896120601119896 isin 1198712 (Ω) infinsum119896=1
12058221198961206012119896 lt infin (7)
It is interesting to note that another very popular ldquointe-gralrdquo fractional Laplacian is defined as
(minusΔ)119904 119906 (119909) = 119875119881 intR119873
119906 (119909) minus 119906 (119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899+2119904 119889119910
= lim120576997888rarr0
intR119873119861120576(119909)
119906 (119909) minus 119906 (119910)1003816100381610038161003816119909 minus 1199101003816100381610038161003816119899+2119904 119889119910(8)
up to a normalization constant which will be omitted forbrevity For differences between the spectral fractional Lapla-cian (6) and the fractional Laplacian (8) see [18ndash20]
In this paper we mainly consider some properties ofisolated blow-up point
Definition 4 Suppose that 119906119894 is a solution to problem (2)Thepoint 119909 isin Ω is called a blow-up point of 119906119894 if there exists asequence of point 119909119894 isin Ω such that 119909119894 997888rarr 119909 and 119906119894(119909119894) 997888rarrinfin
The concept of an isolated blow-up point was firstintroduced by Schoen for more details of the definition ofisolated blow-up point see [17 21]
Definition 5 Let 119906119894 be a solution to problem (2) A point 119909 isinΩ is an isolated blow-up point of 119906119894 if there exist 0 lt 119903 lt119889119894119904119905(119909 120597Ω) 119862 gt 0 and a sequence 119909119894 997888rarr 119909 such that 119909119894is a local maximum point of 119906119894 119906119894(119909119894) 997888rarr infin and for any119909 isin 119861119903(119909119894)
119906119894 (119909) le 119862 1003816100381610038161003816119909 minus 1199091198941003816100381610038161003816minus2120572(119901minus1) (9)
3 Proof of Theorem 1
In this section we give the proof of Theorem 1 which will bedivided into three lemmas
Firstly according to Remark 2 we know that 119909 = 0is the blow-up point in view of Definition 4 there exists asequence of point 119909119894 isin Ω and 119906119894 such that 119909119894 997888rarr 0 and119906119894(119909119894) 997888rarr infin In the following the index 119894 is omitted for thesake of simplicity The following lemma shows that 119909 = 0 isan isolated blow-up point
Lemma 6 Let 119906(119909) be a solution to problem (2) Then thereexists a constant 119862 = 119862(119873) such that
119906 (119909) le 119862|119909|(119873minus2120572)2 119909 isin Ω (10)
Proof Suppose on the contrary there exists a 119909119895 isin Ω suchthat
119872119895 fl 119906 (119909119895) 1003816100381610038161003816100381611990911989510038161003816100381610038161003816(119873minus2120572)2 = sup119909isinΩ
119906 (119909) |119909|(119873minus2120572)2997888rarr infin
(11)
Journal of Function Spaces 3
According to Lemma 31 in [9] there is a constant119862 gt 0 suchthat sup119909isinO(Ω119903)119906(119909) le 119862 where
O (Ω 119903) = 119909 isin Ω 119889119894119904119905 (119909 120597Ω) lt 119903 119903 gt 0 (12)
This fact implies that 119861(119909119895 119903) sub Ω DefineV119895 (119909) = 119906 (119909119895)minus1 119906 (119906 (119909119895)minus2(119873minus2120572) 119909 + 119909119895)
119909 isin 119906 (119909119895)2(119873minus2120572) (Ω minus 119909119895) (13)
By Lemma 33 in [9] we know that up to a subsequence V119895(119909)converges to the function V uniformly on any compact setwhere
V (119909) = ( 1205821 + 1205822 1003816100381610038161003816119909 minus 119909010038161003816100381610038162)
