With appropriate hypotheses on the nonlinearity f, we prove the existence of a ground state solution u for the problem
where V is a bounded potential, not necessarily continuous, and F the primitive of f. We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.
N.Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z.248(2) (2004), 423–443. doi:10.1007/s00209-004-0663-y.
2.
V.Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys.57(5) (2016), 051502, 18 pp.
3.
H.Bueno, O.H.Miyagaki and G.A.Pereira, Remarks about a generalized pseudo-relativistic Hartree equation, J. Differential Equations266 (2019), 876–909. doi:10.1016/j.jde.2018.07.058.
4.
X.Cabré and J.Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math.58(12) (2005), 1678–1732. doi:10.1002/cpa.20093.
5.
L.Caffarelli and L.Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations32(7–9) (2007), 1245–1260. doi:10.1080/03605300600987306.
6.
Y.Cho and T.Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal.38(4) (2006), 1060–1074. doi:10.1137/060653688.
7.
S.Cingolani, M.Clapp and S.Secchi, Intertwining semiclassical solutions to a Schrödinger–Newton system, Discrete Contin. Dyn. Syst. Ser4(6) (2013), 891–908.
8.
S.Cingolani and S.Secchi, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinburgh Sect. A145(1) (2015), 73–90. doi:10.1017/S0308210513000450.
9.
S.Cingolani and S.Secchi, Semiclassical analysis for pseudo-relativistic Hartree equations, J. Differential Equations258(12) (2015), 4156–4179. doi:10.1016/j.jde.2015.01.029.
10.
S.Cingolani and S.Secchi, Intertwining solutions for magnetic relativistic Hartree type equations, Nonlinearity31(5) (2018), 2294–2318. doi:10.1088/1361-6544/aab0be.
11.
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(5) (2010), 973–1009. doi:10.1017/S0308210509000584.
12.
V.Coti Zelati and M.Nolasco, Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.22(1) (2011), 51–72. doi:10.4171/RLM/587.
13.
V.Coti Zelati and M.Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Iberoam.22(4) (2013), 1421–1436. doi:10.4171/RMI/763.
14.
F.Demengel and G.Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, 2012.
15.
E.Di Nezza, G.Palatucci and E.Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math.136(5) (2012), 521–573. doi:10.1016/j.bulsci.2011.12.004.
16.
A.Elgart and B.Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math.60(4) (2007), 500–545. doi:10.1002/cpa.20134.
17.
P.Felmer, A.Quaas and J.Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A142(6) (2012), 1237–1262. doi:10.1017/S0308210511000746.
18.
J.Fröhlich and E.Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, in: Séminaire: “Equations aux Dérivées Partielles”, Sémin. Équ. Dériv. Partielles, Vol. 2003–2004, Exp. No. XIX, 26 pp., École Polytech, Palaiseau, 2004.
19.
M.F.Furtado, L.A.Maia and E.S.Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud.8(2) (2008), 353–373. doi:10.1515/ans-2008-0207.
20.
O.Kavian, Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques, Springer-Verlag, Paris, 1993.
21.
E.Lenzmann, Uniqueness of ground states for pseudo-relativistic Hartree equations, Anal. PDE2(1) (2009), 1–27. doi:10.2140/apde.2009.2.1.
22.
E.H.Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2)118(2) (1983), 349–374. doi:10.2307/2007032.
23.
E.H.Lieb and H.-T.Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys.112(1) (1987), 147–174. doi:10.1007/BF01217684.
24.
P.L.Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana1(1) (1985), 145–201. doi:10.4171/RMI/6.
25.
M.Melgaard and F.Zongo, Multiple solutions of the quasirelativistic Choquard equation, J. Math. Phys.53(3) (2012), 033709, 12 pp.
26.
V.Moroz and J.Van Schaftingen, Semi-classical states for the Choquard equations, Calc. Var. Partial Differential Equations52(1–2) (2015), 199–235. doi:10.1007/s00526-014-0709-x.
27.
P.H.Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys.43(2) (1992), 270–291. doi:10.1007/BF00946631.
28.
S.Secchi, Existence of solutions for a semirelativistic Hartree equation with unbounded potentials, Forum Math.30(1) (2018), 129–140. doi:10.1515/forum-2017-0006.
29.
L.Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Springer, Berlin, 2007.
30.
J.Wei and M.Winter, Strongly interacting bumps for the Schrödinger–Newton equations, J. Math. Phys.50(1) (2009), 012905, 22 pp.
