American Institute of Mathematical Sciences

Advanced
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

Wei Long 1, , Shuangjie Peng 1, and  Jing Yang 2,

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.
Keywords: Fractional Laplacian, reduction method., nonlinear scalar field equation.
Mathematics Subject Classification: 35J20, 35J6.
Citation: Wei Long, Shuangjie Peng, Jing Yang. Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 917-939. doi: 10.3934/dcds.2016.36.917
References:
[1]

L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves,, Phys. D, 40 (1989), 360. doi: 10.1016/0167-2789(89)90050-X. Google Scholar

[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. doi: 10.1007/s00526-013-0694-5. Google Scholar

[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. doi: 10.1081/PDE-200040205. Google Scholar

[4]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[5]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[6]

D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency,, Math. Ann., 336 (2006), 925. doi: 10.1007/s00208-006-0021-y. Google Scholar

[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. doi: 10.1080/03605302.2011.562954. Google Scholar

[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. doi: 10.1007/s00526-004-0293-6. Google Scholar

[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. doi: 10.1002/cpa.21410. Google Scholar

[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 (): 1070. doi: 10.1137/050648389. Google Scholar

[11]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[12]

G. Chen and Y. Zhang, Concentration phenomenon for fractionsl nonlinear Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 2359. doi: 10.3934/cpaa.2014.13.2359. Google Scholar

[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. doi: 10.1016/j.anihpc.2009.01.002. Google Scholar

[14]

J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for fractional nonlinear Schrödinger equation,, J. Differerntial Equations, 256 (2014), 858. doi: 10.1016/j.jde.2013.10.006. Google Scholar

[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. doi: 10.1007/BF01189950. Google Scholar

[16]

G. Devillanova and S. Solimini, Min-max solutions to some scalar field equations,, Adv. Nonlinear Stud., 12 (2012), 173. Google Scholar

[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. doi: 10.1007/BF00282336. Google Scholar

[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. doi: 10.1017/S0308210511000746. Google Scholar

[19]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbbR$,, Acta Math., 210 (2013), 261. doi: 10.1007/s11511-013-0095-9. Google Scholar

[20]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, , (). Google Scholar

[21]

A. Elgart and B. Schlein, Mean field dynamics of boson stars,, Comm. Pure Appl. Math., 60 (2007), 500. doi: 10.1002/cpa.20134. Google Scholar

[22]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, Adv. Differential Equations, 5 (2000), 899. Google Scholar

[23]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbbR^n$,, Arch. Ration. Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502. Google Scholar

[24]

N. Laskin, Fractional quantum mechanics and Lévy path integrals,, Phys. Lett. A, 268 (2000), 298. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar

[25]

N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar

[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. doi: 10.1007/BF02679124. Google Scholar

[27]

M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation,, Nonlinear Anal., 51 (2002), 1073. doi: 10.1016/S0362-546X(01)00880-X. Google Scholar

[28]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem,, J. Lond. Math. Soc., 62 (2000), 213. doi: 10.1112/S002461070000898X. Google Scholar

[29]

E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems,, Proc. London Math. Soc., 76 (1998), 427. doi: 10.1112/S0024611598000148. Google Scholar

[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. doi: 10.1007/BF01217684. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[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. doi: 10.1007/s00526-013-0656-y. Google Scholar

[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. doi: 10.1016/j.jfa.2009.01.020. Google Scholar

[36]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21. doi: 10.1007/s00526-010-0378-3. Google Scholar

[37]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Comm. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar

[38]

J. Wei and S. Yan, Infinite many positive solutions for the nonlinear Schrödinger equation in $\mathbbR^n$,, Calc. Var. Partial Differential Equations, 37 (2010), 423. doi: 10.1007/s00526-009-0270-1. Google Scholar

[39]

J. Wei and S. Yan, Infinite many positive solutions for the prescribed scalar curvature problem on $\mathbbS^N$,, J. Funct. Anal., 258 (2010), 3048. doi: 10.1016/j.jfa.2009.12.008. Google Scholar

[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. doi: 10.1080/03605308708820522. Google Scholar

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. doi: 10.1016/0167-2789(89)90050-X. Google Scholar

[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. doi: 10.1007/s00526-013-0694-5. Google Scholar

[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. doi: 10.1081/PDE-200040205. Google Scholar

[4]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[5]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[6]

D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency,, Math. Ann., 336 (2006), 925. doi: 10.1007/s00208-006-0021-y. Google Scholar

[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. doi: 10.1080/03605302.2011.562954. Google Scholar

[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. doi: 10.1007/s00526-004-0293-6. Google Scholar

[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. doi: 10.1002/cpa.21410. Google Scholar

[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 (): 1070. doi: 10.1137/050648389. Google Scholar

[11]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[12]