(119873minus2120572)2
(14)
Obviously V(0) = 1 then for fixed 120582 ge 1 120582 = 1 + 1205822|1199090|2which implies that |1199090| le 1 Note that V119894 997888rarr V in 1198621(119861(1199090 119903))for 119903 gt 0 and 1199090 is the extreme point of V Then there exists asequence of points119909119894 isin 119861(1199090 119903) such thatnablaV119894(119909119894) = 0 Takinginto account Remark 2 we find
119906 (119909119894)minus2(119873minus2120572) 119909 + 119909119894 = 0 (15)
This fact together with (11) implies that10038161003816100381610038161199091198941003816100381610038161003816 = 119906 (119909119894)2(119873minus2120572) 10038161003816100381610038161199091198941003816100381610038161003816 997888rarr infin (16)
which contradicts the fact that 119909119894 isin 119861(1199090 119903) sub Ω This provesthe validity of this lemma
Lemma 7 Let 119906(119909) be a solution of problem (2) Then thereexist two positive constants 119862 and 120575 such that119906 (119909) le 119862119906 (119909)119871infin 120575119873minus2120572 119909 isin 119863 fl Ω cap |119909| gt 120575 (17)
Remark 8 It can be easily seen that lim119894997888rarrinfin119906119894(119909) = 0uniformly for 119909 isin Ω cap |119909| gt 120575Proof Analysis similar to that in the proof of Proposition 49in [21] shows that there exists a positive constant 119862 such thatfor any |119909 minus 119909119894| le 1
119906119894 (119909) le 119862119906minus1119894 (119909119894) 1003816100381610038161003816119909 minus 11990911989410038161003816100381610038162120572minus119873 (18)
This fact implies that there exist 120575 gt 0 and 119862 gt 0 such thatfor any |119909| le 120575
119906119894 (119909) le 119862 10038171003817100381710038171199061198941003817100381710038171003817minus1 |119909|2120572minus119873 (19)
Particularly
119906119894 (119909) le 119862 10038171003817100381710038171199061198941003817100381710038171003817minus1 1205752120572minus119873 |119909| = 120575 (20)
Now we argue by contradiction to show (17) holds Supposethat there exists a point 119909119894 isin 119863 say the maximum point of 119906119894in119863 such that
119906119894 (119909) gt 119862 1003817100381710038171003817119906119894 (119909119894)1003817100381710038171003817minus1 1205752120572minus119873 gt 0 (21)
It can be easily seen that 119909119894 isin 119863 Thus nabla119906119894(119909119894) = 0Consequently 119909119894 = 0 this contradicts the fact that 119909119894 isin119863
Remark 9 It is worth pointing out that in [21] Jin et alconsider the integral fractional Laplacian but the argumentsof Propositions 44 and 49 in [21] still hold for spectralfractional Laplacian
Lemma 10 Let 119906119894(119909) be a solution to problem (2) Then
lim119894997888rarrinfin
intΩ119906119894 (119909)119901+1 119889119909 = int
R119873119880 (119909)119901+1 119889119909 (22)
Proof Decompose Ω as Ω = Ω1 cup Ω2 where Ω1 = 119909 isin Ω |119909| le 120575 and Ω2 = 119909 isin Ω |119909| gt 120575 120575 is the constant whichappears in Lemma 7 we get
intΩ119906119894 (119909)119901+1 119889119909 = int
Ω1
119906119894 (119909)119901+1 119889119909 + intΩ2
119906119894 (119909)119901+1 119889119909 (23)
By (17) it is obvious that intΩ2119906119894(119909)119901+1119889119909 997888rarr 0 On the other
hand for any |119910| lt 120575119906119894(0)(1minus119901)2120572 define119906119894 (119910) = 119906119894 (0)minus1 119906119894 (119906119894 (0)(1minus119901)2120572 119910) (24)
