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Transactions of the American Mathematical Society

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Elliptic equations with critical growth and a large set of boundary singularities


Authors: Nassif Ghoussoub and Frédéric Robert
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
Published electronically: April 17, 2009
MathSciNet review: 2506429
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Abstract: We solve variationally certain equations of stellar dynamics of the form $ -\sum_i\partial_{ii} u(x) =\frac{\vert u\vert^{p-2}u(x)}{{\rm 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):

$\displaystyle 0<\mu_{s,\mathcal{P}}(\Omega) =\inf\left\{\int_{\Omega}\vert\nabl... ...\frac{\vert u(x)\vert^{2^{\star}(s)}}{\vert\pi(x)\vert^s} dx=1\right.\right\},$

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 $ \hbox{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}$.


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  • 1. Badiale, M.; Tarantello, G. A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal., 163 (2002), 259-293. MR 1918928 (2003j:35079)
  • 2. Atkinson, F.V.; Peletier, L.A. Elliptic equations with nearly critical growth. J. Differential Equations, 70 (1987), 349-365. MR 915493 (89e:35054)
  • 3. Brézis, H.; Peletier, L.A. Asymptotics for elliptic equations involving critical Sobolev exponent. Partial differential equations and the calculus of variations, Vol. I, 149-192. Progr. Nonlinear Differential Equations Appl., 1, Birkhäuser Boston, Boston, MA, 1989. MR 1034005 (91a:35030)
  • 4. Caffarelli, L.; Gidas, B.; Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math., 42 (1989), 271-297. MR 982351 (90c:35075)
  • 5. Druet, O. The best constants problem in Sobolev inequalities. Math. Ann., 314 (1999), 327-346. MR 1697448 (2000d:58033)
  • 6. Druet, O. Elliptic equations with critical Sobolev exponent in dimension 3. Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 125-142. MR 1902741 (2003f:35104)
  • 7. Druet, O.; Hebey, E. Elliptic equations of Yamabe type. IMRS Int. Math. Res. Surv., 2005, 1-113. MR 2148873 (2006b:53046)
  • 8. Druet, O.; Hebey, E.; Robert, F. Blow up theory for elliptic PDE's in Riemannian geometry. Mathematical Notes, 45. Princeton University Press, Princeton, NJ, 2004. Announcement in ``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 (2004c:58046)
  • 9. Caffarelli, L.; Kohn, R.; Nirenberg, L. First order interpolation inequality with weights, Compositio Math., 53 (1984), 259-275. MR 768824 (86c:46028)
  • 10. Egnell, H., Positive solutions of semilinear equations in cones. Trans. Amer. Math. Soc., 330 (1992), 191-201. MR 1034662 (92f:35018)
  • 11. Ghoussoub, N.; Kang, X.S. Hardy-Sobolev Critical Elliptic Equations with Boundary Singularities. Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 767-793. MR 2097030 (2005i:35086)
  • 12. Ghoussoub, N.; Robert, F. The effect of curvature on the best constant in the Hardy-Sobolev inequalities.Geom. Funct. Anal., 16 (2006), no. 6, 1201-1245. MR 2276538 (2007k:35085)
  • 13. Ghoussoub, N.; Robert, F. Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth. IMRP Int. Math. Res. Pap., (2006), 21867, 1-85. MR 2210661 (2006k:35094)
  • 14. Ghoussoub, N.; Yuan, C. Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Amer. Math. Soc., 352 (2000), 5703-5743. MR 1695021 (2001b:35109)
  • 15. Gidas, B.; Ni, W.M.; Nirenberg, L. Symmetry and related properties via the maximum principle. Comm. Math. Phys., 68 (1979), 209-243. MR 544879 (80h:35043)
  • 16. Gilbarg, G.; Trudinger, N.S. Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • 17. Han, Z.C. Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. H.Poincaré. Anal. Non Linéaire, 8 (1991), 159-174. MR 1096602 (92c:35047)
  • 18. Hebey, E.; Vaugon, M. The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds. Duke Math. J., 79 (1995), 235-279. MR 1340298 (96c:53057)
  • 19. Hebey, E.; Vaugon, M. From best constants to critical functions. Math. Z., 237 (2001), 737-767. MR 1854089 (2002h:58061)
  • 20. Maz'ya, V.G. Sobolev spaces. Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. MR 817985 (87g:46056)
  • 21. Robert, F. Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent. The radial case II. NoDEA Nonlinear Differential Equations Appl., 9 (2002), 361-384. MR 1917379 (2003i:35058)
  • 22. Robert, F. Critical functions and optimal Sobolev inequalities. Math. Z., 249 (2005), 485-492. MR 2121735 (2007m:53035)
  • 23. Schoen, R.; Zhang, D. Prescribed scalar curvature on the $ n$-sphere. Calc. Var. Partial Differential Equations, 4 (1996), 1-25. MR 1379191 (97j:58027)
  • 24. Struwe, M. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, 34. Springer-Verlag, Berlin, 2000. MR 1736116 (2000i:49001)

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Additional Information

Nassif Ghoussoub
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada
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

DOI: https://doi.org/10.1090/S0002-9947-09-04655-8
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.
Article copyright: © Copyright 2009 American Mathematical Society

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