### Semiclassical states for critical Choquard equations with critical frequency

DOI: http://dx.doi.org/10.12775/TMNA.2020.001

#### Abstract

#### Keywords

#### References

N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), 423–443.

C.O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in R2 , J. Differential Equations 261 (2016), 1933–1972.

C.O. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations 263 (2017), 3943–3988.

C.O. Alves, A.B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations 55 (2016), 28 pp.

C.O. Alves and M. Yang, Multiplicity and concentration behavior of solutions for a quasilinear Choquard equation via penalization method, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 23–58.

C.O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations 257 (2014), 4133–4164.

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schödinger equations, Arch. Rat. Mech. Anal. 140 (1997), 285–300.

V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274 (1982), 533–572.

V. Benci and G. Cerami, Existence of positive solutions of the equation −∆u + a(x)u = u(N +2)/(N −2) in RN , J. Funct. Anal. 88 (1990), 90–117.

B. Buffoni, L. Jeanjean and C.A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993), 179–186.

J. Byeon and Z.Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 165 (2002), 295–316.

J. Byeon and Z.Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differential Equations 18 (2003), 207–219.

D. Cao and S. Peng, Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations 34 (2009), 1566–1591.

D. Cassani and J. Zhang, Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth, Adv. Nonlinear Anal. 8 (2019), 1184–1212.

J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differential Equations 3 (1995), 493–512.

S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys. 63 (2012), 233–248.

S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 973–1009.

Y. Ding, F. Gao and M. Yang, Semiclassical states for a class of Choquard type equations with critical growth: critical frequency case, arXiv:1710.05255

Y. Ding and F. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations 30 (2007), 231–249.

A. Floer and A. Weinstein, Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal. 69 (1986), 397–408.

F. Gao, E. Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, Proc. Roy. Soc. Edinburgh Sect. A, DOI:10.1017/prm.2018.13.

F. Gao and M. Yang, On the Brezis–Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math. 61 (2018), 1219–1242.

F. Gao and M. Yang, On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents, J. Math. Anal. Appl. 448 (2017), 1006–1041.

F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy–Littlewood–Sobolev inequality, Commun.Contemp. Math. 20 (2018), 1750037, 22 pp.

F. Gao, M. Yang, C.A. Santos and J. Zhou, Infinitely many solutions for a class of critical Choquard equation, Topol. Methods Nonlinear Anal. 54 (2019), 219–232.

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1976/77), 93–105.

E. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, AMS, Providence, 2001.

P.L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063–1072.

H. Luo, Ground state solutions of Pohozaev type and Nehari type for a class of nonlinear Choquard equations, J. Math. Anal. Appl. 467 (2018), 842–862.

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455–467.

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153–184.

V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557–6579.

V. Moroz, and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations 52 (2015), 199–235.

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), 1550005, 12 pp.

V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl. 19 (2017), 773–813.

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

R. Penrose, On gravity’s role in quantum state reduction, Gen. Relativ. Gravitat. 28 (1996), 581–600.

M. del Pino and P. Felmer, Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137.

J. Seok, Nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett. 76 (2018), 148–156.

B. Sirakov, Standing wave solutions of the nonlinear Schrödinger equations in RN , Ann. Mat. Pura Appl. 183 (2002), 73–83.

J. Van Schaftingen and J. Xia, Standing waves with a critical frequency for nonlinear Choquard equations, Nonlinear Anal. 161 (2017), 87–107.

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger–Newton equations, J Math. Phys. 50 (2009), 012905.

M. Willem, Minimax Theorems, Birkhäuser, 1996.

M. Yang and Y. Ding, Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part, Comm. Pure Appl. Anal. 12 (2013), 771–783.

M. Yang, J. Zhang and Y. Zhang, Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity, Comm. Pure Appl. Anal. 16 (2017), 493–512.

### Refbacks

- There are currently no refbacks.