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On the classical solvability of near field reflector problems
Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations
1. | School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China |
2. | Department of Mathematics, Huazhong Normal University,Wuhan, 430079 |
References:
[1] |
L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D, 40 (1989), 360-392.
doi: 10.1016/0167-2789(89)90050-X. |
[2] |
W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potentials, Calc. Var. Partial Differential Equations, 51 (2014), 761-798.
doi: 10.1007/s00526-013-0694-5. |
[3] |
J. Byeon and Y. Oshita, Existence of multi-bump stading waves with a critical frequency for nonlinear schrödinger equations, Comm. Partial Differential Equations, 29 (2005), 1877-1904.
doi: 10.1081/PDE-200040205. |
[4] |
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. |
[5] |
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. |
[6] |
D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency, Math. Ann., 336 (2006), 925-948.
doi: 10.1007/s00208-006-0021-y. |
[7] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[8] |
G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations, 23 (2005), 139-168.
doi: 10.1007/s00526-004-0293-6. |
[9] |
G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equation with non-symmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.
doi: 10.1002/cpa.21410. |
[10] |
S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM. J. Math. Anal., 39 (2007/08), 1070-1111.
doi: 10.1137/050648389. |
[11] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[12] |
G. Chen and Y. Zhang, Concentration phenomenon for fractionsl nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.
doi: 10.3934/cpaa.2014.13.2359. |
[13] |
T. D'Aprile and A. Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1423-1451.
doi: 10.1016/j.anihpc.2009.01.002. |
[14] |
J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for fractional nonlinear Schrödinger equation, J. Differerntial Equations, 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[15] |
M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[16] |
G. Devillanova and S. Solimini, Min-max solutions to some scalar field equations, Adv. Nonlinear Stud., 12 (2012), 173-186. |
[17] |
W. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[18] |
P. Felmer, A. Quass and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[19] |
R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[20] |
R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, arXiv:1302.2652. |
[21] |
A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.
doi: 10.1002/cpa.20134. |
[22] |
X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928. |
[23] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[24] |
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[25] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[26] |
A. J. Majda, D. W. McLaughlin and E. G. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.
doi: 10.1007/BF02679124. |
[27] |
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. |
[28] |
E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.
doi: 10.1112/S002461070000898X. |
[29] |
E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc., 76 (1998), 427-452.
doi: 10.1112/S0024611598000148. |
[30] |
E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.
doi: 10.1007/BF01217684. |
[31] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109-145. |
[32] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 223-283. |
[33] |
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. |
[34] |
G. Palatucci and A. Pisante, Improved sobolev embeddings, profile decomposition and concentration-compactness for fractional sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[35] |
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.
doi: 10.1016/j.jfa.2009.01.020. |
[36] |
J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.
doi: 10.1007/s00526-010-0378-3. |
[37] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[38] |
J. Wei and S. Yan, Infinite many positive solutions for the nonlinear Schrödinger equation in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439.
doi: 10.1007/s00526-009-0270-1. |
[39] |
J. Wei and S. Yan, Infinite many positive solutions for the prescribed scalar curvature problem on $\mathbb{S}^{n}$, J. Funct. Anal., 258 (2010), 3048-3081.
doi: 10.1016/j.jfa.2009.12.008. |
[40] |
M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, 12 (1987), 1133-1173.
doi: 10.1080/03605308708820522. |
show all references
References:
[1] |
L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D, 40 (1989), 360-392.
doi: 10.1016/0167-2789(89)90050-X. |
[2] |
W. Ao and J. Wei, Infinitely many positive solutions for nonlinear equations with non-symmetric potentials, Calc. Var. Partial Differential Equations, 51 (2014), 761-798.
doi: 10.1007/s00526-013-0694-5. |
[3] |
J. Byeon and Y. Oshita, Existence of multi-bump stading waves with a critical frequency for nonlinear schrödinger equations, Comm. Partial Differential Equations, 29 (2005), 1877-1904.
doi: 10.1081/PDE-200040205. |
[4] |
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. |
[5] |
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. |
[6] |
D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency, Math. Ann., 336 (2006), 925-948.
doi: 10.1007/s00208-006-0021-y. |
[7] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[8] |
G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations, 23 (2005), 139-168.
doi: 10.1007/s00526-004-0293-6. |
[9] |
G. Cerami, D. Passaseo and S. Solimini, Infinitely many positive solutions to some scalar field equation with non-symmetric coefficients, Comm. Pure Appl. Math., 66 (2013), 372-413.
doi: 10.1002/cpa.21410. |
[10] |
S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM. J. Math. Anal., 39 (2007/08), 1070-1111.
doi: 10.1137/050648389. |
[11] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[12] |
G. Chen and Y. Zhang, Concentration phenomenon for fractionsl nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.
doi: 10.3934/cpaa.2014.13.2359. |
[13] |
T. D'Aprile and A. Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1423-1451.
doi: 10.1016/j.anihpc.2009.01.002. |
[14] |
J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for fractional nonlinear Schrödinger equation, J. Differerntial Equations, 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[15] |
M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[16] |
G. Devillanova and S. Solimini, Min-max solutions to some scalar field equations, Adv. Nonlinear Stud., 12 (2012), 173-186. |
[17] |
W. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91 (1986), 283-308.
doi: 10.1007/BF00282336. |
[18] |
P. Felmer, A. Quass and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[19] |
R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[20] |
R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, arXiv:1302.2652. |
[21] |
A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.
doi: 10.1002/cpa.20134. |
[22] |
X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5 (2000), 899-928. |
[23] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[24] |
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[25] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[26] |
A. J. Majda, D. W. McLaughlin and E. G. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.
doi: 10.1007/BF02679124. |
[27] |
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. |
[28] |
E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. Lond. Math. Soc., 62 (2000), 213-227.
doi: 10.1112/S002461070000898X. |
[29] |
E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc., 76 (1998), 427-452.
doi: 10.1112/S0024611598000148. |
[30] |
E. H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174.
doi: 10.1007/BF01217684. |
[31] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 109-145. |
[32] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II., Ann. Inst. H. Poincaré Anal. Non Lineaire, 1 (1984), 223-283. |
[33] |
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. |
[34] |
G. Palatucci and A. Pisante, Improved sobolev embeddings, profile decomposition and concentration-compactness for fractional sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[35] |
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.
doi: 10.1016/j.jfa.2009.01.020. |
[36] |
J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.
doi: 10.1007/s00526-010-0378-3. |
[37] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[38] |
J. Wei and S. Yan, Infinite many positive solutions for the nonlinear Schrödinger equation in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 37 (2010), 423-439.
doi: 10.1007/s00526-009-0270-1. |
[39] |
J. Wei and S. Yan, Infinite many positive solutions for the prescribed scalar curvature problem on $\mathbb{S}^{n}$, J. Funct. Anal., 258 (2010), 3048-3081.
doi: 10.1016/j.jfa.2009.12.008. |
[40] |
M. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, 12 (1987), 1133-1173.
doi: 10.1080/03605308708820522. |
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