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Existence of solutions for a semirelativistic Hartree equation with unbounded potentials. (English) Zbl 1391.35139

The paper deals with the existence of a solution to the semirelativistic Hartree equation \[ \sqrt{-\Delta+m^2 u }+ V(x) u = A(x)( W * | u|^p) | u|^{p-2} u \] under suitable growth assumption on the potential functions \(V\) and \(A\), which can be unbounded from above. The author introduces a class of function spaces and gives a compact embedding result which is the main ingredient to prove the existence result.

MSC:

35J60 Nonlinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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[1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), 423-443. · Zbl 1059.35037
[2] R. A. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996.
[3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063
[4] X. Cabré and J. Solà-Morales, Layers solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (2005), 1678-1732. · Zbl 1102.35034
[5] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052-2093. · Zbl 1198.35286
[6] Y. H. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity 29 (2016), 1827-1842. · Zbl 1381.35213
[7] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal. 38 (2006), no. 4, 1060-1074. · Zbl 1122.35119
[8] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys. 63 (2012), 233-248. · Zbl 1247.35141
[9] S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), 891-908. · Zbl 1260.35198
[10] S. Cingolani and S. Secchi, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), 73-90. · Zbl 1320.35300
[11] S. Cingolani and S. Secchi, Semiclassical analysis for pseudo-relativistic Hartree equations, J. Differential Equations 258 (2015), 4156-4179. · Zbl 1319.35204
[12] S. Cingolani, S. Secchi and M. Squassina, Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 973-1009. · Zbl 1215.35146
[13] V. Coti Zelati and M. Nolasco, Existence of ground state for nonlinear, pseudorelativistic Schrödinger equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22 (2011), 51-72. · Zbl 1219.35292
[14] V. Coti Zelati and M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Iberoam. 29 (2013), 1421-1436. · Zbl 1283.35121
[15] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. · Zbl 1252.46023
[16] A. Elgart, B. and Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math. 60 (2007), 500-545. · Zbl 1113.81032
[17] M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5827-5867. · Zbl 1336.35356
[18] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1237-1262. · Zbl 1290.35308
[19] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: {L^{p}} Spaces, Springer, New York, 2007. · Zbl 1153.49001
[20] J. Fröhlich, J. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys. 274 (2007), 1-30. · Zbl 1126.35064
[21] J. Fröhlich and E. Lenzmann, Mean-field limit of quantum Bose gases and nonlinear Hartree equation, Sémin. Équ. Dériv. 2003-2004 (2004), Exposé No. 18. · Zbl 1292.81040
[22] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93-105. · Zbl 0369.35022
[23] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349-374. · Zbl 0527.42011
[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. 4 (1980), 1063-1073. · Zbl 0453.47042
[26] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455-467. · Zbl 1185.35260
[27] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity 15 (1998), 2733-2742. · Zbl 0936.83037
[28] Y. J. Park, Fractional Gagliardo-Nirenberg inequality, J. Chungcheong Math. Soc. 24 (2011), no. 3, 583-586.
[29] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), 1927-1939. · Zbl 1152.81659
[30] R. Penrose, The road to reality. A complete guide to the laws of the universe, Alfred A. Knopf, New York, 2005. · Zbl 1188.00007
[31] S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal. 72 (2010), 3842-3856. · Zbl 1187.35254
[32] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in {\mathbb{R}^{N}}, J. Math. Phys. 54 (2013), no. 3, Article No. 031501. · Zbl 1281.81034
[33] S. Secchi, On some nonlinear fractional equations involving the Bessel operator, J. Dynam. Differential Equations (2016), 10.1007/s10884-016-9521-y. · Zbl 1377.35107
[34] S. Secchi, Concave-convex nonlinearities for some nonlinear fractional equations involving the Bessel operator, Complex Var. Elliptic Equ. 62 (2017), 10.1080/17476933.2016.1234465. · Zbl 1365.35033
[35] B. Sirakov, Existence and multiplicity of solutions of semmi-linear elliptic equations in {\mathbb{R}^{N}}, Cal. Var. Partial Differential Equations 11 (2000), 119-142. · Zbl 0977.35049
[36] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. · Zbl 0207.13501
[37] R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48-79. · Zbl 0515.58037
[38] P. Tod, The ground state energy of the Schrödinger-Newton equation, Phys. Lett. A 280 (2001), 173-176. · Zbl 0984.81024
[39] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation, J. Math. Phys. 50 (2009), Article ID 012905. · Zbl 1189.81061
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