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On a fractional harmonic replacement
Multi-bump solutions for Schrödinger equation involving critical growth and potential wells
1. | Department of Mathematics, Tsinghua University, Beijing, 100084 |
2. | School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875 |
References:
[1] |
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch.Ration.Mech.Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[2] |
A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch.Ration.Mech.Anal., 159 (2001), 253-271.
doi: 10.1007/s002050100152. |
[3] |
T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential, Discrete Contin. Dyn. Syst, 33 (2013), 7-26. |
[4] |
T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math.Phys., 51 (2000), 366-384.
doi: 10.1007/PL00001511. |
[5] |
V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{\frac{N+2}{N-2}}$ in $\mathbbR^N,$ J. Funct. Anal., 88 (1990), 90-117.
doi: 10.1016/0022-1236(90)90120-A. |
[6] |
J. Byeon and Z. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316.
doi: 10.1007/s00205-002-0225-6. |
[7] |
J. Byeon and Z. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II, Calc.Var.P. D. E., 18 (2003), 207-219.
doi: 10.1007/s00526-002-0191-8. |
[8] |
A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann.Inst.Henri Poincaré, 2 (1985), 463-470. |
[9] |
J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugaliae Mathematica., 57 (2000), 273-284. |
[10] |
J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent, Z. Angew. Math. Phys., 49 (1998), 276-293.
doi: 10.1007/PL00001485. |
[11] |
S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff.Equat., 160 (2000), 118-138.
doi: 10.1006/jdeq.1999.3662. |
[12] |
S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Royal Soc. Edinburgh., 128 (1998), 1249-1260.
doi: 10.1017/S030821050002730X. |
[13] |
M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, Ann.Inst.Henri Poincaré, 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[14] |
M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.
doi: 10.1006/jfan.1996.3085. |
[15] |
Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal., 251 (2007), 546-572.
doi: 10.1016/j.jfa.2007.07.005. |
[16] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[17] |
M. Grossi, A nondegeneracy result for a nonlinear elliptic equation, Nonlinear Differ. Equ. Appl., 12 (2005), 227-241.
doi: 10.1007/s00030-005-0010-y. |
[18] |
W. M. Ni, X. Pan and I. Takagi, Singular behavior of least energy solutions of a smilinear Neumannn problem involving critcial Sobolev exponents, Duke Math.J., 67 (1992), 1-20.
doi: 10.1215/S0012-7094-92-06701-9. |
[19] |
Y.-G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.
doi: 10.1007/BF02161413. |
[20] |
Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $(V)_a$, Comm. Part. Diff. Equat., 13 (1988), 1499-1519.
doi: 10.1080/03605308808820585. |
[21] |
O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52.
doi: 10.1016/0022-1236(90)90002-3. |
[22] |
Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Commun. Pure Appl. Anal., 13 (2014), 237-248.
doi: 10.3934/cpaa.2014.13.237. |
[23] |
J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, Preprint. |
show all references
References:
[1] |
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch.Ration.Mech.Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[2] |
A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch.Ration.Mech.Anal., 159 (2001), 253-271.
doi: 10.1007/s002050100152. |
[3] |
T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential, Discrete Contin. Dyn. Syst, 33 (2013), 7-26. |
[4] |
T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math.Phys., 51 (2000), 366-384.
doi: 10.1007/PL00001511. |
[5] |
V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{\frac{N+2}{N-2}}$ in $\mathbbR^N,$ J. Funct. Anal., 88 (1990), 90-117.
doi: 10.1016/0022-1236(90)90120-A. |
[6] |
J. Byeon and Z. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316.
doi: 10.1007/s00205-002-0225-6. |
[7] |
J. Byeon and Z. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II, Calc.Var.P. D. E., 18 (2003), 207-219.
doi: 10.1007/s00526-002-0191-8. |
[8] |
A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann.Inst.Henri Poincaré, 2 (1985), 463-470. |
[9] |
J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugaliae Mathematica., 57 (2000), 273-284. |
[10] |
J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent, Z. Angew. Math. Phys., 49 (1998), 276-293.
doi: 10.1007/PL00001485. |
[11] |
S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff.Equat., 160 (2000), 118-138.
doi: 10.1006/jdeq.1999.3662. |
[12] |
S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Royal Soc. Edinburgh., 128 (1998), 1249-1260.
doi: 10.1017/S030821050002730X. |
[13] |
M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, Ann.Inst.Henri Poincaré, 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[14] |
M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.
doi: 10.1006/jfan.1996.3085. |
[15] |
Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal., 251 (2007), 546-572.
doi: 10.1016/j.jfa.2007.07.005. |
[16] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[17] |
M. Grossi, A nondegeneracy result for a nonlinear elliptic equation, Nonlinear Differ. Equ. Appl., 12 (2005), 227-241.
doi: 10.1007/s00030-005-0010-y. |
[18] |
W. M. Ni, X. Pan and I. Takagi, Singular behavior of least energy solutions of a smilinear Neumannn problem involving critcial Sobolev exponents, Duke Math.J., 67 (1992), 1-20.
doi: 10.1215/S0012-7094-92-06701-9. |
[19] |
Y.-G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.
doi: 10.1007/BF02161413. |
[20] |
Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $(V)_a$, Comm. Part. Diff. Equat., 13 (1988), 1499-1519.
doi: 10.1080/03605308808820585. |
[21] |
O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1990), 1-52.
doi: 10.1016/0022-1236(90)90002-3. |
[22] |
Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Commun. Pure Appl. Anal., 13 (2014), 237-248.
doi: 10.3934/cpaa.2014.13.237. |
[23] |
J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, Preprint. |
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