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Multiplicity results for a class of elliptic problems with nonlinear boundary condition
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Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part
1. | Department of Mathematics, Zhejiang Normal University, Jinhua, 321004 |
2. | Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080 |
References:
[1] |
N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443. |
[2] |
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[3] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. |
[4] |
L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas, 7 (2000), 210-230. |
[5] |
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. |
[6] |
F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. |
[7] |
Y. H. Ding and F. H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, 30 (2007), 231-249.
doi: 10.1007/s00526-007-0091-z. |
[8] |
Y. H. Ding and J. C. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Func. Anal., 251 (2007), 546-572. |
[9] |
A. Floer and A. Weinstein, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[10] |
I. Ianni, Solutions of the Schrödinger-Poisson system concentrating on spheres, part II: existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910. |
[11] |
I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlin. Studies, 8 (2008), 573-595. |
[12] |
E. Lieb and M. Loss, "Analysis," Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001. |
[13] |
P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. |
[14] |
A. G. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232. |
[15] |
G. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301. |
[16] |
Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_\alpha$, Comm. Part. Diff. Eqs., 13 (1988), 1499-1519. |
[17] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.
doi: 10.1007/BF02161413. |
[18] |
M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations, Ann. IHP, Analyse Nonlineaire, 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[19] |
M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.
doi: 10.1007/s002080200327. |
[20] |
P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Ang. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[21] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. |
[22] |
B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$, Annali di Matematica, 183 (2002), 73-83. |
[23] |
J. Wei and M. Winter, Strongly Interacting Bumps for the Schrodinger-Newton Equations, Journal of Mathematical Physics, 50 (2009), 012905.
doi: 10.1063/1.3060169. |
show all references
References:
[1] |
N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443. |
[2] |
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[3] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. |
[4] |
L. Bergé and A. Couairon, Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas, 7 (2000), 210-230. |
[5] |
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. |
[6] |
F. Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. |
[7] |
Y. H. Ding and F. H. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, 30 (2007), 231-249.
doi: 10.1007/s00526-007-0091-z. |
[8] |
Y. H. Ding and J. C. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Func. Anal., 251 (2007), 546-572. |
[9] |
A. Floer and A. Weinstein, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[10] |
I. Ianni, Solutions of the Schrödinger-Poisson system concentrating on spheres, part II: existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910. |
[11] |
I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlin. Studies, 8 (2008), 573-595. |
[12] |
E. Lieb and M. Loss, "Analysis," Gradute Studies in Mathematics, AMS, Providence, Rhode island, 2001. |
[13] |
P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. |
[14] |
A. G. Litvak, Self-focusing of powerful light beams by thermal effects, JETP Lett., 4 (1966), 230-232. |
[15] |
G. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301. |
[16] |
Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_\alpha$, Comm. Part. Diff. Eqs., 13 (1988), 1499-1519. |
[17] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.
doi: 10.1007/BF02161413. |
[18] |
M. del Pino and P. Felmer, Multipeak bound states of nonlinear Schrödinger equations, Ann. IHP, Analyse Nonlineaire, 15 (1998), 127-149.
doi: 10.1016/S0294-1449(97)89296-7. |
[19] |
M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.
doi: 10.1007/s002080200327. |
[20] |
P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Ang. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[21] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674. |
[22] |
B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in $R^N$, Annali di Matematica, 183 (2002), 73-83. |
[23] |
J. Wei and M. Winter, Strongly Interacting Bumps for the Schrodinger-Newton Equations, Journal of Mathematical Physics, 50 (2009), 012905.
doi: 10.1063/1.3060169. |
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