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

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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

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

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.
[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.
[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.
[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.
[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.
[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.
[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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