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A Serrin-type regularity criterion for the Navier-Stokes equations via one velocity component
A refined result on sign changing solutions for a critical elliptic problem
1. | Laboratoire d'Analyse et de Mathématiques Appliquées, CNRS UMR 8050, Département de Mathématiques, Université Paris Est Créteil, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France |
2. | Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago |
3. | Dipartimento di Metodi e Modelli Matematici, Università di Roma "La Sapienza", Via Scarpa, 16 - 00166 Roma |
4. | University of Washington |
References:
[1] |
A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.
doi: 10.1002/cpa.3160410302. |
[2] |
A. Bahri, Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Differ. Equ., 3 (1995), 67-93.
doi: 10.1007/BF01190892. |
[3] |
T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Top. Meth. Nonlin. Anal., 22 (2003), 1-14. |
[4] |
T. Bartsch, A. M. Micheletti and A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Differential Equationsm, 26 (2006), 265-282.
doi: 10.1007/s00526-006-0004-6. |
[5] |
H. Brezis and L. Nirenberg, Positive solutions of non linear elliptic equations involving critical Sobolev expronents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[6] |
A. Byde, Gluing theorems for constant scalar curvature manifolds, Indiana Univ. Math. J., 52 (2003), 1147-1199.
doi: 10.1512/iumj.2003.52.2109. |
[7] |
M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain, Calc. Var. Partial Differential Equations, 21 (2004), 1-14.
doi: 10.1007/s00526-003-0241-x. |
[8] |
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGill-Hill, New York, 1955. |
[9] |
J. M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sr. I Math., 299 (1984), 209-212. |
[10] |
M. del Pino, J. Dolbeault and M. Musso, "Bubble-tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Differ. Equation, 193 (2003), 280-306.
doi: 10.1016/S0022-0396(03)00151-7. |
[11] |
M. del Pino, J. Dolbeault and M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures Appl., 83 (2004), 1405-1456.
doi: 10.1016/j.matpur.2004.02.007. |
[12] |
M. del Pino, M. Musso and A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 45-82.
doi: 10.1016/j.anihpc.2004.05.001. |
[13] |
M. del Pino, P. Felmer and M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes, J. Differ. Equation, 182 (2002), 511-540.
doi: 10.1006/jdeq.2001.4098. |
[14] |
M. del Pino, P. Felmer and M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries, Bull. London Math. Soc., 35 (2003), 513-521.
doi: 10.1112/S0024609303001942. |
[15] |
Y. Ge, R. Jing and F. Pacard, Bubble towers for supercritical semilinear elliptic equations, J. Funct. Anal., 221 (2005), 251-302.
doi: 10.1016/j.jfa.2004.09.011. |
[16] |
Y. Ge, R. Jing and F. Zhou, Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains, DCDS-A, 17 (2007), 751-770.
doi: 10.3934/dcds.2007.17.751. |
[17] |
Y. Ge, M. Musso and A. Pistoia, Sign changing tower of bubbles for an elliptic problem at the critical exponent in pierced non-symmetric domains, Communications in Partial Differential Equations, 35 (2010), 1419-1457.
doi: 10.1080/03605302.2010.490286. |
[18] |
E. Hebey and M. Vaugon, Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth, J. Funct. Anal., 119 (1994), 298-318.
doi: 10.1006/jfan.1994.1012. |
[19] |
J. Kazdan, F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597.
doi: 10.1002/cpa.3160280502. |
[20] |
Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I., J. Differ. Equations, 120 (1995), 319-410.
doi: 10.1006/jdeq.1995.1115. |
[21] |
R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities, J. Duke Math., 99 (1999), 353-418.
doi: 10.1215/S0012-7094-99-09913-1. |
[22] |
A. M. Micheletti and A. Pistoia, On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth, Nonlinearity, 17 (2004), 851-866.
doi: 10.1088/0951-7715/17/3/007. |
[23] |
M. Musso and A. Pistoia, Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains, J. Math. Pures Appl., 86 (2006), 510-528.
doi: 10.1016/j.matpur.2006.10.006. |
[24] |
M. Musso and A. Pistoia, Sign changing solutions to a Bahri-Coron's problem in pierced domains, Discrete Contin. Dyn. Syst., 21 (2008), 295-306.
doi: 10.3934/dcds.2008.21.295. |
[25] |
M. Musso and A. Pistoia, Persistence of Coron's solution in nearly critical problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 331-357. |
[26] |
M. Musso and A. Pistoia, Tower of Bubbles for almost critical problems in general domains, J. Math. Pure Appl., 93 (2010), 1-40.
doi: 10.1016/j.matpur.2009.08.001. |
[27] |
F. Pacard and T. Rivière, "Linear and Nonlinear Aspects of Vortices. The Ginzburg-Landau Model," Progress in Nonlinear Differential Equations and their Applications, 39 Boston, Birkhäuser, 2000.
doi: 10.1007/978-1-4612-1386-4. |
[28] |
A. Pistoia and T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340.
doi: 10.1016/j.anihpc.2006.03.002. |
[29] |
S. I. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. |
[30] |
O. Rey, On a variational problem with lack of compactness: the effect of small holes in the domain, C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 349-352. |
[31] |
O. Rey, The role of the Green function in a nonlinear elliptic equation involving the critical Sobolev exponetion, J. Funct. Anal., 89 (1990), 1-52.
