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Existence of solutions for a semirelativistic Hartree equation with unbounded potentials

Simone Secchi
From the journal Forum Mathematicum

Abstract

We prove the existence of a solution to the semirelativistic Hartree equation

-Δ+m2u+V(x)u=A(x)(W*|u|p)|u|p-2u

under suitable growth assumption on the potential functions V and A. In particular, both can be unbounded from above.

MSC 2010: 35J60; 35Q55; 35S05

Communicated by Christopher D. Sogge


Funding statement: The author is supported by the MIUR 2015 PRIN project “Variational methods, with applications to problems in mathematical physics and geometry”.

References

[1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), 423–443. Search in Google Scholar

[2] R. A. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996. Search in Google Scholar

[3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381. 10.1016/0022-1236(73)90051-7Search in Google Scholar

[4] X. Cabré and J. Solà-Morales, Layers solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (2005), 1678–1732. 10.1002/cpa.20093Search in Google Scholar

[5] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093. 10.1016/j.aim.2010.01.025Search in Google Scholar

[6] Y. H. Chen and C. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity 29 (2016), 1827–1842. 10.1088/0951-7715/29/6/1827Search in Google Scholar

[7] Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal. 38 (2006), no. 4, 1060–1074. 10.1137/060653688Search in Google Scholar

[8] S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys. 63 (2012), 233–248. 10.1007/s00033-011-0166-8Search in Google Scholar

[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. Search in Google Scholar

[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. 10.1017/S0308210513000450Search in Google Scholar

[11] S. Cingolani and S. Secchi, Semiclassical analysis for pseudo-relativistic Hartree equations, J. Differential Equations 258 (2015), 4156–4179. 10.1016/j.jde.2015.01.029Search in Google Scholar

[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. 10.1017/S0308210509000584Search in Google Scholar

[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. Search in Google Scholar

[14] V. Coti Zelati and M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Iberoam. 29 (2013), 1421–1436. 10.4171/RMI/763Search in Google Scholar

[15] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar

[16] A. Elgart and B. Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math. 60 (2007), 500–545. 10.1002/cpa.20134Search in Google Scholar

[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. 10.3934/dcds.2015.35.5827Search in Google Scholar

[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. 10.1017/S0308210511000746Search in Google Scholar

[19] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer, New York, 2007. Search in Google Scholar

[20] J. Fröhlich, J. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys. 274 (2007), 1–30. 10.1007/s00220-007-0272-9Search in Google Scholar

[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. Search in Google Scholar

[22] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105. 10.1002/sapm197757293Search in Google Scholar

[23] E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349–374. 10.2307/2007032Search in Google Scholar

[24] E. H. Lieb and B. Simon, The Hartree–Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977), 185–194. 10.1007/BF01609845Search in Google Scholar

[25] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063–1073. 10.1016/0362-546X(80)90016-4Search in Google Scholar

[26] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455–467. 10.1007/s00205-008-0208-3Search in Google Scholar

[27] I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger–Newton equations, Classical Quantum Gravity 15 (1998), 2733–2742. 10.1088/0264-9381/15/9/019Search in Google Scholar

[28] Y. J. Park, Fractional Gagliardo–Nirenberg inequality, J. Chungcheong Math. Soc. 24 (2011), no. 3, 583–586. Search in Google Scholar

[29] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), 1927–1939. 10.1098/rsta.1998.0256Search in Google Scholar

[30] R. Penrose, The road to reality. A complete guide to the laws of the universe, Alfred A. Knopf, New York, 2005. Search in Google Scholar

[31] S. Secchi, A note on Schrödinger–Newton systems with decaying electric potential, Nonlinear Anal. 72 (2010), 3842–3856. 10.1016/j.na.2010.01.021Search in Google Scholar

[32] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in N, J. Math. Phys. 54 (2013), no. 3, Article No. 031501. Search in Google Scholar

[33] S. Secchi, On some nonlinear fractional equations involving the Bessel operator, J. Dynam. Differential Equations (2016), 10.1007/s10884-016-9521-y. 10.1007/s10884-016-9521-ySearch in Google Scholar

[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. 10.1080/17476933.2016.1234465Search in Google Scholar

[35] B. Sirakov, Existence and multiplicity of solutions of semmi-linear elliptic equations in N, Cal. Var. Partial Differential Equations 11 (2000), 119–142. Search in Google Scholar

[36] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. Search in Google Scholar

[37] R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983), 48–79. 10.1016/0022-1236(83)90090-3Search in Google Scholar

[38] P. Tod, The ground state energy of the Schrödinger–Newton equation, Phys. Lett. A 280 (2001), 173–176. 10.1016/S0375-9601(01)00059-7Search in Google Scholar

[39] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger–Newton equation, J. Math. Phys. 50 (2009), Article ID 012905. Search in Google Scholar

Received: 2017-01-11
Published Online: 2017-04-20
Published in Print: 2018-01-01

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