Sage Journals: Discover world-class research

Abstract

This paper is concerned with the following Choquard equation $\begin{matrix} \{\begin{matrix} - Δ u + (V (x) - \frac{μ}{| x |^{2}}) u = (\int_{R^{N}} \frac{F (u (y))}{| x - y |^{α}} d y) f (u), \\ u \in H^{1} (R^{N}), \end{matrix} \end{matrix}$ where $N ⩾ 3$ , $0 ⩽ μ < \frac{{(N - 2)}^{2}}{4}$ , $0 < α < N$ , V is 1-periodic in each of $x_{1}, x_{2}, \dots, x_{N}$ and F is the primitive function of f. Under some mild assumptions on V and f, we establish the existence and asymptotical behavior of ground state solutions by variational methods.

Keywords

Choquard equation ground state solutions asymptotical behavior inverse square potential

Get full access to this article

View all access options for this article.

References

Abdellaoui ,

Felli and

Peral , Existence and multiplicity for perturbations of equation involving Hardy inequality and critical Sobolev exponent in the whole

R^{N}

, Adv. Differential Equations 9 (2004), 481–508.

C.O.

Alves and

M.B.

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.

S.T.

Chen and

X.H.

Tang , Improved results for Klein–Gordon–Maxwell systems with general nonlinearity, Discrete. Contin. Dyn. Syst. 38 (2018), 2333–2348.

S.T.

Chen and

X.H.

Tang , Ground state solutions of Schröinger–Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl. 73 (2019), 87–111. doi:10.1016/j.jmaa.2018.12.037.

Choquard ,

Stubbe and

Vuffray , Stationary solutions of the Schrödinger–Newton model – an ODE approach, Differential Integral Equations 21 (2008), 665–679.

Deng ,

Jin and

Peng , Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differential Equations 253(5) (2012), 1376–1398. doi:10.1016/j.jde.2012.05.009.

Felli , On the existence of ground state solutions to nonlinear Schrödinger equations with multisingular inverse-square anisotropic potentials, J. Anal. Math. 108 (2009), 189–217. doi:10.1007/s11854-009-0023-2.

Felli ,

Marchini and

Terracini , On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal. 250(2) (2007), 265–316. doi:10.1016/j.jfa.2006.10.019.

Felli and

Terracini , Elliptic equations with multi-singular inverse-square potentials and critical nonlineaity, Comm. Partial Differential Equations 31(1–3) (2006), 469–495. doi:10.1080/03605300500394439.

10.

Guo and

Mederski , Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials, J. Differential Equations 260 (2016), 4180–4202. doi:10.1016/j.jde.2015.11.006.

11.

E.H.

Lieb , Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105. doi:10.1002/sapm197757293.

12.

P.L.

Lions , The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063–1072. doi:10.1016/0362-546X(80)90016-4.

13.

G.P.

Menzala , On regular solutions of a nonlinear equation of Choquard’s type, Proc. Roy. Soc. Edinburgh Sect. A 86(3–4) (1980), 291–301. doi:10.1017/S0308210500012191.

14.

I.M.

Moroz ,

Penrose and

Tod , Spherically-symmetric solutions of the Schrödinger–Newton equations, Classical Quantum Gravity 15 (1998), 2733–2742. doi:10.1088/0264-9381/15/9/019.

15.

Moroz and

van Schaftingen , Existence of groundstate for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557–6579. doi:10.1090/S0002-9947-2014-06289-2.

16.

N.S.

Papageorgiou and

V.D.

Rădulescu , Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity, in: Recent Trends in Nonlinear Partial Differential Equations. II. Stationary Problems, Contemp. Math., Vol. 595, Amer. Math. Soc., Providence, RI, 2013, pp. 293–315. doi:10.1090/conm/595/11801.

17.

N.S.

Papageorgiou and

V.D.

Rădulescu , Multiplicity of solutions for resonant Neumann problems with indefinite and unbounded potential, Trans. Amer. Math. Soc. 367 (2015), 8723–8756. doi:10.1090/S0002-9947-2014-06518-5.

18.

N.S.

Papageorgiou and

V.D.

Rădulescu , An infinity of nodal solutions for superlinear Robin problems with an indefinite and unbounded potential, Bull. Sci. math. 141 (2017), 251–266. doi:10.1016/j.bulsci.2017.03.001.

19.

N.S.

Papageorgiou ,

V.D.

Rădulescu and

D.D.

Repovš , Nonlinear Analysis – Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019.

20.

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

21.

Reed and

Simon , Methods of Modern Mathematical Physics, Analysis of Operators, Vol. IV, Academic Press, New York, 1978.

22.

Ruiz and

Willem , Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations 190(2) (2003), 524–538. doi:10.1016/S0022-0396(02)00178-X.

23.

Schechter , A variation of the mountain pass lemma and applications, J. London Math. Soc. 44(2) (1991), 491–502. doi:10.1112/jlms/s2-44.3.491.

24.

Smets , Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc. 357(7) (2005), 2909–2938. doi:10.1090/S0002-9947-04-03769-9.

25.

Szulkin and

Weth , Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257(12) (2009), 3802–3822. doi:10.1016/j.jfa.2009.09.013.

26.

X.H.

Tang , Non-Nehari manifold method for superlinear Schrödinger equation, Taiwanese J. Math. 18 (2014), 1957–1979. doi:10.11650/tjm.18.2014.3541.

27.

X.H.

Tang , Non-Nehari-Manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc. 98 (2015), 104–116. doi:10.1017/S144678871400041X.

28.

X.H.

Tang , Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math. 58 (2015), 715–728. doi:10.1007/s11425-014-4957-1.

29.

X.H.

Tang and

S.T.

Chen , Ground state solutions of Nehari–Pohoz˘aev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations 56 (2017), 110–134. doi:10.1007/s00526-017-1214-9.

30.

X.H.

Tang and

B.T.

Cheng , Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations 261 (2016), 2384–2402. doi:10.1016/j.jde.2016.04.032.

Ground state solutions for nonlinear Choquard equations with inverse-square potentials 1

Abstract

Keywords

Get full access to this article

References