doi: 10.1016/0022-1236(90)90002-3. |
show all references
References:
[1] |
A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.
doi: 10.1002/cpa.3160410302. |
[2] |
A. Bahri, Y. Li and O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Differ. Equ., 3 (1995), 67-93.
doi: 10.1007/BF01190892. |
[3] |
T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Top. Meth. Nonlin. Anal., 22 (2003), 1-14. |
[4] |
T. Bartsch, A. M. Micheletti and A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Differential Equationsm, 26 (2006), 265-282.
doi: 10.1007/s00526-006-0004-6. |
[5] |
H. Brezis and L. Nirenberg, Positive solutions of non linear elliptic equations involving critical Sobolev expronents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[6] |
A. Byde, Gluing theorems for constant scalar curvature manifolds, Indiana Univ. Math. J., 52 (2003), 1147-1199.
doi: 10.1512/iumj.2003.52.2109. |
[7] |
M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric domain, Calc. Var. Partial Differential Equations, 21 (2004), 1-14.
doi: 10.1007/s00526-003-0241-x. |
[8] |
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGill-Hill, New York, 1955. |
[9] |
J. M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sr. I Math., 299 (1984), 209-212. |
[10] |
M. del Pino, J. Dolbeault and M. Musso, "Bubble-tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Differ. Equation, 193 (2003), 280-306.
doi: 10.1016/S0022-0396(03)00151-7. |
[11] |
M. del Pino, J. Dolbeault and M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures Appl., 83 (2004), 1405-1456.
doi: 10.1016/j.matpur.2004.02.007. |
[12] |
M. del Pino, M. Musso and A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 45-82.
doi: 10.1016/j.anihpc.2004.05.001. |
[13] |
M. del Pino, P. Felmer and M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes, J. Differ. Equation, 182 (2002), 511-540.
doi: 10.1006/jdeq.2001.4098. |
[14] |
M. del Pino, P. Felmer and M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries, Bull. London Math. Soc., 35 (2003), 513-521.
doi: 10.1112/S0024609303001942. |
[15] |
Y. Ge, R. Jing and F. Pacard, Bubble towers for supercritical semilinear elliptic equations, J. Funct. Anal., 221 (2005), 251-302.
doi: 10.1016/j.jfa.2004.09.011. |
[16] |
Y. Ge, R. Jing and F. Zhou, Bubble tower solutions of slightly supercritical elliptic equations and application in symmetric domains, DCDS-A, 17 (2007), 751-770.
doi: 10.3934/dcds.2007.17.751. |
[17] |
Y. Ge, M. Musso and A. Pistoia, Sign changing tower of bubbles for an elliptic problem at the critical exponent in pierced non-symmetric domains, Communications in Partial Differential Equations, 35 (2010), 1419-1457.
doi: 10.1080/03605302.2010.490286. |
[18] |
E. Hebey and M. Vaugon, Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth, J. Funct. Anal., 119 (1994), 298-318.
doi: 10.1006/jfan.1994.1012. |
[19] |
J. Kazdan, F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math., 28 (1975), 567-597.
doi: 10.1002/cpa.3160280502. |
[20] |
Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I., J. Differ. Equations, 120 (1995), 319-410.
doi: 10.1006/jdeq.1995.1115. |
[21] |
R. Mazzeo and F. Pacard, Constant scalar curvature metrics with isolated singularities, J. Duke Math., 99 (1999), 353-418.
doi: 10.1215/S0012-7094-99-09913-1. |
[22] |
A. M. Micheletti and A. Pistoia, On the effect of the domain geometry on the existence of sign changing solutions to elliptic problems with critical and supercritical growth, Nonlinearity, 17 (2004), 851-866.
doi: 10.1088/0951-7715/17/3/007. |
[23] |
M. Musso and A. Pistoia, Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains, J. Math. Pures Appl., 86 (2006), 510-528.
doi: 10.1016/j.matpur.2006.10.006. |
[24] |
M. Musso and A. Pistoia, Sign changing solutions to a Bahri-Coron's problem in pierced domains, Discrete Contin. Dyn. Syst., 21 (2008), 295-306.
doi: 10.3934/dcds.2008.21.295. |
[25] |
M. Musso and A. Pistoia, Persistence of Coron's solution in nearly critical problems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 331-357. |
[26] |
M. Musso and A. Pistoia, Tower of Bubbles for almost critical problems in general domains, J. Math. Pure Appl., 93 (2010), 1-40.
doi: 10.1016/j.matpur.2009.08.001. |
[27] |
F. Pacard and T. Rivière, "Linear and Nonlinear Aspects of Vortices. The Ginzburg-Landau Model," Progress in Nonlinear Differential Equations and their Applications, 39 Boston, Birkhäuser, 2000.
doi: 10.1007/978-1-4612-1386-4. |
[28] |
A. Pistoia and T. Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340.
doi: 10.1016/j.anihpc.2006.03.002. |
[29] |
S. I. Pohozaev, Eigenfunction of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. |
[30] |
O. Rey, On a variational problem with lack of compactness: the effect of small holes in the domain, C. R. Acad. Sci. Paris Sér. I Math., 308 (1989), 349-352. |
[31] |
O. Rey, The role of the Green function in a nonlinear elliptic equation involving the critical Sobolev exponetion, J. Funct. Anal., 89 (1990), 1-52.
doi: 10.1016/0022-1236(90)90002-3. |
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