Skip to content
Licensed Unlicensed Requires Authentication Published by Oldenbourg Wissenschaftsverlag September 18, 2019

Hardy–Sobolev inequality with higher dimensional singularity

  • El Hadji Abdoulaye Thiam EMAIL logo
From the journal Analysis

Abstract

For N4, we let Ω be a smooth bounded domain of N, Γ a smooth closed submanifold of Ω of dimension k, with 1kN-2, and h a continuous function defined on Ω. We denote by ρΓ():=dist(,Γ) the distance function to Γ. For σ(0,2), we study the existence of positive solutions uH01(Ω) to the nonlinear equation

-Δu+hu=ρΓ-σu2*(σ)-1in Ω,

where 2*(σ):=2(N-σ)N-2 is the critical Hardy–Sobolev exponent. In particular, we prove the existence of solution under the influence of the local geometry of Γ and the potential h.

Funding statement: This work is supported by the German Academic Exchange Service (DAAD).

Acknowledgements

I wish to thanks my supervisor Mouhamed Moustapha Fall for useful discussions and remarks.

References

[1] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598. 10.4310/jdg/1214433725Search in Google Scholar

[2] M. Badiale and G. Tarantello, A Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), no. 4, 259–293. 10.1007/s002050200201Search in Google Scholar

[3] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. 10.1090/S0002-9939-1983-0699419-3Search in Google Scholar

[4] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. 10.1002/cpa.3160360405Search in Google Scholar

[5] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469. Search in Google Scholar

[6] J.-L. Chern and C.-S. Lin, Minimizers of Caffarelli–Kohn–Nirenberg inequalities with the singularity on the boundary, Arch. Ration. Mech. Anal. 197 (2010), no. 2, 401–432. 10.1007/s00205-009-0269-ySearch in Google Scholar

[7] A. V. Dem’yanov and A. I. Nazarov, On the solvability of the Dirichlet problem for the semilinear Schrödinger equation with a singular potential (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 336 (2006), 25–45, 274; translation in J. Math. Sci. (N.Y.) 143 (2007), no. 2, 2857-2868. 10.1007/s10958-007-0171-9Search in Google Scholar

[8] O. Druet, The best constants problem in Sobolev inequalities, Math. Ann. 314 (1999), no. 2, 327–346. 10.1007/s002080050297Search in Google Scholar

[9] O. Druet, Elliptic equations with critical Sobolev exponents in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), no. 2, 125–142. 10.1016/S0294-1449(02)00095-1Search in Google Scholar

[10] O. Druet, Optimal Sobolev inequalities and extremal functions. The three-dimensional case, Indiana Univ. Math. J. 51 (2002), no. 1, 69–88. 10.1512/iumj.2002.51.2111Search in Google Scholar

[11] H. Egnell, Positive solutions of semilinear equations in cones, Trans. Amer. Math. Soc. 330 (1992), no. 1, 191–201. 10.1090/S0002-9947-1992-1034662-5Search in Google Scholar

[12] M. M. Fall, I. A. Minlend and E. H. A. Thiam, The role of the mean curvature in a Hardy–Sobolev trace inequality, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 5, 1047–1066. 10.1007/s00030-015-0313-6Search in Google Scholar

[13] M. M. Fall and E. h. A. Thiam, Hardy–Sobolev inequality with singularity a curve, Topol. Methods Nonlinear Anal. 51 (2018), no. 1, 151–181. 10.12775/TMNA.2017.045Search in Google Scholar

[14] N. Ghoussoub and X. S. Kang, Hardy–Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 6, 767–793. 10.1016/j.anihpc.2003.07.002Search in Google Scholar

[15] N. Ghoussoub and F. Robert, Concentration estimates for Emden–Fowler equations with boundary singularities and critical growth, IMRP Int. Math. Res. Pap. 2006 (2006), Article ID 21867. 10.1155/IMRP/2006/21867Search in Google Scholar

[16] N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy–Sobolev inequalities, Geom. Funct. Anal. 16 (2006), no. 6, 1201–1245. 10.1007/s00039-006-0579-2Search in Google Scholar

[17] N. Ghoussoub and F. Robert, Elliptic equations with critical growth and a large set of boundary singularities, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4843–4870. 10.1090/S0002-9947-09-04655-8Search in Google Scholar

[18] N. Ghoussoub and F. Robert, On the Hardy–Schrödinger operator with a boundary singularity, preprint (2014), https://arxiv.org/abs/1410.1913. Search in Google Scholar

[19] N. Ghoussoub and F. Robert, Sobolev inequalities for the Hardy–Schrödinger operator: Extremals and critical dimensions, Bull. Math. Sci. 6 (2016), no. 1, 89–144. 10.1007/s13373-015-0075-9Search in Google Scholar

[20] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5703–5743. 10.1090/S0002-9947-00-02560-5Search in Google Scholar

[21] A. Gray, Tubes, 2nd ed., Progr. Math. 221, Birkhäuser, Basel, 2004. 10.1007/978-3-0348-7966-8Search in Google Scholar

[22] E. Hebey and M. Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J. 79 (1995), no. 1, 235–279. 10.1215/S0012-7094-95-07906-XSearch in Google Scholar

[23] E. Hebey and M. Vaugon, Meilleures constantes dans le théorème d’inclusion de Sobolev, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 1, 57–93. 10.1016/S0294-1449(16)30097-XSearch in Google Scholar

[24] H. Jaber, Hardy–Sobolev equations on compact Riemannian manifolds, Nonlinear Anal. 103 (2014), 39–54. 10.1016/j.na.2014.02.011Search in Google Scholar

[25] H. Jaber, Optimal Hardy–Sobolev inequalities on compact Riemannian manifolds, J. Math. Anal. Appl. 421 (2015), no. 2, 1869–1888. 10.1016/j.jmaa.2014.07.075Search in Google Scholar

[26] Y. Y. Li and C.-S. Lin, A nonlinear elliptic PDE and two Sobolev–Hardy critical exponents, Arch. Ration. Mech. Anal. 203 (2012), no. 3, 943–968. 10.1007/s00205-011-0467-2Search in Google Scholar

[27] E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. 10.2307/2007032Search in Google Scholar

[28] C.-S. Lin and H. Wadade, On the attainability for the best constant of the Sobolev–Hardy type inequality, RIMS Kôkyûroku 1740 (2011), 141–157. Search in Google Scholar

[29] C.-S. Lin and H. Wadade, Minimizing problems for the Hardy–Sobolev type inequality with the singularity on the boundary, Tohoku Math. J. (2) 64 (2012), no. 1, 79–103. 10.2748/tmj/1332767341Search in Google Scholar

[30] G. Mancini, I. Fabbri and K. Sandeep, Classification of solutions of a critical Hardy–Sobolev operator, J. Differential Equations 224 (2006), no. 2, 258–276. 10.1016/j.jde.2005.07.001Search in Google Scholar

[31] M. Struwe, Variational Methods: Applications to nonlinear Partial Differential Equations and Hamiltonian Systems, 4th ed., Ergeb. Math. Grenzgeb. (3) 34, Springer, Berlin, 2008. Search in Google Scholar

[32] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. 10.1007/BF02418013Search in Google Scholar

[33] E. H. A. Thiam, Weighted Hardy inequality on Riemannian manifolds, Commun. Contemp. Math. 18 (2016), no. 6, Article ID 1550072. 10.1142/S0219199715500728Search in Google Scholar

[34] E. H. A. Thiam, Hardy and Hardy–Sobolev inequalities on Riemannian manifolds, IMHOTEP J. Afr. Math. Pures Appl. 2 (2017), no. 1, 14–35. Search in Google Scholar

[35] E. H. A. Thiam, The role of the mean curvature in a mixed Hardy–Sobolev trace inequality, preprint. 10.1007/978-3-030-57336-2_12Search in Google Scholar

Received: 2018-01-24
Revised: 2018-10-01
Accepted: 2019-07-08
Published Online: 2019-09-18
Published in Print: 2019-10-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 25.4.2024 from https://www.degruyter.com/document/doi/10.1515/anly-2018-0006/html
Scroll to top button