By a similar argument as Proposition 44 in [21] we derivethat 119906119894(119910) 997888rarr 119880(119910) in1198622119897119900119888(R119873)Therefore by the dominatedconvergence theorem we find
intΩ1
119906119894 (119909)119901+1 119889119909 = int|119910|lt119906119894
(119901minus1)2120572119906119894 (119910)119901+1 119889119910
997888rarr intR119873119880 (119909)119901+1 119889119909
(25)
This finishes the proof of (22)
Proof ofTheorem 1 Since 119906119894(119909) is the solution to problem (2)therefore
intΩ
10038161003816100381610038161003816A12120572 119906100381610038161003816100381610038162 119889119909 = intΩA12120572 119906 sdotA12120572 119906 119889119909
= 119873 (119873 minus 2120572)intΩ119906119901+1119889119909
+ 120576intΩ1199062119889119909
(26)
According to the Holder inequality we find
intΩ1199062119889119909 le |Ω|(119901minus1)(119901+1) (int
Ω119906119901+1119889119909)2(119901+1) (27)
Thus taking into account (26) and (27) we obtain
intΩ
10038161003816100381610038161003816A12120572 119906100381610038161003816100381610038162 119889119909(intΩ119906119901+1119889119909)2(119901+1)
le 119873 (119873 minus 2120572) (intΩ119906119901+1119889119909)(119901minus1)(119901+1)
+ 120576 |Ω|(119901minus1)(119901+1)
(28)
This fact combined with (22) shows that (3) holds Thiscompletes the proof of Theorem 1
4 Journal of Function Spaces
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All the authors equally contributed in this work
Acknowledgments
This work is partially supported by the National Sci-ence Foundation of China (No 11761059) General Top-ics of National Education and Scientific Research (NoZXYB18019) Fundamental Research Funds for the CentralUniversities (No 31920170001) and Key Subject of GansuProvince
References
[1] H Brezis and L Nirenberg ldquoPositive solutions of nonlinearelliptic equations involving critical Sobolev exponentsrdquo Com-munications on Pure and Applied Mathematics vol 36 no 4pp 437ndash477 1983
[2] A Capozzi D Fortunato and G Palmieri ldquoAn existenceresult for nonlinear elliptic problems involving critical Sobolevexponentrdquo Annales de lrsquoInstitut Henri Poincare (C) Analyse NonLineaire vol 2 no 6 pp 463ndash470 1985
[3] O Rey ldquoThe role of the Greenrsquos function in a nonlinear ellipticequation involving the critical Sobolev exponentrdquo Journal ofFunctional Analysis vol 89 no 1 pp 1ndash52 1990
[4] Z Han ldquoAsymptotic approach to singular solutions for non-linear elliptic equations involving critical Sobolev exponentrdquoAnnales de lrsquoInstitut Henri Poincare (C) Analyse Non Lineairevol 8 no 2 pp 159ndash174 1991
[5] B Barrios E Colorado A de Pablo and U Sanchez ldquoOn somecritical problems for the fractional Laplacian operatorrdquo Journalof Differential Equations vol 252 no 11 pp 6133ndash6162 2012
[6] M Bhakta and D Mukherjee ldquoNonlocal scalar field equationsQualitative properties asymptotic profiles and local uniquenessof solutionsrdquo Journal of Differential Equations vol 266 no 11pp 6985ndash7037 2019
[7] C Bucur and M Medina ldquoA fractional elliptic problem in RNwith critical growth and convex nonlinearitiesrdquo ManuscriptaMathematica vol 158 no 3-4 pp 371ndash400 2019
[8] W Choi and S Kim ldquoClassification of finite energy solutionsto the fractional LanendashEmdenndashFowler equations with slightlysubcritical exponentsrdquoAnnali di Matematica Pura ed Applicatavol 196 no 1 pp 269ndash308 2017
