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The Lin-Ni’s problem for mean convex domains
About this Title
Olivier Druet, Ecole normale supérieure de Lyon, Département de Mathématiques - UMPA, 46 allée d’Italie, 69364 Lyon cedex 07, France, Frédéric Robert, Institut Élie Cartan, Université Henri Poincaré Nancy 1, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France and Juncheng Wei, Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 218, Number 1027
ISBNs: 978-0-8218-6909-3 (print); 978-0-8218-9016-5 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00646-5
Published electronically: November 30, 2011
Keywords: Neumann elliptic problem,
critical exponent,
blow-up
MSC: Primary 35J20, 35J60
Table of Contents
Chapters
- Introduction
- 1. $L^\infty -$bounded solutions
- 2. Smooth domains and extensions of solutions to elliptic equations
- 3. Exhaustion of the concentration points
- 4. A first upper-estimate
- 5. A sharp upper-estimate
- 6. Asymptotic estimates in $C^1\left (\Omega \right )$
- 7. Convergence to singular harmonic functions
- 8. Estimates of the interior blow-up rates
- 9. Estimates of the boundary blow-up rates
- 10. Proof of Theorems and
- A. Construction and estimates on the Green’s function
- B. Projection of the test functions
Abstract
We prove some refined asymptotic estimates for positive blow-up solutions to $\Delta u+\epsilon u=n(n-2)u^{\frac {n+2}{n-2}}$ on $\Omega$, $\partial _\nu u=0$ on $\partial \Omega$, $\Omega$ being a smooth bounded domain of $\mathbb {R}^n$, $n\geq 3$. In particular, we show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, we prove the validity of the Lin-Ni’s conjecture in dimension $n=3$ and $n\geq 7$ for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition.- Adimurthi, G. Mancini, and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent, Comm. Partial Differential Equations 20 (1995), no. 3-4, 591–631. MR 1318082, DOI 10.1080/03605309508821110
- Adimurthi, Filomena Pacella, and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal. 113 (1993), no. 2, 318–350. MR 1218099, DOI 10.1006/jfan.1993.1053
- Adimurthi and S. L. Yadava, Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents, Arch. Rational Mech. Anal. 115 (1991), no. 3, 275–296. MR 1106295, DOI 10.1007/BF00380771
- Adimurthi and S. L. Yadava, Nonexistence of positive radial solutions of a quasilinear Neumann problem with a critical Sobolev exponent, Arch. Rational Mech. Anal. 139 (1997), no. 3, 239–253. MR 1480241, DOI 10.1007/s002050050052
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
- 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
- E. N. Dancer and Shusen Yan, Interior and boundary peak solutions for a mixed boundary value problem, Indiana Univ. Math. J. 48 (1999), no. 4, 1177–1212. MR 1757072, DOI 10.1512/iumj.1999.48.1827
- Manuel del Pino, Monica Musso, and Angela Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 1, 45–82 (English, with English and French summaries). MR 2114411, DOI 10.1016/j.anihpc.2004.05.001
- Olivier Druet, From one bubble to several bubbles: the low-dimensional case, J. Differential Geom. 63 (2003), no. 3, 399–473. MR 2015469
- Olivier Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not. 23 (2004), 1143–1191. MR 2041549, DOI 10.1155/S1073792804133278
- Olivier Druet and Emmanuel Hebey, Stability and instability for Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds, Math. Z. 263 (2009), no. 1, 33–67. MR 2529487, DOI 10.1007/s00209-008-0409-3
- Olivier Druet, Emmanuel Hebey, and Frédéric Robert, Blow-up theory for elliptic PDEs in Riemannian geometry, Mathematical Notes, vol. 45, Princeton University Press, Princeton, NJ, 2004. MR 2063399
- M. D. Faddeev, Asymptotic behavior of the Green function for the Neumann problem near a boundary point, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 131 (1983), 142–147 (Russian, with English summary). Questions in quantum field theory and statistical physics, 4. MR 718685
- Changfeng Gui and Nassif Ghoussoub, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z. 229 (1998), no. 3, 443–474. MR 1658569, DOI 10.1007/PL00004663
- 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
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- Georges Giraud, Sur le problème de Dirichlet généralisé (deuxième mémoire), Ann. Sci. École Norm. Sup. (3) 46 (1929), 131–245 (French). MR 1509295
- Changfeng Gui and Chang-Shou Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem, J. Reine Angew. Math. 546 (2002), 201–235. MR 1900999, DOI 10.1515/crll.2002.044
- Changfeng Gui and Juncheng Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000), no. 3, 522–538. MR 1758231, DOI 10.4153/CJM-2000-024-x
- Changfeng Gui, Juncheng Wei, and Matthias Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 1, 47–82 (English, with English and French summaries). MR 1743431, DOI 10.1016/S0294-1449(99)00104-3
- Hebey, E.; Robert, F. Asymptotic analysis for fourth order Paneitz equations with critical growth. Advances in the Calculus of Variations, to appear.
