Elliptic equations with critical growth and a large set of boundary singularities
HTML articles powered by AMS MathViewer
- by Nassif Ghoussoub and Frédéric Robert PDF
- Trans. Amer. Math. Soc. 361 (2009), 4843-4870 Request permission
Abstract:
We solve variationally certain equations of stellar dynamics of the form $-\sum _i\partial _{ii} u(x) =\frac {|u|^{p-2}u(x)}{\textrm {dist} (x,{\mathcal A} )^s}$ in a domain $\Omega$ of $\mathbb {R}^n$, where ${\mathcal A}$ is a proper linear subspace of $\mathbb {R}^n$. Existence problems are related to the question of attainability of the best constant in the following inequality due to Maz’ya (1985): \[ 0<\mu _{s,\mathcal {P}}(\Omega ) =\inf \left \{\int _{\Omega }|\nabla u|^2 dx\; \left |\; u\in H_{1,0}^2(\Omega ) \;\mathrm { and }\; \int _{\Omega }\frac {|u(x)|^{2^{\star }(s)}}{|\pi (x)|^s} dx=1\right .\right \},\] where $0<s<2$, $2^{\star }(s) =\frac {2(n-s)}{n-2}$ and where $\pi$ is the orthogonal projection on a linear space $\mathcal {P}$, where $\operatorname {dim}_{\mathbb {R}}\mathcal {P}\geq 2$ (see also Badiale-Tarantello (2002)). We investigate this question and how it depends on the relative position of the subspace ${\mathcal P}^{\bot }$, the orthogonal of $\mathcal P$, with respect to the domain $\Omega$, as well as on the curvature of the boundary $\partial \Omega$ at its points of intersection with ${\mathcal P}^{\bot }$.References
- Marino Badiale and Gabriella 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. MR 1918928, DOI 10.1007/s002050200201
- F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations 70 (1987), no. 3, 349–365. MR 915493, DOI 10.1016/0022-0396(87)90156-2
- Haïm Brezis and Lambertus A. Peletier, Asymptotics for elliptic equations involving critical growth, Partial differential equations and the calculus of variations, Vol. I, Progr. Nonlinear Differential Equations Appl., vol. 1, Birkhäuser Boston, Boston, MA, 1989, pp. 149–192. MR 1034005
- Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
- Olivier Druet, The best constants problem in Sobolev inequalities, Math. Ann. 314 (1999), no. 2, 327–346. MR 1697448, DOI 10.1007/s002080050297
- Olivier Druet, Elliptic equations with critical Sobolev exponents in dimension 3, Ann. Inst. H. Poincaré C Anal. Non Linéaire 19 (2002), no. 2, 125–142 (English, with English and French summaries). MR 1902741, DOI 10.1016/S0294-1449(02)00095-1
- Olivier Druet and Emmanuel Hebey, Elliptic equations of Yamabe type, IMRS Int. Math. Res. Surv. 1 (2005), 1–113. MR 2148873, DOI 10.1155/imrs.2005.1
- Olivier Druet, Emmanuel Hebey, and Frédéric Robert, A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 19–25. MR 1988868, DOI 10.1090/S1079-6762-03-00108-2
- L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259–275. MR 768824
- Henrik Egnell, Positive solutions of semilinear equations in cones, Trans. Amer. Math. Soc. 330 (1992), no. 1, 191–201. MR 1034662, DOI 10.1090/S0002-9947-1992-1034662-5
- N. Ghoussoub and X. S. Kang, Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincaré C Anal. Non Linéaire 21 (2004), no. 6, 767–793 (English, with English and French summaries). MR 2097030, DOI 10.1016/j.anihpc.2003.07.002
- 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. MR 2276538, DOI 10.1007/s00039-006-0579-2
- N. Ghoussoub and F. Robert, Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth, IMRP Int. Math. Res. Pap. (2006), 21867, 1–85. MR 2210661
- 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. MR 1695021, DOI 10.1090/S0002-9947-00-02560-5
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879, DOI 10.1007/BF01221125
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Zheng-Chao Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré C Anal. Non Linéaire 8 (1991), no. 2, 159–174 (English, with French summary). MR 1096602, DOI 10.1016/S0294-1449(16)30270-0
- Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J. 79 (1995), no. 1, 235–279. MR 1340298, DOI 10.1215/S0012-7094-95-07906-X
- Emmanuel Hebey and Michel Vaugon, From best constants to critical functions, Math. Z. 237 (2001), no. 4, 737–767. MR 1854089, DOI 10.1007/PL00004889
- Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
- Frédéric Robert, Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent: the radial case. II, NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 3, 361–384. MR 1917379, DOI 10.1007/s00030-002-8133-x
- Frédéric Robert, Critical functions and optimal Sobolev inequalities, Math. Z. 249 (2005), no. 3, 485–492. MR 2121735, DOI 10.1007/s00209-004-0708-2
- Richard Schoen and Dong Zhang, Prescribed scalar curvature on the $n$-sphere, Calc. Var. Partial Differential Equations 4 (1996), no. 1, 1–25. MR 1379191, DOI 10.1007/BF01322307
- Michael Struwe, Variational methods, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 34, Springer-Verlag, Berlin, 2000. Applications to nonlinear partial differential equations and Hamiltonian systems. MR 1736116, DOI 10.1007/978-3-662-04194-9
Additional Information
- Nassif Ghoussoub
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
- MR Author ID: 73130
- Email: nassif@math.ubc.ca
- Frédéric Robert
- Affiliation: Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, France
- Email: frobert@math.unice.fr
- Received by editor(s): February 28, 2006
- Received by editor(s) in revised form: October 2, 2007
- Published electronically: April 17, 2009
- Additional Notes: This research was partially supported by the Natural Sciences and Engineering Research Council of Canada. The first author gratefully acknowledges the hospitality and support of the Université de Nice where this work was initiated.
The second author gratefully acknowledges the hospitality and support of the University of British Columbia. - © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 4843-4870
- MSC (2000): Primary 35J35; Secondary 35B40
- DOI: https://doi.org/10.1090/S0002-9947-09-04655-8
- MathSciNet review: 2506429