American Institute of Mathematical Sciences

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 & Applied Analysis, 2017, 16 (2) : 493-512. doi: 10.3934/cpaa.2017025
References:
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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Memoirs of the American Mathematical Society, 229 (2014).  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] P. D'Avenia, A. Pomponio and D. 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Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, 2001. Google Scholar [24] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] M. Macrı 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.  Google Scholar [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.  Google Scholar [38] R. Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred A. Knopf, Inc. , New York, 2005  Google Scholar [39] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar [40] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1997), 149-162.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [44] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] J. Byeon and K. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations, Memoirs of the American Mathematical Society, 229 (2014).  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] E. P. Gross, Physics of Many-Particle Systems, Vol. 1, Gordon Breach, New York, 1996. Google Scholar [22] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equationl, Stud. Appl. Math., 57 (1977), 93-105.  Google Scholar [23] E. H. Lieb and M. Loss, Analysis, 2nd edition, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, 2001. Google Scholar [24] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] M. Macrı 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] R. Penrose, On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.  Google Scholar [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.  Google Scholar [38] R. Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred A. Knopf, Inc. , New York, 2005  Google Scholar [39] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar [40] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1997), 149-162.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [44] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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