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The topological contribution of the critical points at infinity for critical fractional Yamabe-type equations. (English) Zbl 1458.35161

Authors’ abstract: In this paper, we study the critical fractional nonlinear PDE: \((-\Delta)^su=u^\frac{n+2s}{n-2s},\) \(u>0\) in \(\Omega\) and \(u=0\) on \(\partial \Omega,\) where \(\Omega\) is a thin annuli-domain of \(\mathbb{R}^n,\) \(n\geq2.\) We compute the evaluation of the difference of topology induced by the critical points at infinity between the level sets of the associated variational function. Our Theorem can be seen as a nonlocal analog of the result of [M. Ahmedou and K. El Mehdi, Duke Math. J. 94, No. 2, 215–229 (1998; Zbl 0966.35043)] on the classical Yamabe-type equation.

MSC:

35J60 Nonlinear elliptic equations
35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs

Citations:

Zbl 0966.35043
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References:

[1] Ahmedou, M.; El Mehdi, K., Computation of the difference of topology at infinity for Yamabe type problems on annuli domains (Part I), Duke Math. J., 9, 4, 215-229 (1998) · Zbl 0966.35043 · doi:10.1215/S0012-7094-98-09411-X
[2] Abdelhedi, W.; Chtioui, H.; Hajaiej, H., A complete study of the lack of compactness and existence results of a fractional Nirenberg equation via a flatness hypothesis: Part I, Anal. PDE, 9, 6, 1285-1315 (2016) · Zbl 1366.35211 · doi:10.2140/apde.2016.9.1285
[3] Abdelhedi, W.; Chtioui, H.; Hajaiej, H., The Bahri-Coron theorem for fractional Yamabe-type problems, Adv. Nonlinear Stud., 18, 2, 393-407 (2018) · Zbl 1388.35064 · doi:10.1515/ans-2017-6035
[4] Abdullah Sharaf, K.; Chtioui, H., Conformal metrics with prescribed fractional Q-curvatures on the standard n-dimensional sphere, Differ. Geom. Appl., 68, 101562 (2020) · Zbl 1437.35318 · doi:10.1016/j.difgeo.2019.101562
[5] Alghanemi, A.; Chtioui, H., Prescribing scalar curvaturs on n-dimensional manifolds, C. R. Acad. Bulg. Sci., 73, 2, 163-169 (2020) · Zbl 1463.58005
[6] Alghanemi, A.; Chtioui, H., Perturbation theorems for fractional critical equations on bounded domains, J. Aust. Math. Soc. (2020) · Zbl 1473.35113 · doi:10.1017/S144678871900048X
[7] Bahri, A.: Critical point at infinity in some variational problems. In: Pitman Research Notes in Mathematics Series, vol. 182. Longman Scientific and Technical, Harlow (1989) · Zbl 0676.58021
[8] Bahri, A., An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimensions, Duke Math. J., 81, 323-466 (1996) · Zbl 0856.53028 · doi:10.1215/S0012-7094-96-08116-8
[9] Ben Ayed, M.; Chen, Y.; Chtioui, H.; Hammami, M., On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J., 84, 633-677 (1996) · Zbl 0862.53034 · doi:10.1215/S0012-7094-96-08420-3
[10] Brändle, C.; Colorado, E.; de Pablo, A.; Sánchez, U., A concave-convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 143, 39-71 (2013) · Zbl 1290.35304 · doi:10.1017/S0308210511000175
[11] Cabré, X.; Tan, J., Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224, 2052-2093 (2010) · Zbl 1198.35286 · doi:10.1016/j.aim.2010.01.025
[12] Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Commun. Partial. Differ. Equ., 32, 1245-1260 (2007) · Zbl 1143.26002 · doi:10.1080/03605300600987306
[13] Chang, A.; Gonzalez, M., Fractional Laplacian in conformal geometry, Adv. Math., 226, 2, 1410-1432 (2011) · Zbl 1214.26005 · doi:10.1016/j.aim.2010.07.016
[14] Chtioui, H.; Rigane, A., On the prescribed Q-curvature problem on \(S^n\), J. Funct. Anal., 261, 2999-3043 (2011) · Zbl 1233.53007 · doi:10.1016/j.jfa.2011.07.017
[15] Gonzalez, M.; Mazzeo, R.; Sire, Y., Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22, 845-863 (2012) · Zbl 1255.53037 · doi:10.1007/s12220-011-9217-9
[16] Gonzalez, M.; Qing, J., Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE, 6, 1535-1576 (2013) · Zbl 1287.35039 · doi:10.2140/apde.2013.6.1535
[17] Graham, CR; Zworski, M., Scattering matrix in conformal geometry, Invent. Math., 152, 89-118 (2003) · Zbl 1030.58022 · doi:10.1007/s00222-002-0268-1
[18] Jin, T.; Li, Y.; Xiong, J., On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16, 1111-1171 (2014) · Zbl 1300.53041 · doi:10.4171/JEMS/456
[19] Jin, T.; Li, Y.; Xiong, J., On a fractional Nirenberg problem, part II: existence of solutions, Int. Math. Res. Not., 6, 1555-1589 (2015) · Zbl 1319.53031
[20] Jin, T.; Li, Y.; Xiong, J., The Nirenberg problem and its generalizations: a unified approach, Math. Ann., 369, 1-2, 109-151 (2017) · Zbl 1384.53039 · doi:10.1007/s00208-016-1477-z
[21] Musina, R.; Nazarov, A., On fractional Laplacians, Commun. Partial Differ. Equ., 39, 9, 1780-1790 (2014) · Zbl 1304.47061 · doi:10.1080/03605302.2013.864304
[22] Qing, J., Raske, D.: On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds. Int. Math. Res. Not. Art. ID 94172 (2006) · Zbl 1115.53028
[23] Rey, O., The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89, 1-52 (1990) · Zbl 0786.35059 · doi:10.1016/0022-1236(90)90002-3
[24] Schoen, R.; Zhang, D., Prescribed scalar curvature on the n-sphere, Calc. Var. Partial Differ. Equ., 4, 1-25 (1996) · Zbl 0843.53037 · doi:10.1007/BF01322307
[25] Tan, J., The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differ. Equ., 42, 21-41 (2011) · Zbl 1248.35078 · doi:10.1007/s00526-010-0378-3
[26] Tan, J., Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33, 837-859 (2013) · Zbl 1276.35092 · doi:10.3934/dcds.2013.33.837
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