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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

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

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 $M_1\cup \dots \cup M_K$.
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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