March  2017, 16(2): 493-512. doi: 10.3934/cpaa.2017025

Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

2. 

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

3. 

Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China

*Corresponding author

Received  April 2016 Revised  November 2016 Published  January 2017

In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential V. Moreover, the monotonicity of f(s)=s and the so-called Ambrosetti-Rabinowitz condition are not required.

Citation: Minbo Yang, Jianjun Zhang, Yimin Zhang. Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (2) : 493-512. doi: 10.3934/cpaa.2017025
References:
[1]

C. O. Alves, J. Marcos do O and M. A. S. Souto, Local mountain-pass for a class of elliptic problems in ${{\mathbb{R}}^{N}}$ involving critical growth, Nonlinear Anal., 46 (2001), 495-510. doi: 10.1016/S0362-546X(00)00125-5.

[2]

C. O. Alves and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh, 146A (2016), 23-58. doi: 10.1017/S0308210515000311.

[3]

C. O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations, 257 (2014), 4133-4164. doi: 10.1016/j.jde.2014.08.004.

[4]

C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation inR2, J. Differential Equations, 261 (2016), 1933-1972. doi: 10.1016/j.jde.2016.04.021.

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346. doi: 10.1007/BF00250555.

[6]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. doi: 10.1007/s00205-006-0019-3.

[7]

J. Byeon and L. Jeanjean, Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity, Discrete Contin. Dynam. Syst., 19 (2007), 255-269. doi: 10.3934/dcds.2007.19.255.

[8]

J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc., 15 (2013), 1859-1899. doi: 10.4171/JEMS/407.

[9]

J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc., 362 (2010), 1981-2001. doi: 10.1090/S0002-9947-09-04746-1.

[10]

J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Memoirs of the American Mathematical Society, 229 (2014).

[11]

J. Byeon and Z.-Q. Wang, Standing waves with critical frequency for nonlinear Schrödinger equations Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8.

[12]

C. Bonanno, P. d'Avenia, M. Ghimenti and M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math.Anal.Appl., 417 (2014), 180-199. doi: 10.1016/j.jmaa.2014.02.063.

[13]

W. X. Chen, C. M. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure. Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[14]

S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh, 140A (2010), 973-1009. doi: 10.1017/S0308210509000584.

[15]

M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137. doi: 10.1007/BF01189950.

[16]

M. del Pino and P. Felmer, Multi-peak bound states of nonlinear Schrödinger equations, Annales Inst. H. Poincaré Analyse Non Linéaire, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7.

[17]

M. del Pino and P. L. Felmer, Spike-layered solutions of singularlyly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999) 883-898. doi: 10.1512/iumj.1999.48.1596.

[18]

P. D'Avenia, A. Pomponio and D. Ruiz, Semi-classical states for the Nonlinear Schrödinger Equation on saddle points of the potential via variational methods, J. Funct. Anal., 262 (2012), 4600-4633. doi: 10.1016/j.jfa.2012.03.009.

[19]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.

[21]

E. P. Gross, Physics of Many-Particle Systems, Vol. 1, Gordon Breach, New York, 1996.

[22]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equationl, Stud. Appl. Math., 57 (1977), 93-105.

[23]

E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, 2001.

[24]

E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.

[25]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal. TMA, 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4.

[26]

P. L. Lions, Compactness and topological methods for some nonlinear variational problems of mathematical physics, in Nonlinear Problems: Present and Future (A. Bishop, D. Campbell and B. Nicolaenko eds. ), North Holland (1982), 17-34.

[27]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case Ⅰ. Ⅱ, Annales Inst. H. Poincaré Analyse Non Linéaire, 1 (1984), 109-145,223-283.

[28]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.

[29]

M. Macr`ı and M. Nolasco, Stationary solutions for the non-linear Hartree equation with a slowly varying potential, NoDEA, 16 (2009), 681-715. doi: 10.1007/s00030-009-0030-0.

[30]

V. Moroz and J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579. doi: 10.1090/S0002-9947-2014-06289-2.

[31]

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52 (2015), 199-235. doi: 10.1007/s00526-014-0709-x.

[32]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007.

[33]

W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Commun. Pure Appl. Math., 48 (1995) 731-768. doi: 10.1002/cpa.3160480704.

[34]

M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential, Comm. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.

[35]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V)a, Comm. Partial Differential Equations, 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.

[36]

R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.

[37]

R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.

[38]

R. Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred A. Knopf, Inc. , New York, 2005

[39]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[40]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1997), 149-162.

[41]

S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.

[42]

X. Sun and Y. Zhang, Multi-peak solution for nonlinear magnetic Choquard type equation, J. Math. Phys., 55 (2014), 031508. doi: 10.1063/1.4868481.

[43]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation, J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.

[44]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.

[45]

M. Yang and Y. Ding, Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part, Comm. Pure Appl. Anal., 12 (2013), 771-783. doi: 10.3934/cpaa.2013.12.771.

[46]

V. C. Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.2307/2939286.

[47]

J. J. Zhang, Z. J. Chen, W. M. Zou, Standing Waves for nonlinear Schrödinger Equations involving critical growth, J. Lond. Math. Soc., 90 (2014), 827-844. doi: 10.1112/jlms/jdu054.

[48]

J. J. Zhang and W. M. Zou, Solutions concentrating around the saddle points of the potential for Schrödinger equations involving critical growth, Calc. Var. Partial Differ. Equ., 54 (2015), 4119-4142. doi: 10.1007/s00526-015-0933-z.

show all references

References:
[1]

C. O. Alves, J. Marcos do O and M. A. S. Souto, Local mountain-pass for a class of elliptic problems in ${{\mathbb{R}}^{N}}$ involving critical growth, Nonlinear Anal., 46 (2001), 495-510. doi: 10.1016/S0362-546X(00)00125-5.

[2]

C. O. Alves and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh, 146A (2016), 23-58. doi: 10.1017/S0308210515000311.

[3]

C. O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations, 257 (2014), 4133-4164. doi: 10.1016/j.jde.2014.08.004.

[4]

C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation inR2, J. Differential Equations, 261 (2016), 1933-1972. doi: 10.1016/j.jde.2016.04.021.

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346. doi: 10.1007/BF00250555.

[6]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. doi: 10.1007/s00205-006-0019-3.

[7]

J. Byeon and L. Jeanjean, Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity, Discrete Contin. Dynam. Syst., 19 (2007), 255-269. doi: 10.3934/dcds.2007.19.255.

[8]

J. Byeon and K. Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential, J. Eur. Math. Soc., 15 (2013), 1859-1899. doi: 10.4171/JEMS/407.

[9]

J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc., 362 (2010), 1981-2001. doi: 10.1090/S0002-9947-09-04746-1.

[10]

J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Memoirs of the American Mathematical Society, 229 (2014).

[11]

J. Byeon and Z.-Q. Wang, Standing waves with critical frequency for nonlinear Schrödinger equations Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8.

[12]

C. Bonanno, P. d'Avenia, M. Ghimenti and M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math.Anal.Appl., 417 (2014), 180-199. doi: 10.1016/j.jmaa.2014.02.063.

[13]

W. X. Chen, C. M. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure. Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.

[14]

S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh, 140A (2010), 973-1009. doi: 10.1017/S0308210509000584.

[15]

M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137. doi: 10.1007/BF01189950.

[16]

M. del Pino and P. Felmer, Multi-peak bound states of nonlinear Schrödinger equations, Annales Inst. H. Poincaré Analyse Non Linéaire, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7.

[17]

M. del Pino and P. L. Felmer, Spike-layered solutions of singularlyly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999) 883-898. doi: 10.1512/iumj.1999.48.1596.

[18]

P. D'Avenia, A. Pomponio and D. Ruiz, Semi-classical states for the Nonlinear Schrödinger Equation on saddle points of the potential via variational methods, J. Funct. Anal., 262 (2012), 4600-4633. doi: 10.1016/j.jfa.2012.03.009.

[19]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.

[21]

E. P. Gross, Physics of Many-Particle Systems, Vol. 1, Gordon Breach, New York, 1996.

[22]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equationl, Stud. Appl. Math., 57 (1977), 93-105.

[23]

E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, 2001.

[24]

E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.

[25]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal. TMA, 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4.

[26]

P. L. Lions, Compactness and topological methods for some nonlinear variational problems of mathematical physics, in Nonlinear Problems: Present and Future (A. Bishop, D. Campbell and B. Nicolaenko eds. ), North Holland (1982), 17-34.

[27]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case Ⅰ. Ⅱ, Annales Inst. H. Poincaré Analyse Non Linéaire, 1 (1984), 109-145,223-283.

[28]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.

[29]

M. Macr`ı and M. Nolasco, Stationary solutions for the non-linear Hartree equation with a slowly varying potential, NoDEA, 16 (2009), 681-715. doi: 10.1007/s00030-009-0030-0.

[30]

V. Moroz and J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579. doi: 10.1090/S0002-9947-2014-06289-2.

[31]

V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52 (2015), 199-235. doi: 10.1007/s00526-014-0709-x.

[32]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007.

[33]

W. M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Commun. Pure Appl. Math., 48 (1995) 731-768. doi: 10.1002/cpa.3160480704.

[34]

M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential, Comm. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.

[35]

Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V)a, Comm. Partial Differential Equations, 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.

[36]

R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.

[37]

R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.

[38]

R. Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred A. Knopf, Inc. , New York, 2005

[39]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.

[40]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1997), 149-162.

[41]

S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.

[42]

X. Sun and Y. Zhang, Multi-peak solution for nonlinear magnetic Choquard type equation, J. Math. Phys., 55 (2014), 031508. doi: 10.1063/1.4868481.

[43]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation, J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.

[44]

X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.

[45]

M. Yang and Y. Ding, Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part, Comm. Pure Appl. Anal., 12 (2013), 771-783. doi: 10.3934/cpaa.2013.12.771.

[46]

V. C. Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.2307/2939286.

[47]

J. J. Zhang, Z. J. Chen, W. M. Zou, Standing Waves for nonlinear Schrödinger Equations involving critical growth, J. Lond. Math. Soc., 90 (2014), 827-844. doi: 10.1112/jlms/jdu054.

[48]

J. J. Zhang and W. M. Zou, Solutions concentrating around the saddle points of the potential for Schrödinger equations involving critical growth, Calc. Var. Partial Differ. Equ., 54 (2015), 4119-4142. doi: 10.1007/s00526-015-0933-z.

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