[9] W Choi S Kim and K Lee ldquoAsymptotic behavior of solutionsfor nonlinear elliptic problems with the fractional LaplacianrdquoJournal of Functional Analysis vol 266 no 11 pp 6531ndash65982014
[10] S Huang and Q Tian ldquoMarcinkiewicz estimates for solution tofractional elliptic Laplacian equationrdquo Computers ampMathemat-ics with Applications 2019
[11] R Servadei and E Valdinoci ldquoThe Brezis-Nirenberg resultfor the fractional Laplacianrdquo Transactions of the AmericanMathematical Society vol 367 no 1 pp 67ndash102 2015
[12] C Brandle E Colorado A de Pablo and U Sanchez ldquoAconcavemdashconvex elliptic problem involving the fractionalLaplacianrdquo Proceedings of the Royal Society of Edinburgh Sec-tion A Mathematics vol 143 no 1 pp 39ndash71 2013
[13] S Yan J Yang and X Yu ldquoEquations involving fractionallaplacian operator compactness and applicationrdquo Journal ofFunctional Analysis vol 269 no 1 pp 47ndash79 2015
[14] L A Caffarelli and L Silvestre ldquoAn extension problem related tothe fractional LaplacianrdquoCommunications in PartialDifferentialEquations vol 32 no 7ndash9 pp 1245ndash1260 2007
[15] X Cabre and J Tan ldquoPositive solutions of nonlinear problemsinvolving the square root of the Laplacianrdquo Advances in Mathe-matics vol 224 no 5 pp 2052ndash2093 2010
[16] M Bhakta D Mukherjee and S Santra ldquoProfile of solutionsfor nonlocal equations with critical and supercritical nonlinear-itiesrdquo Communications in Contemporary Mathematics vol 21no 01 Article ID 1750099 2019
[17] K Cerqueti and M Grossi ldquoLocal estimates for a semilinearelliptic equation with Sobolev critical exponent and applicationto a uniqueness resultrdquo Nonlinear Differential Equations andApplications NoDEA vol 8 no 3 pp 251ndash283 2001
[18] N Abatangelo and E Valdinoci ldquoGetting acquainted with thefractional laplacian contemporary research in elliptic PDEs andrelated topicsrdquo vol 33 of Springer INdAM Ser Springer ChamSwitzerland 2019
[19] R Musina and A I Nazarov ldquoOn fractional laplaciansrdquo Com-munications in Partial Differential Equations vol 39 no 9 pp1780ndash1790 2014
[20] R Servadei and E Valdinoci ldquoOn the spectrum of two differentfractional operatorsrdquo Proceedings of the Royal Society of Edin-burgh Section AMathematics vol 144 no 4 pp 831ndash855 2014
[21] T Jin Y Li and J Xiong ldquoOn a fractional Nirenberg problempart I blow up analysis and compactness of solutionsrdquo Journalof the EuropeanMathematical Society vol 16 no 6 pp 1111ndash11712014
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Journal of Function Spaces 3
According to Lemma 31 in [9] there is a constant119862 gt 0 suchthat sup119909isinO(Ω119903)119906(119909) le 119862 where
O (Ω 119903) = 119909 isin Ω 119889119894119904119905 (119909 120597Ω) lt 119903 119903 gt 0 (12)
This fact implies that 119861(119909119895 119903) sub Ω DefineV119895 (119909) = 119906 (119909119895)minus1 119906 (119906 (119909119895)minus2(119873minus2120572) 119909 + 119909119895)
119909 isin 119906 (119909119895)2(119873minus2120572) (Ω minus 119909119895) (13)
By Lemma 33 in [9] we know that up to a subsequence V119895(119909)converges to the function V uniformly on any compact setwhere
V (119909) = ( 1205821 + 1205822 1003816100381610038161003816119909 minus 119909010038161003816100381610038162)
(119873minus2120572)2
(14)
