Abstract
It is established existence and nonexistence of solutions to nonlocal elliptic problems involving the generalized pseudo-relativistic Hartree equation. Our arguments are based on variational methods together with a fine analysis on the Pohozaev identity.
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References
Ambrosio, V.: Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator. J. Math. Phys. 57(5), 051502 (2016)
Ambrosio, V.: Existence of heteroclinic solutions for a pseudo-relativistic Allen–Cahn type equation. Adv. Nonlinear Stud. 15(2), 395–414 (2015)
Ambrosio, V.: Nonlinear Fractional Schrödinger Equations in \({\mathbb{R}}^N\), Frontiers in Elliptic and Parabolic Problems, p. xvii+662. Birkhäuser/Springer, Cham (2021)
Ambrosio, V.: The nonlinear fractional relativistic Schrödinger equation: existence, multiplicity, decay and concentration results. Discrete Cont. Dyn. Syst. 41(12), 5659–5705 (2021)
Belchior, P., Bueno, H., Miyagaki, O.H., Pereira, G.: Asymptotic behavior of ground states of generalized pseudo-relativistic Hartree equation. Asymptot. Anal. 118(4), 269–295 (2020)
Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7(9), 981–1012 (1983)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
Cassani, D., Zhang, J.: Ground states and semiclassical states of nonlinear Choquard equations involving Hardy–Littlewood–Sobolev critical growth. Adv. Nonlinear Anal. 8(1), 1184–1212 (2019)
Cho, Y.G., Ozawa, T.: On the semi relativistic Hartree-type equation. SIAM J. Math. Anal. 38(4), 1060–1074 (2006)
Choi, W., Seok, J.: Non relativistic limit of standing waves for pseudo-relativistic nonlinear Schrödinger equations. J. Math. Phys. 57(2), 021510 (2016)
Cingolani, S., Clapp, M., Secchi, S.: Intertwining semiclassical solutions to a Schrödinger–Newton system. Discrete Contin. Dyn. Syst. Ser. S 6(4), 891–908 (2013)
Cingolani, S., Secchi, S.: Ground states for the pseudo-relativistic Hartree equation with external potential. Proc. Roy. Soc. Edinb. Sect. A 145(1), 73–90 (2015)
Cingolani, S., Secchi, S.: Semiclassical analysis for pseudo-relativistic Hartree equations. J. Differ. Equ. 258(12), 4156–4179 (2015)
Cingolani, S., Secchi, S., Squassina, M.: Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. R. Soc. Edinb. Sect. A 140(5), 973–1009 (2010)
Costa, D.G., Magalhães, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23(11), 1401–1412 (1994)
Costa, D.G., Tehrani, H.: On a class of asymptotically linear elliptic problems in \({\mathbb{R}}^N\). J. Differ. Equ. 173(2), 470–494 (2001)
Coti Zelati, V., Nolasco, M.: Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22(1), 51–72 (2011)
Demengel, F., Demengel, G.: Functional Spaces for the Theory of Elliptic Partial Differential Equations. Springer, London (2012)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhikers guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Ekeland, I.: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin (1990)
Elgart, A., Schlein, B.: Mean field dynamics of boson stars. Commun. Pure Appl. Math. 60(4), 500–545 (2007)
Fall, M.M., Felli, V.: Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete Contin. Dyn. Syst. 35(12), 5827–5867 (2015)
Fröhlich, J., Lenzmann, E.: Mean-field limit of quantum Bose gases and nonlinear Hartree equation, Séminaire: Équations aux Dérivées Partielles. 2003–2004, Exp. No. XIX, 26 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau (2004)
Gao, F., Yang, M.: The Brezis–Nirenberg type critical problem for the nonlinear Choquard equation. Sci. China Math. 61(7), 1219–1242 (2018)
Jeanjean, L., Tanaka, K.: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 21(3), 287–318 (2004)
Lehrer, R., Maia, L.A.: Positive solutions of asymptotically linear equations via Pohozaev manifold. J. Funct. Anal. 266(1), 213–246 (2014)
Lehrer, R., Maia, L.A., Ruviaro, R.: Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential. NoDEA Nonlinear Differ. Equ. Appl. 22(4), 651–672 (2015)
Lenzmann, E.: Uniqueness of ground states for pseudo relativistic Hartree equations. Anal. PDE 2(1), 1–27 (2009)
Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. (2) 118(2), 349–374 (1983)
Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112(1), 147–174 (1987)
Melgaard, M., Zongo, F.: Multiple solutions of the quasi relativistic Choquard equation. J. Math. Phys. 53(3), 033709 (2012)
Moroz, V., Van Schaftingen, J.: Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265(2), 153–184 (2013)
Moroz, V., Van Schaftingen, J.: Semi-classical states for the Choquard equations. Calc. Var. Partial Differ. Equ. 52(1–2), 199–235 (2015)
Qin, Dongdong, Chen, Jing, Tang, XianHua: Existence and non-existence of nontrivial solutions for Schrödinger systems via Nehari–Pohozaev manifold. Comput. Math. Appl. 74(12), 3141–3160 (2017)
Tartar, L.: An introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3. Springer, Berlin; UMI, Bologna (2007)
Wei, J., Winter, M.: Strongly interacting bumps for the Schrödinger–Newton equation. J. Math. Phys. 50(1), 012905 (2009)
Willem, M.: Minimax Theorems. Birkhäser, Boston (1996)
Acknowledgements
The authors would like to thank Aldo H. S. Medeiros for many useful conversations and suggestions.
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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.
Research supported in part by INCTmat/MCT/Brazil, CNPq and CAPES/Brazil. The third author was partially supported by CNPq with Grants 309026/2020-2 and 429955/2018-9.
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Bueno, H., Pereira, G.A., Silva, E.D. et al. Existence and nonexistence of solutions to nonlocal elliptic problems. Partial Differ. Equ. Appl. 3, 8 (2022). https://doi.org/10.1007/s42985-021-00142-3
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DOI: https://doi.org/10.1007/s42985-021-00142-3