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Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents
1. | School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
2. | Centro de Modelamiento Matemático (CMM), Universidad de Chile, Beauchef 851, Santiago |
3. | Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago |
References:
[1] |
Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity,, A tribute in honour of G. Prodi. Scuola Norm. Sup. Pisa, (1991), 9.
|
[2] |
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), 591.
doi: 10.1080/03605309508821110. |
[3] |
Adimurthi, F. 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), 318.
doi: 10.1006/jfan.1993.1053. |
[4] |
Adimurthi, F. Pacella and S. L. Yadava, Characterization of concentration points and L1- estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent,, Diff. Integ. Equ., 8 (1995), 41.
|
[5] |
W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem,, Journal of Differential Equations, 251 (2011), 881.
doi: 10.1016/j.jde.2011.05.009. |
[6] |
W. Ao, M. Musso and J. Wei, Triple junction solutions for a singularly perturbed Neumann problem,, SIAM J. Math. Anal., 43 (2011), 2519.
doi: 10.1137/100812100. |
[7] |
G. Bianchi and H. Egnell, A note on the Sobolev inequality,, J. Funct. Anal., 100 (1991), 18.
doi: 10.1016/0022-1236(91)90099-Q. |
[8] |
J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity,, J. Differential Equations, 244 (2008), 2473.
doi: 10.1016/j.jde.2008.02.024. |
[9] |
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.
doi: 10.1002/cpa.3160420304. |
[10] |
D. Cao and T. Kupper, On the existence of multipeaked solutions to a semilinear Neumann problem,, Duke Math. J., 97 (1999), 261.
doi: 10.1215/S0012-7094-99-09712-0. |
[11] |
E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem,, Pacific J. Math., 189 (1999), 241.
doi: 10.2140/pjm.1999.189.241. |
[12] |
M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883.
doi: 10.1512/iumj.1999.48.1596. |
[13] |
M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems,, SIAM J. Math. Anal., 31 (2000), 63.
doi: 10.1137/S0036141098332834. |
[14] |
M. del Pino, F. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents,, J. Eur. Math. Soc. (JEMS), 16 (2014), 1687.
doi: 10.4171/JEMS/473. |
[15] |
M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent,, J. Eur. Math. Soc. (JEMS), 12 (2010), 1553.
doi: 10.4171/JEMS/241. |
[16] |
M. del Pino, M. Musso and A. Pistoia, Supercritical boundary bubbling in a semilinear Neumann problem,, Ann. Inst. H. Poincare Anal. Non-Linearie, 22 (2005), 45.
doi: 10.1016/j.anihpc.2004.05.001. |
[17] |
N. Ghoussoub and C. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent,, Math. Z., 229 (1998), 443.
doi: 10.1007/PL00004663. |
[18] |
N. Ghoussoub, C. Gui and M. Zhu, On a singularly perturbed Neumann problem with the critical exponent,, Comm. Partial Differential Equations, 26 (2001), 1929.
doi: 10.1081/PDE-100107812. |
[19] |
M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory,, Calc. Var. Partial Differential Equations, 11 (2000), 143.
doi: 10.1007/PL00009907. |
[20] |
C. Gui, Multi-peak solutions for a semilinear Neumann problem,, Duke Math. J., 84 (1996), 739.
doi: 10.1215/S0012-7094-96-08423-9. |
[21] |
C. Gui and C.-S. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem,, J. Reine Angew. Math., 546 (2002), 201.
doi: 10.1515/crll.2002.044. |
[22] |
C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems,, J. Differential Equations, 158 (1999), 1.
doi: 10.1016/S0022-0396(99)80016-3. |
[23] |
Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition,, Comm. Partial Differential Equations, 23 (1998), 487.
doi: 10.1080/03605309808821354. |
[24] |
C.-S. Lin, Locating the peaks of solutions via the maximum principle, I. The Neumann problem,, Comm. Pure Appl. Math., 54 (2001), 1065.
doi: 10.1002/cpa.1017. |
[25] |
F.-H. Lin, W.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem,, Comm. Pure Appl. Math., 60 (2007), 252.
doi: 10.1002/cpa.20139. |
[26] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Diff. Equat., 72 (1988), 1.
doi: 10.1016/0022-0396(88)90147-7. |
[27] |
F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem,, Adv. Math., 209 (2007), 460.
doi: 10.1016/j.aim.2006.05.014. |
[28] |
F. Mahmoudi, A. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve,, Comm. Pure Appl. Math., 62 (2009), 1155.
doi: 10.1002/cpa.20290. |
[29] |
F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold,, Geom. Funct. Anal., 16 (2006), 924.
doi: 10.1007/s00039-006-0566-7. |
[30] |
F. Mahmoudi, F. S. Sanchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture in higher dimension,, J. Differential Equations, 258 (2015), 243.
doi: 10.1016/j.jde.2014.09.010. |
[31] |
S. Maier-Paape, K. Schmitt and Z. Q. Wang, On Neumann problems for semilinear elliptic equations with critical nonlinearity existence and symmetry of multi-peaked solutions,, Comm. Partial Differential Equations, 22 (1997), 1493.
doi: 10.1080/03605309708821309. |
[32] |
A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507.
doi: 10.1002/cpa.10049. |
[33] |
A. Malchiodi and M. Montenegro, Multidimensional Boundary-layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105.
doi: 10.1215/S0012-7094-04-12414-5. |
[34] |
A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, Geom. Funct. Anal., 15 (2005), 1162.
doi: 10.1007/s00039-005-0542-7. |
[35] |
R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes,, Comm. Anal. Geom., 13 (2005), 633.
doi: 10.4310/CAG.2005.v13.n4.a1. |
[36] |
M. Musso and J. Yang, Curve like concentration layers for a singularly perturbed nonlinear problem with critical exponents,, Comm. Partial Differential Equations, 39 (2014), 1048.
doi: 10.1080/03605302.2013.851215. |
[37] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9.
|
[38] |
W.-M. Ni, Qualitative properties of solutions to elliptic problems,, in Stationary Partial Differential Equations. Handbook Differential Equations, (2004), 157.
doi: 10.1016/S1874-5733(04)80005-6. |
[39] |
W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819.
doi: 10.1002/cpa.3160440705. |
[40] |
W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247.
doi: 10.1215/S0012-7094-93-07004-4. |
[41] |
W.-M. Ni, X.-B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents,, Duke Math. J., 67 (1992), 1.
doi: 10.1215/S0012-7094-92-06701-9. |
[42] |
S. Pohozaev, Eigenfunctions of the equation $\Delta u +\lambda f (u) =0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36.
|
[43] |
O. Rey, An elliptic Neumann problem with critical nonlinearity in three dimensional domains,, Comm. Contemp. Math., 1 (1999), 405.
doi: 10.1142/S0219199799000158. |
[44] |
O. Rey and J. Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity,, J. Eur. Math. Soc. (JEMS), 7 (2005), 449.
doi: 10.4171/JEMS/35. |
[45] |
G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl. (IV), 110 (1976), 353.
doi: 10.1007/BF02418013. |
[46] |
L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture,, Trans. Amer. Math. Soc., 362 (2010), 4581.
doi: 10.1090/S0002-9947-10-04955-X. |
[47] |
Z. Q. Wang, High energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponent,, Proc. Roy. Soc. Edimburgh, 125 (1995), 1003.
doi: 10.1017/S0308210500022617. |
[48] |
X.-J. Wang, Neumann problem of semilinear elliptic equations involving critical Sobolev exponents,, J. Differential Equations, 93 (1991), 283.
doi: 10.1016/0022-0396(91)90014-Z. |
[49] |
J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem,, J. Differential Equations, 134 (1997), 104.
doi: 10.1006/jdeq.1996.3218. |
show all references
References:
[1] |
Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity,, A tribute in honour of G. Prodi. Scuola Norm. Sup. Pisa, (1991), 9.
|
[2] |
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), 591.
doi: 10.1080/03605309508821110. |
[3] |
Adimurthi, F. 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), 318.
doi: 10.1006/jfan.1993.1053. |
[4] |
Adimurthi, F. Pacella and S. L. Yadava, Characterization of concentration points and L1- estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent,, Diff. Integ. Equ., 8 (1995), 41.
|
[5] |
W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem,, Journal of Differential Equations, 251 (2011), 881.
doi: 10.1016/j.jde.2011.05.009. |
[6] |
W. Ao, M. Musso and J. Wei, Triple junction solutions for a singularly perturbed Neumann problem,, SIAM J. Math. Anal., 43 (2011), 2519.
doi: 10.1137/100812100. |
[7] |
G. Bianchi and H. Egnell, A note on the Sobolev inequality,, J. Funct. Anal., 100 (1991), 18.
doi: 10.1016/0022-1236(91)90099-Q. |
[8] |
J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity,, J. Differential Equations, 244 (2008), 2473.
doi: 10.1016/j.jde.2008.02.024. |
[9] |
L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271.
doi: 10.1002/cpa.3160420304. |
[10] |
D. Cao and T. Kupper, On the existence of multipeaked solutions to a semilinear Neumann problem,, Duke Math. J., 97 (1999), 261.
doi: 10.1215/S0012-7094-99-09712-0. |
[11] |
E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem,, Pacific J. Math., 189 (1999), 241.
doi: 10.2140/pjm.1999.189.241. |
[12] |
M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883.
doi: 10.1512/iumj.1999.48.1596. |
[13] |
M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems,, SIAM J. Math. Anal., 31 (2000), 63.
doi: 10.1137/S0036141098332834. |
[14] |
M. del Pino, F. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents,, J. Eur. Math. Soc. (JEMS), 16 (2014), 1687.
doi: 10.4171/JEMS/473. |
[15] |
M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent,, J. Eur. Math. Soc. (JEMS), 12 (2010), 1553.
doi: 10.4171/JEMS/241. |
[16] |
M. del Pino, M. Musso and A. Pistoia, Supercritical boundary bubbling in a semilinear Neumann problem,, Ann. Inst. H. Poincare Anal. Non-Linearie, 22 (2005), 45.
doi: 10.1016/j.anihpc.2004.05.001. |
[17] |
N. Ghoussoub and C. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent,, Math. Z., 229 (1998), 443.
doi: 10.1007/PL00004663. |
[18] |
N. Ghoussoub, C. Gui and M. Zhu, On a singularly perturbed Neumann problem with the critical exponent,, Comm. Partial Differential Equations, 26 (2001), 1929.
doi: 10.1081/PDE-100107812. |
[19] |
M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory,, Calc. Var. Partial Differential Equations, 11 (2000), 143.
doi: 10.1007/PL00009907. |
[20] |
C. Gui, Multi-peak solutions for a semilinear Neumann problem,, Duke Math. J., 84 (1996), 739.
doi: 10.1215/S0012-7094-96-08423-9. |
[21] |
C. Gui and C.-S. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem,, J. Reine Angew. Math., 546 (2002), 201.
doi: 10.1515/crll.2002.044. |
[22] |
C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems,, J. Differential Equations, 158 (1999), 1.
doi: 10.1016/S0022-0396(99)80016-3. |
[23] |
Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition,, Comm. Partial Differential Equations, 23 (1998), 487.
doi: 10.1080/03605309808821354. |
[24] |
C.-S. Lin, Locating the peaks of solutions via the maximum principle, I. The Neumann problem,, Comm. Pure Appl. Math., 54 (2001), 1065.
doi: 10.1002/cpa.1017. |
[25] |
F.-H. Lin, W.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem,, Comm. Pure Appl. Math., 60 (2007), 252.
doi: 10.1002/cpa.20139. |
[26] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Diff. Equat., 72 (1988), 1.
doi: 10.1016/0022-0396(88)90147-7. |
[27] |
F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem,, Adv. Math., 209 (2007), 460.
doi: 10.1016/j.aim.2006.05.014. |
[28] |
F. Mahmoudi, A. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve,, Comm. Pure Appl. Math., 62 (2009), 1155.
doi: 10.1002/cpa.20290. |
[29] |
F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold,, Geom. Funct. Anal., 16 (2006), 924.
doi: 10.1007/s00039-006-0566-7. |
[30] |
F. Mahmoudi, F. S. Sanchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture in higher dimension,, J. Differential Equations, 258 (2015), 243.
doi: 10.1016/j.jde.2014.09.010. |
[31] |
S. Maier-Paape, K. Schmitt and Z. Q. Wang, On Neumann problems for semilinear elliptic equations with critical nonlinearity existence and symmetry of multi-peaked solutions,, Comm. Partial Differential Equations, 22 (1997), 1493.
doi: 10.1080/03605309708821309. |
[32] |
A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507.
doi: 10.1002/cpa.10049. |
[33] |
A. Malchiodi and M. Montenegro, Multidimensional Boundary-layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105.
doi: 10.1215/S0012-7094-04-12414-5. |
[34] |
A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, Geom. Funct. Anal., 15 (2005), 1162.
doi: 10.1007/s00039-005-0542-7. |
[35] |
R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes,, Comm. Anal. Geom., 13 (2005), 633.
doi: 10.4310/CAG.2005.v13.n4.a1. |
[36] |
M. Musso and J. Yang, Curve like concentration layers for a singularly perturbed nonlinear problem with critical exponents,, Comm. Partial Differential Equations, 39 (2014), 1048.
doi: 10.1080/03605302.2013.851215. |
[37] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9.
|
[38] |
W.-M. Ni, Qualitative properties of solutions to elliptic problems,, in Stationary Partial Differential Equations. Handbook Differential Equations, (2004), 157.
doi: 10.1016/S1874-5733(04)80005-6. |
[39] |
W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819.
doi: 10.1002/cpa.3160440705. |
[40] |
W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, Duke Math. J., 70 (1993), 247.
doi: 10.1215/S0012-7094-93-07004-4. |
[41] |
W.-M. Ni, X.-B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents,, Duke Math. J., 67 (1992), 1.
doi: 10.1215/S0012-7094-92-06701-9. |
[42] |
S. Pohozaev, Eigenfunctions of the equation $\Delta u +\lambda f (u) =0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36.
|
[43] |
O. Rey, An elliptic Neumann problem with critical nonlinearity in three dimensional domains,, Comm. Contemp. Math., 1 (1999), 405.
doi: 10.1142/S0219199799000158. |
[44] |
O. Rey and J. Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity,, J. Eur. Math. Soc. (JEMS), 7 (2005), 449.
doi: 10.4171/JEMS/35. |
[45] |
G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl. (IV), 110 (1976), 353.
doi: 10.1007/BF02418013. |
[46] |
L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture,, Trans. Amer. Math. Soc., 362 (2010), 4581.
doi: 10.1090/S0002-9947-10-04955-X. |
[47] |
Z. Q. Wang, High energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponent,, Proc. Roy. Soc. Edimburgh, 125 (1995), 1003.
doi: 10.1017/S0308210500022617. |
[48] |
X.-J. Wang, Neumann problem of semilinear elliptic equations involving critical Sobolev exponents,, J. Differential Equations, 93 (1991), 283.
doi: 10.1016/0022-0396(91)90014-Z. |
[49] |
J. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem,, J. Differential Equations, 134 (1997), 104.
doi: 10.1006/jdeq.1996.3218. |
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