Obviously V(0) = 1 then for fixed 120582 ge 1 120582 = 1 + 1205822|1199090|2which implies that |1199090| le 1 Note that V119894 997888rarr V in 1198621(119861(1199090 119903))for 119903 gt 0 and 1199090 is the extreme point of V Then there exists asequence of points119909119894 isin 119861(1199090 119903) such thatnablaV119894(119909119894) = 0 Takinginto account Remark 2 we find
119906 (119909119894)minus2(119873minus2120572) 119909 + 119909119894 = 0 (15)
This fact together with (11) implies that10038161003816100381610038161199091198941003816100381610038161003816 = 119906 (119909119894)2(119873minus2120572) 10038161003816100381610038161199091198941003816100381610038161003816 997888rarr infin (16)
which contradicts the fact that 119909119894 isin 119861(1199090 119903) sub Ω This provesthe validity of this lemma
Lemma 7 Let 119906(119909) be a solution of problem (2) Then thereexist two positive constants 119862 and 120575 such that119906 (119909) le 119862119906 (119909)119871infin 120575119873minus2120572 119909 isin 119863 fl Ω cap |119909| gt 120575 (17)
Remark 8 It can be easily seen that lim119894997888rarrinfin119906119894(119909) = 0uniformly for 119909 isin Ω cap |119909| gt 120575Proof Analysis similar to that in the proof of Proposition 49in [21] shows that there exists a positive constant 119862 such thatfor any |119909 minus 119909119894| le 1
119906119894 (119909) le 119862119906minus1119894 (119909119894) 1003816100381610038161003816119909 minus 11990911989410038161003816100381610038162120572minus119873 (18)
This fact implies that there exist 120575 gt 0 and 119862 gt 0 such thatfor any |119909| le 120575
119906119894 (119909) le 119862 10038171003817100381710038171199061198941003817100381710038171003817minus1 |119909|2120572minus119873 (19)
Particularly
119906119894 (119909) le 119862 10038171003817100381710038171199061198941003817100381710038171003817minus1 1205752120572minus119873 |119909| = 120575 (20)
Now we argue by contradiction to show (17) holds Supposethat there exists a point 119909119894 isin 119863 say the maximum point of 119906119894in119863 such that
119906119894 (119909) gt 119862 1003817100381710038171003817119906119894 (119909119894)1003817100381710038171003817minus1 1205752120572minus119873 gt 0 (21)
It can be easily seen that 119909119894 isin 119863 Thus nabla119906119894(119909119894) = 0Consequently 119909119894 = 0 this contradicts the fact that 119909119894 isin119863
Remark 9 It is worth pointing out that in [21] Jin et alconsider the integral fractional Laplacian but the argumentsof Propositions 44 and 49 in [21] still hold for spectralfractional Laplacian
Lemma 10 Let 119906119894(119909) be a solution to problem (2) Then
lim119894997888rarrinfin
intΩ119906119894 (119909)119901+1 119889119909 = int
R119873119880 (119909)119901+1 119889119909 (22)
Proof Decompose Ω as Ω = Ω1 cup Ω2 where Ω1 = 119909 isin Ω |119909| le 120575 and Ω2 = 119909 isin Ω |119909| gt 120575 120575 is the constant whichappears in Lemma 7 we get
intΩ119906119894 (119909)119901+1 119889119909 = int
Ω1
119906119894 (119909)119901+1 119889119909 + intΩ2
119906119894 (119909)119901+1 119889119909 (23)
By (17) it is obvious that intΩ2119906119894(119909)119901+1119889119909 997888rarr 0 On the other
hand for any |119910| lt 120575119906119894(0)(1minus119901)2120572 define119906119894 (119910) = 119906119894 (0)minus1 119906119894 (119906119894 (0)(1minus119901)2120572 119910) (24)
By a similar argument as Proposition 44 in [21] we derivethat 119906119894(119910) 997888rarr 119880(119910) in1198622119897119900119888(R119873)Therefore by the dominatedconvergence theorem we find
intΩ1
