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Nonlinear noncoercive Neumann problems
Nodal solutions for nonlinear Schrödinger equations with decaying potential
1. | School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China |
References:
[1] | |
[2] |
A. Amborosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potential vanishing at infinity, J. Eur. Math. Soc. (JEMS), 7 (2005), 117-144.
doi: 10.4171/JEMS/24. |
[3] |
A. Ambrositti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$, Birkhäuser Verlag, 2006. |
[4] |
A. Amborosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potential vanishing at infinity, J. Anal. Math., 8 (2006), 317-348. |
[5] |
A. Amborosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332. |
[6] |
C. O. Alves and S. Soares, On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations, J. Math. Anal. Appl., 296 (2004), 563-577.
doi: 10.1016/j.jmaa.2004.04.022. |
[7] |
S. Bae and J. Byeon, Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity, Commun. Pure Appl. Anal., 12 (2013), 831-850.
doi: 10.3934/cpaa.2013.12.831. |
[8] |
T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation, Proc. London Math. Soc.(3), 91 (2005), 129-152.
doi: 10.1112/S0024611504015187. |
[9] |
T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18.
doi: 10.1007/BF02787822. |
[10] |
T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\mathbbR^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[11] |
H. Berestycki and P. L. Lions, Nonlinear scalar fields equations I, Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[12] |
M.-F. Bidaut-Veron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49.
doi: 10.1007/BF02788105. |
[13] |
J. Byeon and Z.-Q. 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. |
[14] |
J. Byeon and Z.-Q. Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials, J. Eur. Math. Soc.(JEMS), 8 (2006), 217-228.
doi: 10.4171/JEMS/48. |
[15] |
M. Del Pino and P. Felmer, Local mountainpasses for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[16] |
Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math., 112 (2003), 109-135.
doi: 10.1007/s00229-003-0397-x. |
[17] |
A. Farina, On the classification of solutions of the Lane-Emden equation on undounded domains of $\R^N$, J. Math. Pures Appl., (9) 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[18] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. |
[19] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[20] |
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Grundlehren, 224, Berlin-Heidelgerg-New York-Tokyo: Springer, 1983.
doi: 10.1007/978-3-642-61798-0. |
[21] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u +u^p=0$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[22] |
P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case: Parts 1,2, Ann. Inst. H.Poincaré Anal. Non linéaire, 1 (1984), 109-145; Ann. Inst. H. Poincaré Anal. Non linéaire, 2 (1984), 223-283. |
[23] |
V. Moroz and J.V. Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials, Calc. Var. Partial Differential Equations, 37 (2010), 1-27.
doi: 10.1007/s00526-009-0249-y. |
[24] |
Y. G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253. |
[25] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[26] |
H. Yin and P. Zhang, Bound states of nonlinear Schrödinger equations with potential tending to zero at infinity, J. Differential Equations, 247 (2009), 618-647.
doi: 10.1016/j.jde.2009.03.002. |
[27] |
X. Wang and B. Zeng, On the concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.
doi: 10.1137/S0036141095290240. |
[28] |
M. Willem, Minimax Theorems, Birkhäiuser, Basel, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] | |
[2] |
A. Amborosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potential vanishing at infinity, J. Eur. Math. Soc. (JEMS), 7 (2005), 117-144.
doi: 10.4171/JEMS/24. |
[3] |
A. Ambrositti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $R^n$, Birkhäuser Verlag, 2006. |
[4] |
A. Amborosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potential vanishing at infinity, J. Anal. Math., 8 (2006), 317-348. |
[5] |
A. Amborosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations, 18 (2005), 1321-1332. |
[6] |
C. O. Alves and S. Soares, On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations, J. Math. Anal. Appl., 296 (2004), 563-577.
doi: 10.1016/j.jmaa.2004.04.022. |
[7] |
S. Bae and J. Byeon, Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity, Commun. Pure Appl. Anal., 12 (2013), 831-850.
doi: 10.3934/cpaa.2013.12.831. |
[8] |
T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a $p$-Laplacian equation, Proc. London Math. Soc.(3), 91 (2005), 129-152.
doi: 10.1112/S0024611504015187. |
[9] |
T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18.
doi: 10.1007/BF02787822. |
[10] |
T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $\mathbbR^N$, Arch. Ration. Mech. Anal., 124 (1993), 261-276.
doi: 10.1007/BF00953069. |
[11] |
H. Berestycki and P. L. Lions, Nonlinear scalar fields equations I, Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[12] |
M.-F. Bidaut-Veron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49.
doi: 10.1007/BF02788105. |
[13] |
J. Byeon and Z.-Q. 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. |
[14] |
J. Byeon and Z.-Q. Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials, J. Eur. Math. Soc.(JEMS), 8 (2006), 217-228.
doi: 10.4171/JEMS/48. |
[15] |
M. Del Pino and P. Felmer, Local mountainpasses for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[16] |
Y. H. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math., 112 (2003), 109-135.
doi: 10.1007/s00229-003-0397-x. |
[17] |
A. Farina, On the classification of solutions of the Lane-Emden equation on undounded domains of $\R^N$, J. Math. Pures Appl., (9) 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[18] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. |
[19] |
B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[20] |
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Grundlehren, 224, Berlin-Heidelgerg-New York-Tokyo: Springer, 1983.
doi: 10.1007/978-3-642-61798-0. |
[21] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u +u^p=0$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[22] |
P. L. Lions, The concentration-compactness principle in the calculus of variations: The locally compact case: Parts 1,2, Ann. Inst. H.Poincaré Anal. Non linéaire, 1 (1984), 109-145; Ann. Inst. H. Poincaré Anal. Non linéaire, 2 (1984), 223-283. |
[23] |
V. Moroz and J.V. Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials, Calc. Var. Partial Differential Equations, 37 (2010), 1-27.
doi: 10.1007/s00526-009-0249-y. |
[24] |
Y. G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253. |
[25] |
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[26] |
H. Yin and P. Zhang, Bound states of nonlinear Schrödinger equations with potential tending to zero at infinity, J. Differential Equations, 247 (2009), 618-647.
doi: 10.1016/j.jde.2009.03.002. |
[27] |
X. Wang and B. Zeng, On the concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655.
doi: 10.1137/S0036141095290240. |
[28] |
M. Willem, Minimax Theorems, Birkhäiuser, Basel, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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