Standing wave concentrating on compact manifolds for nonlinear
Schrödinger equations

Jaeyoung Byeon; Ohsang Kwon; Yoshihito Oshita

doi:10.3934/cpaa.2015.14.825

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May 2015, 14(3): 825-842. doi: 10.3934/cpaa.2015.14.825

Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

Jaeyoung Byeon ^1, , Ohsang Kwon ^2, and Yoshihito Oshita ^3,

1.	Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu Daejeon, 305-701, South Korea
2.	Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Road, Shanghai, China
3.	Department of Mathematics, Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530, Japan

Received April 2014 Revised December 2014 Published March 2015

Abstract
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For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\} $. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0 $. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.

Keywords: Concentration phenomena, nondegeneracy, nonlinear Schrödinger equation., infinite dimensional reduction.

Mathematics Subject Classification: 35J20, 35Q55, 35B4.

Citation: Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825

References:

[1]	A. Ambrosetti, A. Malchiodi and W.-M Ni, Singularly perturves elliptic equation with symmetry: existence of solutions concentrating on spheres. I,, \emph{Comm. Math. Phys.}, 235 (2003), 427. doi: 10.1007/s00220-003-0811-y. Google Scholar
[2]	A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{J. d'Anayse Math.}, 98 (2006), 317. doi: 10.1007/BF02790279. Google Scholar
[3]	A. Ambrosetti, A. Malchiodi and A. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Rational Mech. Anal.}, 159 (2001), 253. doi: 10.1007/s002050100152. Google Scholar
[4]	A. Ambrosetti and D. Ruiz, Radial solutions concentrating on sphere of nonlinear Schrödinger equations with vanishing potentials,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 136 (2006), 889. doi: 10.1017/S0308210500004789. Google Scholar
[5]	M. Badiale. and T. D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation,, \emph{Nonlinear Anal.}, 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9. Google Scholar
[6]	V. Benci. and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, \emph{J. Diff. Equat.}, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138. Google Scholar
[7]	J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential,, \emph{Nonlinear Anal.}, 50 (2002), 1135. doi: 10.1016/S0362-546X(01)00805-7. Google Scholar
[8]	J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 29 (2004), 1877. doi: 10.1081/PDE-200040205. Google Scholar
[9]	J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138A (2008), 975. doi: 10.1017/S0308210507000236. Google Scholar
[10]	J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Rational Mech. Anal.}, 165 (2002), 295. doi: 10.1007/s00205-002-0225-6. Google Scholar
[11]	J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II,, \emph{Calc. Var. Partial Differential Equations}, 18 (2003), 207. doi: 10.1007/s00526-002-0191-8. Google Scholar
[12]	D. Cao, E. S. Noussair and S. Yan, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, to appear in \emph{Trans. of AMS}., (). doi: 10.1090/S0002-9947-08-04348-1. Google Scholar
[13]	D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency,, \emph{Math. Ann}, 336 (2006), 925. doi: 10.1007/s00208-006-0021-y. Google Scholar
[14]	E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, \emph{J. Differential Equations}, 74 (1998), 120. doi: 10.1016/0022-0396(88)90021-6. Google Scholar
[15]	E. N. Dancer and S. Yan, On the existence of multipeak solutions for nonlinear field equations on $R^N$,, \emph{Discrete Contin. Dynam. Systems, 6 (2000), 39. Google Scholar
[16]	E. N. Dancer, K. Y. Lam and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations,, \emph{Abstr. Apply. Anal.}, 3 (1998), 293. doi: 10.1155/S1085337598000578. Google Scholar
[17]	M. del Pino and P. L. Felmer, Local mountaion passes for semilinear elliptic problems in unbounded domains,, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar
[18]	M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations,, \emph{J. Funct. Anal.}, 149 (1997), 245. doi: 10.1006/jfan.1996.3085. Google Scholar
[19]	M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, \emph{Ann. Inst. Henri Poincar\'e}, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar
[20]	M. del Pino and P. L. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, \emph{Math. Ann.}, 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar
[21]	M. del Pino, P. L. Felmer and O. H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential,, \emph{Nonlinear Anal. TMA, 34 (1998), 979. doi: 10.1016/S0362-546X(97)00593-2. Google Scholar
[22]	M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations,, \emph{Comm. Pur. App. Math.}, LX (2007), 113. doi: 10.1002/cpa.20135. Google Scholar
[23]	A. Floer and A. Weinstein, Non spreading wave packets for the cubic Schrödinger equations with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar
[24]	B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys., 68 (1979), 209. Google Scholar
[25]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar
[26]	C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, \emph{Comm. Partial Differential Equations}, 21 (1996), 787. doi: 10.1080/03605309608821208. Google Scholar
[27]	Y. Kabeya and K. Tanaka, Uniqueness of positive solutions of semilinear elliptic equations in $R^N$ and Séré's non-degeneracy condition,, \emph{Comm. Partial Differential Equations, 24 (1999), 563. doi: 10.1080/03605309908821434. Google Scholar
[28]	X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, \emph{Adv. Differential Equations}, 5 (2000), 899. Google Scholar
[29]	T. Kato, Perturbation Theory for Linear Operators,, second ed., (1976). Google Scholar
[30]	O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 139A (2009), 833. doi: 10.1017/S0308210508000309. Google Scholar
[31]	C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains,, \emph{Comm. Partial Differential Equations, 16 (1996), 585. doi: 10.1080/03605309108820770. Google Scholar
[32]	Y. Y. Li, On a singular perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955. Google Scholar
[33]	F. Mahmoudi and A. Malchiodi, Concentration on minimal submainifolds for a singularly perturbed Neumann problem,, \emph{Adv. Math.}, 209 (2007), 460. doi: 10.1016/j.aim.2006.05.014. Google Scholar
[34]	F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold,, \emph{Geom. Funct. Anal.}, 16 (2006), 924. doi: 10.1007/s00039-006-0566-7. Google Scholar
[35]	F. Mahmoudi, F. S. Sánchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture for general submanifolds,, \emph{J. Differential Equations}, 258 (2015), 243. doi: 10.1016/j.jde.2014.09.010. Google Scholar
[36]	A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, \emph{G.A.F.A.}, 15-16 (2005), 15. doi: 10.1007/s00039-005-0542-7. Google Scholar
[37]	A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, \emph{Comm. Pure Appl. Math.}, 15 (2002), 1507. doi: 10.1002/cpa.10049. Google Scholar
[38]	A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem,, \emph{Duke Math. J.}, 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5. Google Scholar
[39]	S. Minakshisundaram and and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds,, \emph{Canad. J. Math.}, 1 (1949), 242. Google Scholar
[40]	Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$,, \emph{Comm. Partial Differential Equations}, 13 (1988), 1499. doi: 10.1080/03605308808820585. Google Scholar
[41]	Y. G. Oh, Correction to: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$,, \emph{Comm. Partial Differential Equations}, 14 (1989), 833. doi: 10.1080/03605308908820631. Google Scholar
[42]	Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multile well potential,, \emph{Comm. Math. Phys.}, 131 (1990), 223. Google Scholar
[43]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar
[44]	Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency,, \emph{Calc. Var. Partial Differential Equations}, 29 (2007), 365. doi: 10.1007/s00526-006-0070-9. Google Scholar
[45]	B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $R^N$,, \emph{Ann. Mat. Pura Appl.}, 184 (2002), 73. doi: 10.1007/s102310200029. Google Scholar
[46]	W. Strauss, Existence of solitary waves in higher demensions,, \emph{Comm. Math. Phys.}, 55 (1977), 149. Google Scholar
[47]	J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials,, \emph{J. Differential Equations}, 238 (2007), 201. doi: 10.1016/j.jde.2007.03.018. Google Scholar
[48]	L. P. Wang, J. Wei and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture for general hypersurfaces,, \emph{Comm. Partial Differential Equations}, 36 (2011), 2117. doi: 10.1080/03605302.2011.580033. Google Scholar
[49]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 153 (1993), 229. Google Scholar
[50]	Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations,, \emph{J. Differential Equations}, 159 (1999), 102. doi: 10.1006/jdeq.1999.3650. Google Scholar

show all references

References:

[1]	A. Ambrosetti, A. Malchiodi and W.-M Ni, Singularly perturves elliptic equation with symmetry: existence of solutions concentrating on spheres. I,, \emph{Comm. Math. Phys.}, 235 (2003), 427. doi: 10.1007/s00220-003-0811-y. Google Scholar
[2]	A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{J. d'Anayse Math.}, 98 (2006), 317. doi: 10.1007/BF02790279. Google Scholar
[3]	A. Ambrosetti, A. Malchiodi and A. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Rational Mech. Anal.}, 159 (2001), 253. doi: 10.1007/s002050100152. Google Scholar
[4]	A. Ambrosetti and D. Ruiz, Radial solutions concentrating on sphere of nonlinear Schrödinger equations with vanishing potentials,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 136 (2006), 889. doi: 10.1017/S0308210500004789. Google Scholar
[5]	M. Badiale. and T. D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation,, \emph{Nonlinear Anal.}, 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9. Google Scholar
[6]	V. Benci. and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, \emph{J. Diff. Equat.}, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138. Google Scholar
[7]	J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential,, \emph{Nonlinear Anal.}, 50 (2002), 1135. doi: 10.1016/S0362-546X(01)00805-7. Google Scholar
[8]	J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 29 (2004), 1877. doi: 10.1081/PDE-200040205. Google Scholar
[9]	J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138A (2008), 975. doi: 10.1017/S0308210507000236. Google Scholar
[10]	J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Rational Mech. Anal.}, 165 (2002), 295. doi: 10.1007/s00205-002-0225-6. Google Scholar
[11]	J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II,, \emph{Calc. Var. Partial Differential Equations}, 18 (2003), 207. doi: 10.1007/s00526-002-0191-8. Google Scholar
[12]	D. Cao, E. S. Noussair and S. Yan, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, to appear in \emph{Trans. of AMS}., (). doi: 10.1090/S0002-9947-08-04348-1. Google Scholar
[13]	D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency,, \emph{Math. Ann}, 336 (2006), 925. doi: 10.1007/s00208-006-0021-y. Google Scholar
[14]	E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, \emph{J. Differential Equations}, 74 (1998), 120. doi: 10.1016/0022-0396(88)90021-6. Google Scholar
[15]	E. N. Dancer and S. Yan, On the existence of multipeak solutions for nonlinear field equations on $R^N$,, \emph{Discrete Contin. Dynam. Systems, 6 (2000), 39. Google Scholar
[16]	E. N. Dancer, K. Y. Lam and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations,, \emph{Abstr. Apply. Anal.}, 3 (1998), 293. doi: 10.1155/S1085337598000578. Google Scholar
[17]	M. del Pino and P. L. Felmer, Local mountaion passes for semilinear elliptic problems in unbounded domains,, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar
[18]	M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations,, \emph{J. Funct. Anal.}, 149 (1997), 245. doi: 10.1006/jfan.1996.3085. Google Scholar
[19]	M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, \emph{Ann. Inst. Henri Poincar\'e}, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar
[20]	M. del Pino and P. L. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, \emph{Math. Ann.}, 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar
[21]	M. del Pino, P. L. Felmer and O. H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential,, \emph{Nonlinear Anal. TMA, 34 (1998), 979. doi: 10.1016/S0362-546X(97)00593-2. Google Scholar
[22]	M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations,, \emph{Comm. Pur. App. Math.}, LX (2007), 113. doi: 10.1002/cpa.20135. Google Scholar
[23]	A. Floer and A. Weinstein, Non spreading wave packets for the cubic Schrödinger equations with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar
[24]	B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys., 68 (1979), 209. Google Scholar
[25]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar
[26]	C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, \emph{Comm. Partial Differential Equations}, 21 (1996), 787. doi: 10.1080/03605309608821208. Google Scholar
[27]	Y. Kabeya and K. Tanaka, Uniqueness of positive solutions of semilinear elliptic equations in $R^N$ and Séré's non-degeneracy condition,, \emph{Comm. Partial Differential Equations, 24 (1999), 563. doi: 10.1080/03605309908821434. Google Scholar
[28]	X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, \emph{Adv. Differential Equations}, 5 (2000), 899. Google Scholar
[29]	T. Kato, Perturbation Theory for Linear Operators,, second ed., (1976). Google Scholar
[30]	O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 139A (2009), 833. doi: 10.1017/S0308210508000309. Google Scholar
[31]	C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains,, \emph{Comm. Partial Differential Equations, 16 (1996), 585. doi: 10.1080/03605309108820770. Google Scholar
[32]	Y. Y. Li, On a singular perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955. Google Scholar
[33]	F. Mahmoudi and A. Malchiodi, Concentration on minimal submainifolds for a singularly perturbed Neumann problem,, \emph{Adv. Math.}, 209 (2007), 460. doi: 10.1016/j.aim.2006.05.014. Google Scholar
[34]	F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold,, \emph{Geom. Funct. Anal.}, 16 (2006), 924. doi: 10.1007/s00039-006-0566-7. Google Scholar
[35]	F. Mahmoudi, F. S. Sánchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture for general submanifolds,, \emph{J. Differential Equations}, 258 (2015), 243. doi: 10.1016/j.jde.2014.09.010. Google Scholar
[36]	A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, \emph{G.A.F.A.}, 15-16 (2005), 15. doi: 10.1007/s00039-005-0542-7. Google Scholar
[37]	A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, \emph{Comm. Pure Appl. Math.}, 15 (2002), 1507. doi: 10.1002/cpa.10049. Google Scholar
[38]	A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem,, \emph{Duke Math. J.}, 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5. Google Scholar
[39]	S. Minakshisundaram and and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds,, \emph{Canad. J. Math.}, 1 (1949), 242. Google Scholar
[40]	Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$,, \emph{Comm. Partial Differential Equations}, 13 (1988), 1499. doi: 10.1080/03605308808820585. Google Scholar
[41]	Y. G. Oh, Correction to: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$,, \emph{Comm. Partial Differential Equations}, 14 (1989), 833. doi: 10.1080/03605308908820631. Google Scholar
[42]	Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multile well potential,, \emph{Comm. Math. Phys.}, 131 (1990), 223. Google Scholar
[43]	P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar
[44]	Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency,, \emph{Calc. Var. Partial Differential Equations}, 29 (2007), 365. doi: 10.1007/s00526-006-0070-9. Google Scholar
[45]	B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $R^N$,, \emph{Ann. Mat. Pura Appl.}, 184 (2002), 73. doi: 10.1007/s102310200029. Google Scholar
[46]	W. Strauss, Existence of solitary waves in higher demensions,, \emph{Comm. Math. Phys.}, 55 (1977), 149. Google Scholar
[47]	J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials,, \emph{J. Differential Equations}, 238 (2007), 201. doi: 10.1016/j.jde.2007.03.018. Google Scholar
[48]	L. P. Wang, J. Wei and J. Yang, On Ambrosetti-Malchiodi-Ni conjecture for general hypersurfaces,, \emph{Comm. Partial Differential Equations}, 36 (2011), 2117. doi: 10.1080/03605302.2011.580033. Google Scholar
[49]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 153 (1993), 229. Google Scholar
[50]	Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations,, \emph{J. Differential Equations}, 159 (1999), 102. doi: 10.1006/jdeq.1999.3650. Google Scholar

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[19]	Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[20]	Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020030

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American Institute of Mathematical Sciences

Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

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American Institute of Mathematical Sciences

Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

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