G. Chen and Y. Zhang, Concentration phenomenon for fractionsl nonlinear Schrödinger equations,, Commun. Pure Appl. Anal., 13 (2014), 2359. doi: 10.3934/cpaa.2014.13.2359. Google Scholar

[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. doi: 10.1016/j.anihpc.2009.01.002. Google Scholar

[14]

J. Dávila, M. Del Pino and J. Wei, Concentrating standing waves for fractional nonlinear Schrödinger equation,, J. Differerntial Equations, 256 (2014), 858. doi: 10.1016/j.jde.2013.10.006. Google Scholar

[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. doi: 10.1007/BF01189950. Google Scholar

[16]

G. Devillanova and S. Solimini, Min-max solutions to some scalar field equations,, Adv. Nonlinear Stud., 12 (2012), 173. Google Scholar

[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. doi: 10.1007/BF00282336. Google Scholar

[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. doi: 10.1017/S0308210511000746. Google Scholar

[19]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbbR$,, Acta Math., 210 (2013), 261. doi: 10.1007/s11511-013-0095-9. Google Scholar

[20]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, , (). Google Scholar

[21]

A. Elgart and B. Schlein, Mean field dynamics of boson stars,, Comm. Pure Appl. Math., 60 (2007), 500. doi: 10.1002/cpa.20134. Google Scholar

[22]

X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, Adv. Differential Equations, 5 (2000), 899. Google Scholar

[23]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbbR^n$,, Arch. Ration. Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502. Google Scholar

[24]

N. Laskin, Fractional quantum mechanics and Lévy path integrals,, Phys. Lett. A, 268 (2000), 298. doi: 10.1016/S0375-9601(00)00201-2. Google Scholar

[25]

N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar

[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. doi: 10.1007/BF02679124. Google Scholar

[27]

M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation,, Nonlinear Anal., 51 (2002), 1073. doi: 10.1016/S0362-546X(01)00880-X. Google Scholar

[28]

E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem,, J. Lond. Math. Soc., 62 (2000), 213. doi: 10.1112/S002461070000898X. Google Scholar

[29]

E. S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems,, Proc. London Math. Soc., 76 (1998), 427. doi: 10.1112/S0024611598000148. Google Scholar

[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. doi: 10.1007/BF01217684. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar

[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. doi: 10.1007/s00526-013-0656-y. Google Scholar

[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. doi: 10.1016/j.jfa.2009.01.020. Google Scholar

[36]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21. doi: 10.1007/s00526-010-0378-3. Google Scholar

[37]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, Comm. Math. Phys., 153 (1993), 229. doi: 10.1007/BF02096642. Google Scholar

[38]

J. Wei and S. Yan, Infinite many positive solutions for the nonlinear Schrödinger equation in $\mathbbR^n$,, Calc. Var. Partial Differential Equations, 37 (2010), 423. doi: 10.1007/s00526-009-0270-1. Google Scholar

[39]

J. Wei and S. Yan, Infinite many positive solutions for the prescribed scalar curvature problem on $\mathbbS^N$,, J. Funct. Anal., 258 (2010), 3048. doi: 10.1016/j.jfa.2009.12.008. Google Scholar

[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. doi: 10.1080/03605308708820522. Google Scholar

[1]

Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[2]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201
[3]

Francesca Faraci, Alexandru Kristály. One-dimensional scalar field equations involving an oscillatory nonlinear term. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 107-120. doi: 10.3934/dcds.2007.18.107
[4]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[5]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154
[6]

Claudianor O. Alves, Giovany M. Figueiredo, Gaetano Siciliano. Ground state solutions for fractional scalar field equations under a general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2199-2215. doi: 10.3934/cpaa.2019099
[7]

Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015
[8]

Eric Chung, Yalchin Efendiev, Ke Shi, Shuai Ye. A multiscale model reduction method for nonlinear monotone elliptic equations in heterogeneous media. Networks & Heterogeneous Media, 2017, 12 (4) : 619-642. doi: 10.3934/nhm.2017025
[9]

Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011
[10]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729
[11]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[12]

Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141
[13]

Chenchen Mou. Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2335-2362. doi: 10.3934/cpaa.2015.14.2335
[14]

Tadeusz Kulczycki, Robert Stańczy. Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2581-2591. doi: 10.3934/dcdsb.2014.19.2581
[15]

Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857
[16]

Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117
[17]

Jacques Giacomoni, Tuhina Mukherjee, Konijeti Sreenadh. Existence and stabilization results for a singular parabolic equation involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 311-337. doi: 10.3934/dcdss.2019022
[18]

Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108
[19]

Valery Imaikin, Alexander Komech, Herbert Spohn. Scattering theory for a particle coupled to a scalar field. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 387-396. doi: 10.3934/dcds.2004.10.387
[20]

Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences