February  2016, 36(2): 917-939. doi: 10.3934/dcds.2016.36.917

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

Received  March 2014 Revised  January 2015 Published  August 2015

We consider the following nonlinear fractional scalar field equation $$ (-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and $1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we show that this problem has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.
Citation: Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917
References:
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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.

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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.

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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.

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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.

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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.

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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.

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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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