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Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells
1. | School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875 , China |
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] |
D. Applebaum, Lévy processes--from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. |
[3] |
T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384.
doi: 10.1007/s000330050003. |
[4] |
J.L. Bona and Y.A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430.
doi: 10.1016/S0021-7824(97)89957-6. |
[5] |
X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[6] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Part. Diff. Equa., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507.
doi: 10.1063/1.3701574. |
[8] |
S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh., 128 (1998), 1249-1260.
doi: 10.1017/S030821050002730X. |
[9] |
J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Diff. Equa., 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[10] |
A. de Bouard and J. C. Saut, Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal., 28 (1997), 1064-1085.
doi: 10.1137/S0036141096297662. |
[11] |
M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.
doi: 10.1006/jfan.1996.3085. |
[12] |
M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire., 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[13] |
S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216. |
[14] |
P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142A (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[15] |
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. |
[16] |
R.L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}^{N}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[17] |
R.L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure. Appl. Math., to appear. |
[18] |
T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.
doi: 10.4171/JEMS/456. |
[19] |
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), 109-145. |
[20] |
M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlinear Anal., 51 (2002), 1073-1085.
doi: 10.1016/S0362-546X(01)00880-X. |
[21] |
Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253. |
[22] |
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. |
[23] |
J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983.
doi: 10.3934/dcds.2011.31.975. |
[24] |
Z. Tang, On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Comput. Math. Appl., 54 (2007), 627-637.
doi: 10.1016/j.camwa.2006.12.031. |
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] |
D. Applebaum, Lévy processes--from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. |
[3] |
T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384.
doi: 10.1007/s000330050003. |
[4] |
J.L. Bona and Y.A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430.
doi: 10.1016/S0021-7824(97)89957-6. |
[5] |
X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[6] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Part. Diff. Equa., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507.
doi: 10.1063/1.3701574. |
[8] |
S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh., 128 (1998), 1249-1260.
doi: 10.1017/S030821050002730X. |
[9] |
J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Diff. Equa., 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[10] |
A. de Bouard and J. C. Saut, Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal., 28 (1997), 1064-1085.
doi: 10.1137/S0036141096297662. |
[11] |
M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.
doi: 10.1006/jfan.1996.3085. |
[12] |
M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire., 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[13] |
S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216. |
[14] |
P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142A (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[15] |
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. |
[16] |
R.L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}^{N}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[17] |
R.L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure. Appl. Math., to appear. |
[18] |
T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.
doi: 10.4171/JEMS/456. |
[19] |
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), 109-145. |
[20] |
M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlinear Anal., 51 (2002), 1073-1085.
doi: 10.1016/S0362-546X(01)00880-X. |
[21] |
Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253. |
[22] |
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. |
[23] |
J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983.
doi: 10.3934/dcds.2011.31.975. |
[24] |
Z. Tang, On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Comput. Math. Appl., 54 (2007), 627-637.
doi: 10.1016/j.camwa.2006.12.031. |
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