Standing wave concentrating on compact manifolds for nonlinear
Schrödinger equations

Jaeyoung Byeon; Ohsang Kwon; Yoshihito Oshita

doi:10.3934/cpaa.2015.14.825

Previous Article
Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves
CPAA Home
This Issue
Next Article
Uniform stability of the Boltzmann equation with an external force near vacuum

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
Related Papers

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

[1]	Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099
[2]	Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[3]	Liren Lin, Tai-Peng Tsai. Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 295-336. doi: 10.3934/dcds.2017013
[4]	Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359
[5]	Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[6]	Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237
[7]	Hongzi Cong, Lufang Mi, Yunfeng Shi, Yuan Wu. On the existence of full dimensional KAM torus for nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6599-6630. doi: 10.3934/dcds.2019287
[8]	Teresa D'Aprile. Some existence and concentration results for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2002, 1 (4) : 457-474. doi: 10.3934/cpaa.2002.1.457
[9]	Yingying Xie, Jian Su, Liquan Mei. Blowup results and concentration in focusing Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 5001-5017. doi: 10.3934/dcds.2020209
[10]	D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[11]	Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379
[12]	Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054
[13]	Shuya Kanagawa, Ben T. Nohara. The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion. Conference Publications, 2013, 2013 (special) : 415-426. doi: 10.3934/proc.2013.2013.415
[14]	Myeongju Chae, Sunggeum Hong, Sanghyuk Lee. Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 909-928. doi: 10.3934/dcds.2011.29.909
[15]	Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254
[16]	César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535
[17]	Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[18]	Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030
[19]	Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[20]	Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063

2019 Impact Factor: 1.105

American Institute of Mathematical Sciences

Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors