A Complete Study of the Lack of Compactness and Existence Results of a Fractional Nirenberg Equation via a Flatness Hypothesis. Part II

Автор(и)

  • Azeb Alghanemi Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
  • Wael Abdelhedi Sfax University, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia
  • Hichem Chtioui Sfax University, Faculty of Sciences of Sfax, 3018 Sfax, Tunisia

DOI:

https://doi.org/10.15407/mag18.01.003

Ключові слова:

конформна геометрія, часткова кривина, варіаційне обчислення, критичні точки на нескінченності

Анотація

Ця стаття є продовженням досліджень статті [2], де вивчалась задача $\sigma$-кривини на стандартній сфері за умови, що порядок сплощення даної функції у критичних точках належить $(1, n-2\sigma]$. Наведено повний опис відсутності компактності задачі, коли порядок сплощення змінюється в $(1, n)$, і доведено теорему існування на основі формули типу Ейлера-Хопфа. Як наслідок, ми узагальнюємо результати робіт [2, 17, 18] та одержуємо новий.

Mathematics Subject Classification: 35A15, 35J60, 58E30

Посилання

W. Abdelhedi and H. Chtioui, On a Nirenberg-type problem involving the square root of the Laplacian, J. Funct. Anal. 265, (2013), 2937-2955. https://doi.org/10.1016/j.jfa.2013.08.005

W. Abdelhedi, H. Chtioui, and H. Hajaiej, 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 (2016), No. 6, 1285–1315. https://doi.org/10.2140/apde.2016.9.1285

W. Abdelhedi and H. Chtioui, On a fractional Nirenberg problem on n-dimensional spheres: Existence and multiplicity results, Bull. Sc. Math., 140 (2016), No. 6, 617–628. https://doi.org/10.1016/j.bulsci.2015.04.007

A. Bahri, Critical Point at Infinity in Some Variational Problems, Pitman Res. Notes Math, 182, Longman Sci. Tech., Harlow, 1989.

A. Bahri, An invariant for Yamabe-type flows with applications to scalar curvature problems in high dimensions, A celebration of J. F. Nash Jr., Duke Math. J. 81 (1996), 323–466. https://doi.org/10.1215/S0012-7094-96-08116-8

A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of topology of the domain, Comm. Pure Appli. Math. 41 (1988), 255–294. https://doi.org/10.1002/cpa.3160410302

A. Bahri and P. H. Rabinowitz, Periodic solutions of 3-body problems, Ann. Inst. H. Poincaré Anal. Nonlin, 8 (1991), 561–649. https://doi.org/10.1016/s0294-1449(16)30252-9

M. Ben Ayed, Y. Chen, H. Chtioui, and M. Hammami, On the prescribed scalar curvature problem on 4-manifolds, Duke Math. J. 84 (1996), 633–677. https://doi.org/10.1215/S0012-7094-96-08420-3

C. Brändle, E. Colorado, A. de Pablo, and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 39–71. https://doi.org/10.1017/S0308210511000175

X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093. https://doi.org/10.1016/j.aim.2010.01.025

C.C. Chen and C.S. Lin, Prescribing the scalar curvature on S n , I. Apriori estimates J. Differential Geom. 57 (2001), 67–171. https://doi.org/10.4310/jdg/1090348090

W. Chen, C. Li, and B. Ou. Classification of solutions for an integral equation, Communications on Pure and Applied Mathematics, 59 (2006), No. 3, 330–343. https://doi.org/10.1002/cpa.20116

Y. Chen, C. Liu, and Y. Zheng, Existence results for the fractional Nirenberg problem, J. Funct. Anal. 270 (2016), No. 11, 4043–4086. https://doi.org/10.1016/j.jfa.2016.03.013

H. Chtioui, On the Chen-Lin conjecture for the prescribed scalar curvature problem, https://arxiv.org/abs/2009.06262.

A. Fiscella, Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator, 29 (2016), No. 5-6, 513–530.

M. Gonzalez and J. Qing, Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE 6 (2013), 1535–1576. https://doi.org/10.2140/apde.2013.6.1535

T. Jin, Y. Li, and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc. 16 (2014), 1111–1171. https://doi.org/10.4171/JEMS/456

T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part II: existence of solutions, Int. Math. Res. Not. 6 (2015), 1555–1589.

T. Jin, Y. Li, and J. Xiong, The Nirenberg problem and its generalizations: A unified approach, Math. Ann. 369 (2017), 109–151. https://doi.org/10.1007/s00208-016-1477-z

Y.Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc. 6 (2004),153–180. https://doi.org/10.4171/JEMS/6

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383–417. https://doi.org/10.1215/S0012-7094-95-08016-8

C. Liu and Q. Ren, Uniqueness of types of infinitely-many-bump solutions for the fractional Nirenberg problem, J. Math. Anal. Appl., 468 (2018), No. 1, 1–37. https://doi.org/10.1016/j.jmaa.2018.06.039

R. Musina, A. I. Nazarov, On fractional Laplacians. Comm. Partial Diferential Equations, 39 (2014), No. 9, 1780–1790. https://doi.org/10.1080/03605302.2013.864304

E. Di Nezza, J. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math, 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A 144, (2014), no. 4, 831-855. https://doi.org/10.1017/S0308210512001783

K. Sharaf and H. Chtioui, Conformal metrics with prescribed fractional Qcurvatures on the standard n-dimensional sphere, Differential Geom. Appl. 68 (2020), 101562. https://doi.org/10.1016/j.difgeo.2019.101562

M. Struwe, A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z. 187, (1984), 511–517. https://doi.org/10.1007/BF01174186

J.Tan, Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst. 33 (2013), 837–859. https://doi.org/10.3934/dcds.2013.33.837

Downloads

Як цитувати

(1)
Alghanemi, A.; Abdelhedi, W.; Chtioui, H. A Complete Study of the Lack of Compactness and Existence Results of a Fractional Nirenberg Equation via a Flatness Hypothesis. Part II. Журн. мат. фіз. анал. геом. 2022, 18, 3-32.

Номер

Розділ

Статті

Завантаження

Дані завантаження ще не доступні.