119906119894 (119909)119901+1 119889119909 = int|119910|lt119906119894
(119901minus1)2120572119906119894 (119910)119901+1 119889119910
997888rarr intR119873119880 (119909)119901+1 119889119909
(25)
This finishes the proof of (22)
Proof ofTheorem 1 Since 119906119894(119909) is the solution to problem (2)therefore
intΩ
10038161003816100381610038161003816A12120572 119906100381610038161003816100381610038162 119889119909 = intΩA12120572 119906 sdotA12120572 119906 119889119909
= 119873 (119873 minus 2120572)intΩ119906119901+1119889119909
+ 120576intΩ1199062119889119909
(26)
According to the Holder inequality we find
intΩ1199062119889119909 le |Ω|(119901minus1)(119901+1) (int
Ω119906119901+1119889119909)2(119901+1) (27)
Thus taking into account (26) and (27) we obtain
intΩ
10038161003816100381610038161003816A12120572 119906100381610038161003816100381610038162 119889119909(intΩ119906119901+1119889119909)2(119901+1)
le 119873 (119873 minus 2120572) (intΩ119906119901+1119889119909)(119901minus1)(119901+1)
+ 120576 |Ω|(119901minus1)(119901+1)
(28)
This fact combined with (22) shows that (3) holds Thiscompletes the proof of Theorem 1
4 Journal of Function Spaces
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All the authors equally contributed in this work
Acknowledgments
This work is partially supported by the National Sci-ence Foundation of China (No 11761059) General Top-ics of National Education and Scientific Research (NoZXYB18019) Fundamental Research Funds for the CentralUniversities (No 31920170001) and Key Subject of GansuProvince
References
[1] H Brezis and L Nirenberg ldquoPositive solutions of nonlinearelliptic equations involving critical Sobolev exponentsrdquo Com-munications on Pure and Applied Mathematics vol 36 no 4pp 437ndash477 1983
[2] A Capozzi D Fortunato and G Palmieri ldquoAn existenceresult for nonlinear elliptic problems involving critical Sobolevexponentrdquo Annales de lrsquoInstitut Henri Poincare (C) Analyse NonLineaire vol 2 no 6 pp 463ndash470 1985
[3] O Rey ldquoThe role of the Greenrsquos function in a nonlinear ellipticequation involving the critical Sobolev exponentrdquo Journal ofFunctional Analysis vol 89 no 1 pp 1ndash52 1990
[4] Z Han ldquoAsymptotic approach to singular solutions for non-linear elliptic equations involving critical Sobolev exponentrdquoAnnales de lrsquoInstitut Henri Poincare (C) Analyse Non Lineairevol 8 no 2 pp 159ndash174 1991
[5] B Barrios E Colorado A de Pablo and U Sanchez ldquoOn somecritical problems for the fractional Laplacian operatorrdquo Journalof Differential Equations vol 252 no 11 pp 6133ndash6162 2012
[6] M Bhakta and D Mukherjee ldquoNonlocal scalar field equationsQualitative properties asymptotic profiles and local uniquenessof solutionsrdquo Journal of Differential Equations vol 266 no 11pp 6985ndash7037 2019
[7] C Bucur and M Medina ldquoA fractional elliptic problem in RNwith critical growth and convex nonlinearitiesrdquo ManuscriptaMathematica vol 158 no 3-4 pp 371ndash400 2019
[8] W Choi and S Kim ldquoClassification of finite energy solutionsto the fractional LanendashEmdenndashFowler equations with slightlysubcritical exponentsrdquoAnnali di Matematica Pura ed Applicatavol 196 no 1 pp 269ndash308 2017
[9] W Choi S Kim and K Lee ldquoAsymptotic behavior of solutionsfor nonlinear elliptic problems with the fractional LaplacianrdquoJournal of Functional Analysis vol 266 no 11 pp 6531ndash65982014
[10] S Huang and Q Tian ldquoMarcinkiewicz estimates for solution tofractional elliptic Laplacian equationrdquo Computers ampMathemat-ics with Applications 2019
