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Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada |
$\begin{equation*} \begin{cases} Δ_g u+fu=0 \ in\ M\\ \frac{\partial u}{\partial ν}+hu=u^{\frac{n}{n-2}}\ on\ \partial M \end{cases}\end{equation*} $ |
$(M,g)$ |
$n≥ 3$ |
$n≥ 3$ |
References:
[1] |
S. Almaraz,
An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1-22.
doi: 10.2140/pjm.2010.248.1. |
[2] |
S. Almaraz,
A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var. Partial Differential Equations, 41 (2011), 341-386.
doi: 10.1007/s00526-010-0365-8. |
[3] |
S. Almaraz,
Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differential Equations, 251 (2011), 1813-1840.
doi: 10.1016/j.jde.2011.04.013. |
[4] |
S. Brendle,
Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc., 21 (2008), 951-979.
doi: 10.1090/S0894-0347-07-00575-9. |
[5] |
S. Brendle and S. Chen,
An existence theorem for the Yamabe problem on manifolds with boundary, J. Eur. Math. Soc. (JEMS), 16 (2014), 991-1016.
doi: 10.4171/JEMS/453. |
[6] |
S. Brendle and F. Marques,
Blow-up phenomena for the Yamabe equation. Ⅱ, J. Differential Geom., 81 (2009), 225-250.
|
[7] |
S. Chen, Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, preprint, arXiv: 0912.1302. Google Scholar |
[8] |
W. Chen, J. Wei and S. Yan,
Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^N$ with critical growth, J. Differential Equations, 252 (2012), 2425-2447.
doi: 10.1016/j.jde.2011.09.032. |
[9] |
P. Cherrier,
Problémes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal., 57 (1984), 154-206.
doi: 10.1016/0022-1236(84)90094-6. |
[10] |
O. Druet,
From one bubble to several bubbles: the low-dimensional case, J. Differential Geom., 63 (2003), 399-473.
|
[11] |
O. Druet,
Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., (2004), 1143-1191.
doi: 10.1155/S1073792804133278. |
[12] |
O. Druet and E. Hebey,
Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Trans. Amer. Math. Soc., 357 (2005), 1915-1929.
doi: 10.1090/S0002-9947-04-03681-5. |
[13] |
M. Disconzi and M. Khuri,
Compactness and non-compactness for the Yamabe problem on manifolds with boundary, J. Reine Angew. Math., 724 (2017), 145-201.
doi: 10.1515/crelle-2014-0083. |
[14] |
J. Escobar,
Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math. (2), 136 (1992), 1-50.
doi: 10.2307/2946545. |
[15] |
J. Escobar,
The Yamabe problem on manifolds with boundary, J. Differential Geom., 35 (1992), 21-84.
|
[16] |
P. Esposito, A. Pistoia and J. Vétois,
The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511-560.
doi: 10.1007/s00208-013-0971-9. |
[17] |
V. Felli and M. Ahmedou,
Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175-210.
doi: 10.1007/s00209-002-0486-7. |
[18] |
V. Felli and M. Ahmedou,
A geometric equation with critical nonlinearity on the boundary, Pacific J. Math., 218 (2005), 75-99.
doi: 10.2140/pjm.2005.218.75. |
[19] |
M. Ghimenti, A. Micheletti and A. Pistoia,
On Yamabe-type problems on Riemannian manifolds with boundary, Pacific J. Math., 284 (2016), 79-102.
doi: 10.2140/pjm.2016.284.79. |
[20] |
M. Ghimenti, A. Micheletti and A. Pistoia, Linear Perturbations of the Yamabe problem on manifolds with boundary preprint, arXiv: 1611.01336.
doi: 10.2140/pjm.2016.284.79. |
[21] |
Z. Han and Y. Li,
The Yamabe problem on manifolds with boundary: existence and compactness results, Duke Math. J., 99 (1999), 489-542.
doi: 10.1215/S0012-7094-99-09916-7. |
[22] |
E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014.
doi: 10.4171/134. |
[23] |
E. Hebey and J. Wei,
Resonant states for the static Klein-Gordon-Maxwell-Proca system, Math. Res. Lett., 19 (2012), 953-967.
doi: 10.4310/MRL.2012.v19.n4.a18. |
[24] |
M. Khuri, F. Marques and R. Schoen,
A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.
|
[25] |
Y. Li and L. Zhang,
A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations, 20 (2004), 133-151.
doi: 10.1007/s00526-003-0224-y. |
[26] |
Y. Li and L. Zhang,
Compactness of solutions to the Yamabe problem. Ⅱ, Calc. Var. Partial Differential Equations, 24 (2005), 185-237.
doi: 10.1007/s00526-004-0320-7. |
[27] |
Y. Li and L. Zhang,
Compactness of solutions to the Yamabe problem. Ⅲ, J. Funct. Anal., 245 (2007), 438-474.
doi: 10.1016/j.jfa.2006.11.010. |
[28] |
Y. Li and M. Zhu,
Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.
doi: 10.1215/S0012-7094-95-08016-8. |
[29] |
Y. Li and M. Zhu,
Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.
doi: 10.1142/S021919979900002X. |
[30] |
F. Marques,
A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.
|
[31] |
F. Marques,
Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599-1620.
doi: 10.1512/iumj.2005.54.2590. |
[32] |
F. Marques,
Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary, Comm. Anal. Geom., 15 (2007), 381-405.
|
[33] |
R. Schoen,
Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.
|
[34] |
R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Lecture Notes in Math., Springer, Berlin, (1989), 120-154.
doi: 10.1007/BFb0089180. |
[35] |
R. Schoen, On the number of constant scalar curvature metrics in a conformal class, in Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, (1991), 311-320. |
[36] |
P. Thizy and J. Vétois, Positive clusters for smooth perturbations of a critical elliptic equation in dimension four and five, preprint, arXiv: 1603.06479. Google Scholar |
[37] |
N. Trudinger,
Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274.
|
[38] |
J. Vétois and S. Wang, Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four, preprint 2017. Google Scholar |
[39] |
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), 4581-4615.
doi: 10.1090/S0002-9947-10-04955-X. |
[40] |
L. Wang, J. Wei and S. Yan,
On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc. (3), 102 (2011), 1099-1126.
doi: 10.1112/plms/pdq051. |
[41] |
J. Wei and S. Yan,
Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb R^N$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439.
doi: 10.1007/s00526-009-0270-1. |
[42] |
J. Wei and S. Yan,
Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb S^N$, J. Funct. Anal., 258 (2010), 3048-3081.
doi: 10.1016/j.jfa.2009.12.008. |
[43] |
J. Wei and S. Yan,
On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 423-457.
|
[44] |
J. Wei and S. Yan,
Infinitely many positive solutions for an elliptic problem with critical or supercritical growth, J. Math. Pures Appl. (9), 96 (2011), 307-333.
doi: 10.1016/j.matpur.2011.01.006. |
show all references
References:
[1] |
S. Almaraz,
An existence theorem of conformal scalar-flat metrics on manifolds with boundary, Pacific J. Math., 248 (2010), 1-22.
doi: 10.2140/pjm.2010.248.1. |
[2] |
S. Almaraz,
A compactness theorem for scalar-flat metrics on manifolds with boundary, Calc. Var. Partial Differential Equations, 41 (2011), 341-386.
doi: 10.1007/s00526-010-0365-8. |
[3] |
S. Almaraz,
Blow-up phenomena for scalar-flat metrics on manifolds with boundary, J. Differential Equations, 251 (2011), 1813-1840.
doi: 10.1016/j.jde.2011.04.013. |
[4] |
S. Brendle,
Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc., 21 (2008), 951-979.
doi: 10.1090/S0894-0347-07-00575-9. |
[5] |
S. Brendle and S. Chen,
An existence theorem for the Yamabe problem on manifolds with boundary, J. Eur. Math. Soc. (JEMS), 16 (2014), 991-1016.
doi: 10.4171/JEMS/453. |
[6] |
S. Brendle and F. Marques,
Blow-up phenomena for the Yamabe equation. Ⅱ, J. Differential Geom., 81 (2009), 225-250.
|
[7] |
S. Chen, Conformal deformation to scalar flat metrics with constant mean curvature on the boundary in higher dimensions, preprint, arXiv: 0912.1302. Google Scholar |
[8] |
W. Chen, J. Wei and S. Yan,
Infinitely many solutions for the Schrödinger equations in $\mathbb{R}^N$ with critical growth, J. Differential Equations, 252 (2012), 2425-2447.
doi: 10.1016/j.jde.2011.09.032. |
[9] |
P. Cherrier,
Problémes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal., 57 (1984), 154-206.
doi: 10.1016/0022-1236(84)90094-6. |
[10] |
O. Druet,
From one bubble to several bubbles: the low-dimensional case, J. Differential Geom., 63 (2003), 399-473.
|
[11] |
O. Druet,
Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., (2004), 1143-1191.
doi: 10.1155/S1073792804133278. |
[12] |
O. Druet and E. Hebey,
Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Trans. Amer. Math. Soc., 357 (2005), 1915-1929.
doi: 10.1090/S0002-9947-04-03681-5. |
[13] |
M. Disconzi and M. Khuri,
Compactness and non-compactness for the Yamabe problem on manifolds with boundary, J. Reine Angew. Math., 724 (2017), 145-201.
doi: 10.1515/crelle-2014-0083. |
[14] |
J. Escobar,
Conformal deformation of a Riemannian metric to a scalar flat metric with constant mean curvature on the boundary, Ann. of Math. (2), 136 (1992), 1-50.
doi: 10.2307/2946545. |
[15] |
J. Escobar,
The Yamabe problem on manifolds with boundary, J. Differential Geom., 35 (1992), 21-84.
|
[16] |
P. Esposito, A. Pistoia and J. Vétois,
The effect of linear perturbations on the Yamabe problem, Math. Ann., 358 (2014), 511-560.
doi: 10.1007/s00208-013-0971-9. |
[17] |
V. Felli and M. Ahmedou,
Compactness results in conformal deformations of Riemannian metrics on manifolds with boundaries, Math. Z., 244 (2003), 175-210.
doi: 10.1007/s00209-002-0486-7. |
[18] |
V. Felli and M. Ahmedou,
A geometric equation with critical nonlinearity on the boundary, Pacific J. Math., 218 (2005), 75-99.
doi: 10.2140/pjm.2005.218.75. |
[19] |
M. Ghimenti, A. Micheletti and A. Pistoia,
On Yamabe-type problems on Riemannian manifolds with boundary, Pacific J. Math., 284 (2016), 79-102.
doi: 10.2140/pjm.2016.284.79. |
[20] |
M. Ghimenti, A. Micheletti and A. Pistoia, Linear Perturbations of the Yamabe problem on manifolds with boundary preprint, arXiv: 1611.01336.
doi: 10.2140/pjm.2016.284.79. |
[21] |
Z. Han and Y. Li,
The Yamabe problem on manifolds with boundary: existence and compactness results, Duke Math. J., 99 (1999), 489-542.
doi: 10.1215/S0012-7094-99-09916-7. |
[22] |
E. Hebey, Compactness and Stability for Nonlinear Elliptic Equations Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2014.
doi: 10.4171/134. |
[23] |
E. Hebey and J. Wei,
Resonant states for the static Klein-Gordon-Maxwell-Proca system, Math. Res. Lett., 19 (2012), 953-967.
doi: 10.4310/MRL.2012.v19.n4.a18. |
[24] |
M. Khuri, F. Marques and R. Schoen,
A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.
|
[25] |
Y. Li and L. Zhang,
A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations, 20 (2004), 133-151.
doi: 10.1007/s00526-003-0224-y. |
[26] |
Y. Li and L. Zhang,
Compactness of solutions to the Yamabe problem. Ⅱ, Calc. Var. Partial Differential Equations, 24 (2005), 185-237.
doi: 10.1007/s00526-004-0320-7. |
[27] |
Y. Li and L. Zhang,
Compactness of solutions to the Yamabe problem. Ⅲ, J. Funct. Anal., 245 (2007), 438-474.
doi: 10.1016/j.jfa.2006.11.010. |
[28] |
Y. Li and M. Zhu,
Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.
doi: 10.1215/S0012-7094-95-08016-8. |
[29] |
Y. Li and M. Zhu,
Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., 1 (1999), 1-50.
doi: 10.1142/S021919979900002X. |
[30] |
F. Marques,
A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.
|
[31] |
F. Marques,
Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J., 54 (2005), 1599-1620.
doi: 10.1512/iumj.2005.54.2590. |
[32] |
F. Marques,
Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary, Comm. Anal. Geom., 15 (2007), 381-405.
|
[33] |
R. Schoen,
Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.
|
[34] |
R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Lecture Notes in Math., Springer, Berlin, (1989), 120-154.
doi: 10.1007/BFb0089180. |
[35] |
R. Schoen, On the number of constant scalar curvature metrics in a conformal class, in Pitman Monogr. Surveys Pure Appl. Math., Longman Sci. Tech., Harlow, (1991), 311-320. |
[36] |
P. Thizy and J. Vétois, Positive clusters for smooth perturbations of a critical elliptic equation in dimension four and five, preprint, arXiv: 1603.06479. Google Scholar |
[37] |
N. Trudinger,
Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 265-274.
|
[38] |
J. Vétois and S. Wang, Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four, preprint 2017. Google Scholar |
[39] |
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), 4581-4615.
doi: 10.1090/S0002-9947-10-04955-X. |
[40] |
L. Wang, J. Wei and S. Yan,
On Lin-Ni's conjecture in convex domains, Proc. Lond. Math. Soc. (3), 102 (2011), 1099-1126.
doi: 10.1112/plms/pdq051. |
[41] |
J. Wei and S. Yan,
Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb R^N$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439.
doi: 10.1007/s00526-009-0270-1. |
[42] |
J. Wei and S. Yan,
Infinitely many solutions for the prescribed scalar curvature problem on $\mathbb S^N$, J. Funct. Anal., 258 (2010), 3048-3081.
doi: 10.1016/j.jfa.2009.12.008. |
[43] |
J. Wei and S. Yan,
On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9 (2010), 423-457.
|
[44] |
J. Wei and S. Yan,
Infinitely many positive solutions for an elliptic problem with critical or supercritical growth, J. Math. Pures Appl. (9), 96 (2011), 307-333.
doi: 10.1016/j.matpur.2011.01.006. |
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