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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. |
[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. |
[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. |
[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. |
[28] |
X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928. |
[29] |
T. Kato, Perturbation Theory for Linear Operators, second ed., Grundlehren Math. Wiss., Band 132, Springer, Berlin-New York, (1976). |
[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. |
[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. |
[32] |
Y. Y. Li, On a singular perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[43] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[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. |
[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. |
[46] |
W. Strauss, Existence of solitary waves in higher demensions, Comm. Math. Phys., 55 (1977), 149-162. |
[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. |
[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. |
[49] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. |
[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. |
[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. |
[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. |
[28] |
X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928. |
[29] |
T. Kato, Perturbation Theory for Linear Operators, second ed., Grundlehren Math. Wiss., Band 132, Springer, Berlin-New York, (1976). |
[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. |
[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. |
[32] |
Y. Y. Li, On a singular perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[43] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[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. |
[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. |
[46] |
W. Strauss, Existence of solitary waves in higher demensions, Comm. Math. Phys., 55 (1977), 149-162. |
[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. |
[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. |
[49] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. |
[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. |
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