[11] R Servadei and E Valdinoci ldquoThe Brezis-Nirenberg resultfor the fractional Laplacianrdquo Transactions of the AmericanMathematical Society vol 367 no 1 pp 67ndash102 2015
[12] C Brandle E Colorado A de Pablo and U Sanchez ldquoAconcavemdashconvex elliptic problem involving the fractionalLaplacianrdquo Proceedings of the Royal Society of Edinburgh Sec-tion A Mathematics vol 143 no 1 pp 39ndash71 2013
[13] S Yan J Yang and X Yu ldquoEquations involving fractionallaplacian operator compactness and applicationrdquo Journal ofFunctional Analysis vol 269 no 1 pp 47ndash79 2015
[14] L A Caffarelli and L Silvestre ldquoAn extension problem related tothe fractional LaplacianrdquoCommunications in PartialDifferentialEquations vol 32 no 7ndash9 pp 1245ndash1260 2007
[15] X Cabre and J Tan ldquoPositive solutions of nonlinear problemsinvolving the square root of the Laplacianrdquo Advances in Mathe-matics vol 224 no 5 pp 2052ndash2093 2010
[16] M Bhakta D Mukherjee and S Santra ldquoProfile of solutionsfor nonlocal equations with critical and supercritical nonlinear-itiesrdquo Communications in Contemporary Mathematics vol 21no 01 Article ID 1750099 2019
[17] K Cerqueti and M Grossi ldquoLocal estimates for a semilinearelliptic equation with Sobolev critical exponent and applicationto a uniqueness resultrdquo Nonlinear Differential Equations andApplications NoDEA vol 8 no 3 pp 251ndash283 2001
[18] N Abatangelo and E Valdinoci ldquoGetting acquainted with thefractional laplacian contemporary research in elliptic PDEs andrelated topicsrdquo vol 33 of Springer INdAM Ser Springer ChamSwitzerland 2019
[19] R Musina and A I Nazarov ldquoOn fractional laplaciansrdquo Com-munications in Partial Differential Equations vol 39 no 9 pp1780ndash1790 2014
[20] R Servadei and E Valdinoci ldquoOn the spectrum of two differentfractional operatorsrdquo Proceedings of the Royal Society of Edin-burgh Section AMathematics vol 144 no 4 pp 831ndash855 2014
[21] T Jin Y Li and J Xiong ldquoOn a fractional Nirenberg problempart I blow up analysis and compactness of solutionsrdquo Journalof the EuropeanMathematical Society vol 16 no 6 pp 1111ndash11712014
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
4 Journal of Function Spaces
Data Availability
No data were used to support this study
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All the authors equally contributed in this work
Acknowledgments
This work is partially supported by the National Sci-ence Foundation of China (No 11761059) General Top-ics of National Education and Scientific Research (NoZXYB18019) Fundamental Research Funds for the CentralUniversities (No 31920170001) and Key Subject of GansuProvince
References
[1] H Brezis and L Nirenberg ldquoPositive solutions of nonlinearelliptic equations involving critical Sobolev exponentsrdquo Com-munications on Pure and Applied Mathematics vol 36 no 4pp 437ndash477 1983
[2] A Capozzi D Fortunato and G Palmieri ldquoAn existenceresult for nonlinear elliptic problems involving critical Sobolevexponentrdquo Annales de lrsquoInstitut Henri Poincare (C) Analyse NonLineaire vol 2 no 6 pp 463ndash470 1985
[3] O Rey ldquoThe role of the Greenrsquos function in a nonlinear ellipticequation involving the critical Sobolev exponentrdquo Journal ofFunctional Analysis vol 89 no 1 pp 1ndash52 1990
[4] Z Han ldquoAsymptotic approach to singular solutions for non-linear elliptic equations involving critical Sobolev exponentrdquoAnnales de lrsquoInstitut Henri Poincare (C) Analyse Non Lineairevol 8 no 2 pp 159ndash174 1991
[5] B Barrios E Colorado A de Pablo and U Sanchez ldquoOn somecritical problems for the fractional Laplacian operatorrdquo Journalof Differential Equations vol 252 no 11 pp 6133ndash6162 2012
[6] M Bhakta and D Mukherjee ldquoNonlocal scalar field equationsQualitative properties asymptotic profiles and local uniquenessof solutionsrdquo Journal of Differential Equations vol 266 no 11pp 6985ndash7037 2019
[7] C Bucur and M Medina ldquoA fractional elliptic problem in RNwith critical growth and convex nonlinearitiesrdquo ManuscriptaMathematica vol 158 no 3-4 pp 371ndash400 2019
[8] W Choi and S Kim ldquoClassification of finite energy solutionsto the fractional LanendashEmdenndashFowler equations with slightlysubcritical exponentsrdquoAnnali di Matematica Pura ed Applicatavol 196 no 1 pp 269ndash308 2017
[9] W Choi S Kim and K Lee ldquoAsymptotic behavior of solutionsfor nonlinear elliptic problems with the fractional LaplacianrdquoJournal of Functional Analysis vol 266 no 11 pp 6531ndash65982014
[10] S Huang and Q Tian ldquoMarcinkiewicz estimates for solution tofractional elliptic Laplacian equationrdquo Computers ampMathemat-ics with Applications 2019
[11] R Servadei and E Valdinoci ldquoThe Brezis-Nirenberg resultfor the fractional Laplacianrdquo Transactions of the AmericanMathematical Society vol 367 no 1 pp 67ndash102 2015
[12] C Brandle E Colorado A de Pablo and U Sanchez ldquoAconcavemdashconvex elliptic problem involving the fractionalLaplacianrdquo Proceedings of the Royal Society of Edinburgh Sec-tion A Mathematics vol 143 no 1 pp 39ndash71 2013
[13] S Yan J Yang and X Yu ldquoEquations involving fractionallaplacian operator compactness and applicationrdquo Journal ofFunctional Analysis vol 269 no 1 pp 47ndash79 2015
[14] L A Caffarelli and L Silvestre ldquoAn extension problem related tothe fractional LaplacianrdquoCommunications in PartialDifferentialEquations vol 32 no 7ndash9 pp 1245ndash1260 2007
[15] X Cabre and J Tan ldquoPositive solutions of nonlinear problemsinvolving the square root of the Laplacianrdquo Advances in Mathe-matics vol 224 no 5 pp 2052ndash2093 2010
[16] M Bhakta D Mukherjee and S Santra ldquoProfile of solutionsfor nonlocal equations with critical and supercritical nonlinear-itiesrdquo Communications in Contemporary Mathematics vol 21no 01 Article ID 1750099 2019
[17] K Cerqueti and M Grossi ldquoLocal estimates for a semilinearelliptic equation with Sobolev critical exponent and applicationto a uniqueness resultrdquo Nonlinear Differential Equations andApplications NoDEA vol 8 no 3 pp 251ndash283 2001
[18] N Abatangelo and E Valdinoci ldquoGetting acquainted with thefractional laplacian contemporary research in elliptic PDEs andrelated topicsrdquo vol 33 of Springer INdAM Ser Springer ChamSwitzerland 2019
[19] R Musina and A I Nazarov ldquoOn fractional laplaciansrdquo Com-munications in Partial Differential Equations vol 39 no 9 pp1780ndash1790 2014
[20] R Servadei and E Valdinoci ldquoOn the spectrum of two differentfractional operatorsrdquo Proceedings of the Royal Society of Edin-burgh Section AMathematics vol 144 no 4 pp 831ndash855 2014
[21] T Jin Y Li and J Xiong ldquoOn a fractional Nirenberg problempart I blow up analysis and compactness of solutionsrdquo Journalof the EuropeanMathematical Society vol 16 no 6 pp 1111ndash11712014
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom