American Institute of Mathematical Sciences

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January  2018, 17(1): 209-230. doi: 10.3934/cpaa.2018013

Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary

Shaodong Wang

McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, Quebec H3A 0B9, Canada

Received  April 2017 Revised  June 2017 Published  September 2017

Fund Project: The author is supported by the China Scholarship Council.

We consider the Yamabe-type problem
$\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*} $
when
$(M,g)$
is the standard half sphere of dimensions
$n≥ 3$
. We establish existence results of positive blowing-up solutions with unbounded energy to this problem for all dimensions
$n≥ 3$
.
Keywords: Blow-up solutions, Yamabe-type problem, half space, Neumann problem, half sphere.
Mathematics Subject Classification: Primary:35J25, 35J65;Secondary:53A30.
Citation: Shaodong Wang. Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary. Communications on Pure & Applied Analysis, 2018, 17 (1) : 209-230. doi: 10.3934/cpaa.2018013
References:
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J. Escobar, The Yamabe problem on manifolds with boundary, J. Differential Geom., 35 (1992), 21-84.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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M. Khuri, F. Marques and R. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[30]

F. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.   Google Scholar

[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.  Google Scholar

[32]

F. Marques, Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary, Comm. Anal. Geom., 15 (2007), 381-405.   Google Scholar

[33]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. Brendle and F. Marques, Blow-up phenomena for the Yamabe equation. Ⅱ, J. Differential Geom., 81 (2009), 225-250.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

O. Druet, From one bubble to several bubbles: the low-dimensional case, J. Differential Geom., 63 (2003), 399-473.   Google Scholar

[11]

O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not., (2004), 1143-1191.  doi: 10.1155/S1073792804133278.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Escobar, The Yamabe problem on manifolds with boundary, J. Differential Geom., 35 (1992), 21-84.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

M. Khuri, F. Marques and R. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom., 81 (2009), 143-196.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

F. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom., 71 (2005), 315-346.   Google Scholar

[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.  Google Scholar

[32]

F. Marques, Conformal deformations to scalar-flat metrics with constant mean curvature on the boundary, Comm. Anal. Geom., 15 (2007), 381-405.   Google Scholar

[33]

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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