- Emmanuel Hebey, Frédéric Robert, and Yuliang Wen, Compactness and global estimates for a fourth order equation of critical Sobolev growth arising from conformal geometry, Commun. Contemp. Math. 8 (2006), no. 1, 9–65. MR 2208810, DOI 10.1142/S0219199706002027
- M. A. Khuri, F. C. Marques, and R. M. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom. 81 (2009), no. 1, 143–196. MR 2477893
- Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1 (1999), no. 1, 1–50. MR 1681811, DOI 10.1142/S021919979900002X
- Chang Shou Lin and Wei-Ming Ni, On the diffusion coefficient of a semilinear Neumann problem, Calculus of variations and partial differential equations (Trento, 1986) Lecture Notes in Math., vol. 1340, Springer, Berlin, 1988, pp. 160–174. MR 974610, DOI 10.1007/BFb0082894
- C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1–27. MR 929196, DOI 10.1016/0022-0396(88)90147-7
- Changshou Lin, Liping Wang, and Juncheng Wei, Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent, Calc. Var. Partial Differential Equations 30 (2007), no. 2, 153–182. MR 2334936, DOI 10.1007/s00526-006-0082-5
- Fang-Hua Lin, Wei-Ming Ni, and Jun-Cheng Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math. 60 (2007), no. 2, 252–281. MR 2275329, DOI 10.1002/cpa.20139
- Pohozaev, S. Eigenfunctions of the equations $\Delta u + \lambda f(u) = 0$. Soviet. Math. Dokl., 6, (1965), 1408-1411.
- Wei-Ming Ni, Qualitative properties of solutions to elliptic problems, Stationary partial differential equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, 2004, pp. 157–233. MR 2103689, DOI 10.1016/S1874-5733(04)80005-6
- Wei-Ming Ni and Izumi Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), no. 7, 819–851. MR 1115095, DOI 10.1002/cpa.3160440705
- Wei-Ming Ni and Izumi Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), no. 2, 247–281. MR 1219814, DOI 10.1215/S0012-7094-93-07004-4
- Wei-Ming Ni, Xing Bin Pan, and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J. 67 (1992), no. 1, 1–20. MR 1174600, DOI 10.1215/S0012-7094-92-06701-9
- Olivier Rey and Juncheng Wei, Blowing up solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. II. $N\geq 4$, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 4, 459–484 (English, with English and French summaries). MR 2145724, DOI 10.1016/j.anihpc.2004.07.004
- Olivier Rey and Juncheng Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 4, 449–476. MR 2159223, DOI 10.4171/JEMS/35
- Robert, F. Existence et asymptotiques optimales des fonctions de Green des opérateurs elliptiques d’ordre deux (Construction and asymptotics for the Green’s function with Neumann boundary condition). Unpublished informal notes (in French), (2009), available at http://www.iecn.u-nancy.fr/$\raisebox {-0.05cm}{\~{}}$frobert.
- Richard M. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 311–320. MR 1173050
- Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), no. 4, 511–517. MR 760051, DOI 10.1007/BF01174186
- Wang, L.; Wei, J.; Yan, S. A Neumann Problem with Critical Exponent in Non-convex Domains and Lin-Ni’s Conjecture. Transactions of Amer. Math. Soc., 361, (2009), 1189-1208.
- Wang, L.; Wei, J.; Yan, S. On the Lin-Ni’s conjecture in convex domains. Proc. London Math. Soc., to appear.
- Juncheng Wei and Shusen Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth, J. Math. Pures Appl. (9) 88 (2007), no. 4, 350–378 (English, with English and French summaries). MR 2384573, DOI 10.1016/j.matpur.2007.07.001
- Juncheng Wei, Existence and stability of spikes for the Gierer-Meinhardt system, Handbook of differential equations: stationary partial differential equations. Vol. V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 487–585. MR 2497911, DOI 10.1016/S1874-5733(08)80013-7
- Juncheng Wei and Xingwang Xu, Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in $\Bbb R^3$, Pacific J. Math. 221 (2005), no. 1, 159–165. MR 2194150, DOI 10.2140/pjm.2005.221.159
- Meijun Zhu, Uniqueness results through a priori estimates. I. A three-dimensional Neumann problem, J. Differential Equations 154 (1999), no. 2, 284–317. MR 1691074, DOI 10.1006/jdeq.1998.3529