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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, Comm. Math. Phys., 235 (2003), 427-466. 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, J. d'Anayse Math., 98 (2006), 317-348. doi: 10.1007/BF02790279.  Google Scholar

[3]

A. Ambrosetti, A. Malchiodi and A. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Rational Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.  Google Scholar

[4]

A. Ambrosetti and D. Ruiz, Radial solutions concentrating on sphere of nonlinear Schrödinger equations with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 889-907. doi: 10.1017/S0308210500004789.  Google Scholar

[5]

M. Badiale. and T. D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation, Nonlinear Anal., 49 (2002), 947-985. 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, J. Diff. Equat., 184 (2002), 109-138. doi: 10.1006/jdeq.2001.4138.  Google Scholar

[7]

J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential, Nonlinear Anal., 50 (2002), 1135-1151. 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, Comm. Partial Differential Equations, 29 (2004), 1877-1904. doi: 10.1081/PDE-200040205.  Google Scholar

[9]

J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 138A (2008), 975-987. doi: 10.1017/S0308210507000236.  Google Scholar

[10]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 165 (2002), 295-316. 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, Calc. Var. Partial Differential Equations, 18 (2003), 207-219. 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, Math. Ann, 336 (2006), 925-948. 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, J. Differential Equations, 74 (1998), 120-156. 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 $R^N$, Discrete Contin. Dynam. Systems, 6 (2000), 39-50.  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, Abstr. Apply. Anal., 3 (1998), 293-318. doi: 10.1155/S1085337598000578.  Google Scholar

[17]

M. del Pino and P. L. Felmer, Local mountaion passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. doi: 10.1007/BF01189950.  Google Scholar

[18]

M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.  Google Scholar

[19]

M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, 15 (1998), 127-149. 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, Math. Ann., 324 (2002), 1-32. 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, Nonlinear Anal. TMA, 34 (1998), 979-989. 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, Comm. Pur. App. Math., LX (2007), 113-146. 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, J. Funct. Anal., 69 (1986), 397-408. 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, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[25]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Berlin, Heidelberg, New York and Tokyo: Springer. Grundlehren 224. (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, Comm. Partial Differential Equations, 21 (1996), 787-820. 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, Comm. Partial Differential Equations, 24 (1999), 563-598. doi: 10.1080/03605309908821434.  Google Scholar

[28]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928.  Google Scholar

[29]

T. Kato, Perturbation Theory for Linear Operators, second ed., Grundlehren Math. Wiss., Band 132, Springer, Berlin-New York, (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, Proc. Roy. Soc. Edinburgh Sect. A, 139A (2009), 833-852. doi: 10.1017/S0308210508000309.  Google Scholar

[31]

C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations, 16 (1996), 585-615. doi: 10.1080/03605309108820770.  Google Scholar

[32]

Y. Y. Li, On a singular perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980.  Google Scholar

[33]

F. Mahmoudi and A. Malchiodi, Concentration on minimal submainifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525. 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, Geom. Funct. Anal., 16 (2006), 924-958. 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, J. Differential Equations, 258 (2015), 243-280. 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, G.A.F.A., 15-16 (2005), 1162-1222. doi: 10.1007/s00039-005-0542-7.  Google Scholar

[37]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 15 (2002), 1507-1568. doi: 10.1002/cpa.10049.  Google Scholar

[38]

A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143. 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, Canad. J. Math., 1 (1949), 242-256.  Google Scholar

[40]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equations, 13 (1988), 1499-1519. 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 $(V)_a$, Comm. Partial Differential Equations, 14 (1989), 833-834. doi: 10.1080/03605308908820631.  Google Scholar

[42]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multile well potential, Comm. Math. Phys., 131 (1990), 223-253.  Google Scholar

[43]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[44]

Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency, Calc. Var. Partial Differential Equations, 29 (2007), 365-395. doi: 10.1007/s00526-006-0070-9.  Google Scholar

[45]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $R^N$, Ann. Mat. Pura Appl., 184 (2002), 73-83. doi: 10.1007/s102310200029.  Google Scholar

[46]

W. Strauss, Existence of solitary waves in higher demensions, Comm. Math. Phys., 55 (1977), 149-162.  Google Scholar

[47]

J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238 (2007), 201-219. 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, Comm. Partial Differential Equations, 36 (2011), 2117-2161. doi: 10.1080/03605302.2011.580033.  Google Scholar

[49]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  Google Scholar

[50]

Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations, J. Differential Equations, 159 (1999), 102-137. 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, Comm. Math. Phys., 235 (2003), 427-466. 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, J. d'Anayse Math., 98 (2006), 317-348. doi: 10.1007/BF02790279.  Google Scholar

[3]

A. Ambrosetti, A. Malchiodi and A. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Rational Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.  Google Scholar

[4]

A. Ambrosetti and D. Ruiz, Radial solutions concentrating on sphere of nonlinear Schrödinger equations with vanishing potentials, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 889-907. doi: 10.1017/S0308210500004789.  Google Scholar

[5]

M. Badiale. and T. D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation, Nonlinear Anal., 49 (2002), 947-985. 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, J. Diff. Equat., 184 (2002), 109-138. doi: 10.1006/jdeq.2001.4138.  Google Scholar

[7]

J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential, Nonlinear Anal., 50 (2002), 1135-1151. 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, Comm. Partial Differential Equations, 29 (2004), 1877-1904. doi: 10.1081/PDE-200040205.  Google Scholar

[9]

J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 138A (2008), 975-987. doi: 10.1017/S0308210507000236.  Google Scholar

[10]

J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 165 (2002), 295-316. 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, Calc. Var. Partial Differential Equations, 18 (2003), 207-219. 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, Math. Ann, 336 (2006), 925-948. 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, J. Differential Equations, 74 (1998), 120-156. 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 $R^N$, Discrete Contin. Dynam. Systems, 6 (2000), 39-50.  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, Abstr. Apply. Anal., 3 (1998), 293-318. doi: 10.1155/S1085337598000578.  Google Scholar

[17]

M. del Pino and P. L. Felmer, Local mountaion passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137. doi: 10.1007/BF01189950.  Google Scholar

[18]

M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.  Google Scholar

[19]

M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, 15 (1998), 127-149. 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, Math. Ann., 324 (2002), 1-32. 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, Nonlinear Anal. TMA, 34 (1998), 979-989. 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, Comm. Pur. App. Math., LX (2007), 113-146. 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, J. Funct. Anal., 69 (1986), 397-408. 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, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[25]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Berlin, Heidelberg, New York and Tokyo: Springer. Grundlehren 224. (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, Comm. Partial Differential Equations, 21 (1996), 787-820. 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, Comm. Partial Differential Equations, 24 (1999), 563-598. doi: 10.1080/03605309908821434.  Google Scholar

[28]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928.  Google Scholar

[29]

T. Kato, Perturbation Theory for Linear Operators, second ed., Grundlehren Math. Wiss., Band 132, Springer, Berlin-New York, (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, Proc. Roy. Soc. Edinburgh Sect. A, 139A (2009), 833-852. doi: 10.1017/S0308210508000309.  Google Scholar

[31]

C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Comm. Partial Differential Equations, 16 (1996), 585-615. doi: 10.1080/03605309108820770.  Google Scholar

[32]

Y. Y. Li, On a singular perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980.  Google Scholar

[33]

F. Mahmoudi and A. Malchiodi, Concentration on minimal submainifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525. 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, Geom. Funct. Anal., 16 (2006), 924-958. 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, J. Differential Equations, 258 (2015), 243-280. 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, G.A.F.A., 15-16 (2005), 1162-1222. doi: 10.1007/s00039-005-0542-7.  Google Scholar

[37]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 15 (2002), 1507-1568. doi: 10.1002/cpa.10049.  Google Scholar

[38]

A. Malchiodi and M. Montenegro, Multidimensional boundary-layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143. 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, Canad. J. Math., 1 (1949), 242-256.  Google Scholar

[40]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differential Equations, 13 (1988), 1499-1519. 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 $(V)_a$, Comm. Partial Differential Equations, 14 (1989), 833-834. doi: 10.1080/03605308908820631.  Google Scholar

[42]

Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multile well potential, Comm. Math. Phys., 131 (1990), 223-253.  Google Scholar

[43]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar

[44]

Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency, Calc. Var. Partial Differential Equations, 29 (2007), 365-395. doi: 10.1007/s00526-006-0070-9.  Google Scholar

[45]

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equation in $R^N$, Ann. Mat. Pura Appl., 184 (2002), 73-83. doi: 10.1007/s102310200029.  Google Scholar

[46]

W. Strauss, Existence of solitary waves in higher demensions, Comm. Math. Phys., 55 (1977), 149-162.  Google Scholar

[47]

J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations, 238 (2007), 201-219. 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, Comm. Partial Differential Equations, 36 (2011), 2117-2161. doi: 10.1080/03605302.2011.580033.  Google Scholar

[49]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  Google Scholar

[50]

Z.-Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations, J. Differential Equations, 159 (1999), 102-137. doi: 10.1006/jdeq.1999.3650.  Google Scholar

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