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Multi-peak positive solutions for a fractional nonlinear elliptic equation
1. | Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China, China |
References:
[1] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[2] |
D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré, 15 (1998), 73-111.
doi: 10.1016/S0294-1449(99)80021-3. |
[3] |
D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Royal Soc. Edinburgh, 129 (1999), 235-264.
doi: 10.1017/S030821050002134X. |
[4] |
D. Cao and S. Peng, Semi-classical bound states for Schröinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations, 34 (2009), 1566-1591.
doi: 10.1080/03605300903346721. |
[5] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some nonlocal semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[6] |
X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2956-2992.
doi: 10.1016/j.jde.2014.01.027. |
[7] |
G. Chen, Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations,, , ().
|
[8] |
G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.
doi: 10.3934/cpaa.2014.13.2359. |
[9] |
M. Cheng, Bound state for the fractional Schrödinger equations with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp.
doi: 10.1063/1.3701574. |
[10] |
W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.
doi: 10.1016/j.jfa.2014.02.029. |
[11] |
J. Dávila, M. Del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum,, , ().
|
[12] |
J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[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.
doi: 10.4418/2013.68.1.15. |
[14] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[15] |
M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $ (-\Delta)^s u + u = u^p$ in $\mathbbR^N$ when $s$ close to 1, Comm. Math. Phys., 329 (2014), 383-404.
doi: 10.1007/s00220-014-1919-y. |
[16] |
P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[17] |
R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, , ().
|
[18] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[19] |
N. Laskin, Fractional quantum mechanics, Phys. Rev. E., 62 (2000), p. 3135.
doi: 10.1103/PhysRevE.62.3135. |
[20] |
L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689.
doi: 10.1512/iumj.2009.58.3611. |
[21] |
W. Long, S. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schrödinger equations,, , ().
|
[22] |
E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2002), 213-227.
doi: 10.1112/S002461070000898X. |
[23] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^N$, J. Math. Phys., 54 (2013), 031501, 17 pp.
doi: 10.1063/1.4793990. |
[24] |
E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. S$\vec e$MA, 49 (2009), 33-44. |
[25] |
L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation,, , ().
|
show all references
References:
[1] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[2] |
D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré, 15 (1998), 73-111.
doi: 10.1016/S0294-1449(99)80021-3. |
[3] |
D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Royal Soc. Edinburgh, 129 (1999), 235-264.
doi: 10.1017/S030821050002134X. |
[4] |
D. Cao and S. Peng, Semi-classical bound states for Schröinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations, 34 (2009), 1566-1591.
doi: 10.1080/03605300903346721. |
[5] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some nonlocal semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[6] |
X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2956-2992.
doi: 10.1016/j.jde.2014.01.027. |
[7] |
G. Chen, Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations,, , ().
|
[8] |
G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.
doi: 10.3934/cpaa.2014.13.2359. |
[9] |
M. Cheng, Bound state for the fractional Schrödinger equations with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp.
doi: 10.1063/1.3701574. |
[10] |
W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.
doi: 10.1016/j.jfa.2014.02.029. |
[11] |
J. Dávila, M. Del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum,, , ().
|
[12] |
J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[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.
doi: 10.4418/2013.68.1.15. |
[14] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[15] |
M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $ (-\Delta)^s u + u = u^p$ in $\mathbbR^N$ when $s$ close to 1, Comm. Math. Phys., 329 (2014), 383-404.
doi: 10.1007/s00220-014-1919-y. |
[16] |
P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[17] |
R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, , ().
|
[18] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[19] |
N. Laskin, Fractional quantum mechanics, Phys. Rev. E., 62 (2000), p. 3135.
doi: 10.1103/PhysRevE.62.3135. |
[20] |
L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689.
doi: 10.1512/iumj.2009.58.3611. |
[21] |
W. Long, S. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schrödinger equations,, , ().
|
[22] |
E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2002), 213-227.
doi: 10.1112/S002461070000898X. |
[23] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbbR^N$, J. Math. Phys., 54 (2013), 031501, 17 pp.
doi: 10.1063/1.4793990. |
[24] |
E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. S$\vec e$MA, 49 (2009), 33-44. |
[25] |
L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation,, , ().
|
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