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The existence and blow-up criterion of liquid crystals system in critical Besov space
Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials
1. | 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.
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.
doi: 10.1007/s002050100152. |
[3] |
T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential,, Discrete Continuous Dynam. Systems, 33 (2013), 7.
|
[4] |
T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation,, Z. angew. Math. Phys., 51 (2000), 266.
doi: 10.1007/s000330050003. |
[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 $R^N$,, J. Funct. Anal., 88 (1990), 90.
doi: 10.1016/0022-1236(90)90120-A. |
[6] |
J. Byeon and Z. Q. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II,, Calc.Var., 18 (2003), 207.
doi: 10.1007/s00526-002-0191-8. |
[7] |
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.
|
[8] |
J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent,, Portugaliae Mathematica, 57 (2000), 273.
|
[9] |
J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent,, Z. angew. Math. Phys., 49 (1998), 276.
doi: 10.1007/PL00001485. |
[10] |
S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, J. Diff. Equat., 160 (2000), 118.
doi: 10.1006/jdeq.1999.3662. |
[11] |
S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations,, Proc. Royal Soc. Edinburgh, 128 (1998), 1249.
doi: 10.1017/S030821050002730X. |
[12] |
M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations,, Ann. Inst. Henri Poincar$\acutee$, 15 (1998), 127.
doi: 10.1016/S0294-1449(97)89296-7. |
[13] |
M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245.
doi: 10.1006/jfan.1996.3085. |
[14] |
Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials,, J. Funct. Anal., 251 (2007), 546.
doi: 10.1016/j.jfa.2007.07.005. |
[15] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397.
doi: 10.1016/0022-1236(86)90096-0. |
[16] |
C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, Comm. Part. Diff. Equat., 21 (1996), 787.
doi: 10.1080/03605309608821208. |
[17] |
Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, Comm. Math. Phys., 131 (1990), 223.
|
[18] |
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.
doi: 10.1080/03605308808820585. |
[19] |
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals,, Milan J.Math., 73 (2005), 563.
doi: 10.1007/s00032-005-0047-8. |
[20] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, vol IV,", Academic Press, (1978).
|
[21] |
A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems,, J. Funct. Anal., 257 (2009), 3802.
doi: 10.1016/j.jfa.2009.09.013. |
[22] |
J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth,, preprint, (). Google Scholar |
[23] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I,, Ann.Inst.H.Poincaré Anal. Non Linéaire, 1 (1984), 109.
|
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.
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.
doi: 10.1007/s002050100152. |
[3] |
T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential,, Discrete Continuous Dynam. Systems, 33 (2013), 7.
|
[4] |
T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation,, Z. angew. Math. Phys., 51 (2000), 266.
doi: 10.1007/s000330050003. |
[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 $R^N$,, J. Funct. Anal., 88 (1990), 90.
doi: 10.1016/0022-1236(90)90120-A. |
[6] |
J. Byeon and Z. Q. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II,, Calc.Var., 18 (2003), 207.
doi: 10.1007/s00526-002-0191-8. |
[7] |
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.
|
[8] |
J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent,, Portugaliae Mathematica, 57 (2000), 273.
|
[9] |
J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent,, Z. angew. Math. Phys., 49 (1998), 276.
doi: 10.1007/PL00001485. |
[10] |
S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, J. Diff. Equat., 160 (2000), 118.
doi: 10.1006/jdeq.1999.3662. |
[11] |
S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations,, Proc. Royal Soc. Edinburgh, 128 (1998), 1249.
doi: 10.1017/S030821050002730X. |
[12] |
M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations,, Ann. Inst. Henri Poincar$\acutee$, 15 (1998), 127.
doi: 10.1016/S0294-1449(97)89296-7. |
[13] |
M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, J. Funct. Anal., 149 (1997), 245.
doi: 10.1006/jfan.1996.3085. |
[14] |
Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials,, J. Funct. Anal., 251 (2007), 546.
doi: 10.1016/j.jfa.2007.07.005. |
[15] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397.
doi: 10.1016/0022-1236(86)90096-0. |
[16] |
C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, Comm. Part. Diff. Equat., 21 (1996), 787.
doi: 10.1080/03605309608821208. |
[17] |
Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, Comm. Math. Phys., 131 (1990), 223.
|
[18] |
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.
doi: 10.1080/03605308808820585. |
[19] |
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals,, Milan J.Math., 73 (2005), 563.
doi: 10.1007/s00032-005-0047-8. |
[20] |
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, vol IV,", Academic Press, (1978).
|
[21] |
A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems,, J. Funct. Anal., 257 (2009), 3802.
doi: 10.1016/j.jfa.2009.09.013. |
[22] |
J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth,, preprint, (). Google Scholar |
[23] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I,, Ann.Inst.H.Poincaré Anal. Non Linéaire, 1 (1984), 109.
|
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