Abstract
In this paper, we investigate the following nonlinear Choquard equation with prescribed \(L^2\)-norm constraint
where \(a>0\), \(\lambda \in \mathbb R\) appears as an unknown Lagrange multiplier and \(\Omega \subset \mathbb R^3\) is an exterior domain with smooth boundary \(\partial \Omega \ne \emptyset \) such that \(\mathbb R^3\backslash \Omega \) is bounded. By using the splitting lemma for the unconstrained problem in exterior domains, we prove the compactness of Palais–Smale sequences corresponding to the above problem at higher energy levels. Then combining the barycentric function and Brouwer degree theory, we establish the existence of positive normalized bound states for any \(a>0\) provided that \(\mathbb R^3\backslash \Omega \) is contained in a small ball and explain that the restriction on domain \(\Omega \) can be equivalently transferred to a. In addition, under the radial setting of domain \(\Omega \), we use genus theory to obtain the existence and multiplicity of radial normalized solutions for any \(a>0\). Finally, we point out that the main results can be extended to a general mass subcritical Choquard equation.
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Benci, V., Cerami, G.: Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Ration. Mech. Anal. 99, 283–300 (1987)
Cazenave, T., Lions, P.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549–561 (1982)
Chen, P., Liu, X.: Positive solutions for a Choquard equation in exterior domain. Commun. Pure Appl. Anal. 20, 2237–2256 (2021)
Chang, K.: Methods in Nonlinear Analysis. Springer, Berlin (2005)
Cingolani, S., Tanaka, K.: Ground state solutions for the nonlinear Choquard equation with prescribed mass. Geom. Prop. Parabol. Elliptic PDE’s 47, 23–41 (2021)
Cingolani, S., Gallo, M., Tanaka, K.: Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities. Calc. Var. Partial Differ. Equ. 61, 1–34 (2022)
Correia, J., Oliveira, C.: Positive solution for a class of Choquard equations with Hardy–Littlewood–Sobolev critical exponent in exterior domains. Complex Var. Elliptic Equ. (2022). https://doi.org/10.1080/17476933.2022.2056888
Ghoussoub, N.: Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press, Cambridge (1993)
Jeanjean, L., Lu, S.: Nonradial normalized solutions for nonlinear scalar field equations. Nonlinearity 32, 4942–4966 (2019)
Jia, H., Luo, X.: Prescribed mass standing waves for energy critical Hartree equations. Calc. Var. Partial Differ. Equ. 62, 71 (2023)
Lei, C., Yang, M., Zhang, B.: Sufficient and necessary nonditions for mormalized solutions to a Choquard equation. J. Geom. Anal. 33, 109 (2023)
Lieb, E., Loss, M.: Analysis. Amer Math Soc, Providence (2001)
Lieb, E.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977)
Lions, P.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)
Li, G., Tang, C.: Existence of ground state solutions for Choquard equation involving the general upper critical Hardy–Littlewood–Sobolev nonlinear term, Commun. Pure. Appl. Anal. 18, 285–300 (2018)
Li, G., Tang, C.: Existence of a ground state solution for Choquard equation with the upper critical exponent. Comput. Math. Appl. 76, 2635–2647 (2018)
Li, G., Ye, H.: The existence of positive solutions with prescribed \(L^2\)-norm for nonlinear Choquard equations. J. Math. Phys. 55, 121501 (2014)
Long, L., Li, F., Zhu, X.: Normalized solutions to nonlinear scalar field equations with doubly nonlocal terms and critical exponent. J. Math. Anal. Appl. 524, 127142 (2023)
Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)
Moroz, V., Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)
Moroz, V., Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)
Moroz, V., Schaftingen, J.: Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent. Commun. Contemp. Math. 17, 1550005 (2015)
Moroz, I., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger–Newton equations. Classical Quantum Gravity 15, 2733–2742 (1998)
Palais, R.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979)
Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954)
Ruiz, D.: The Schrödinger–Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655–674 (2006)
Shang, X., Ma, P.: Normalized solutions to the nonlinear Choquard equations with Hardy–Littlewood–Sobolev upper critical exponent. J. Math. Anal. Appl. 521, 126916 (2023)
Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)
Wang, T., Yi, T.: Uniqueness of positive solutions of the Choquard type equations. Appl. Anal. 96, 3 (2016)
Wei, J., Winter, M.: Strongly interacting bumps for the Schrödinger–Newton equations. J. Math. Phys. 50, 012905 (2009)
Xiang, C.: Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions. Calc. Var. Partial Differ. Equ. 55, 134 (2016)
Yang, J., Zhu, L.: Multiple solutions to Choquard equation in exterior domain. J. Math. Anal. Appl. 507, 125726 (2022)
Yao, S., Chen, H., Rădulescu, D., Sun, J.: Normalized Solutions for lower critical Choquard equations with critical Sobolev perturbation. SIAM J. Math. Anal. 54, 3696–3723 (2022)
Yang, X.: Existence of positive solution for the Choquard equation in exterior domain. Complex Var. Elliptic Equ. 67, 2043–2059 (2022)
Ye, H.: Mass minimizers and concentration for nonlinear Choquard equations in \(\mathbb{R} ^N\). Topol. Methods Nonlinear Anal. 48, 393–417 (2016)
Ye, W., Sheng, Z., Yang, M.: Normalized Solutions for a critical Hartree equation with perturbation. J. Geom. Anal. 32, 242 (2022)
Zhang, Z., Zhang, Z.: Normalized solutions of mass subcritical Schrödinger equations in exterior domains. NoDEA Nonlinear Differ. Equ. Appl. 29, 32 (2022)
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This research is supported by National Natural Science Foundation of China (No. 12371120) and Chongqing Postdoctoral Science Foundation Project (No. CSTB2023NSCQ-BHX0226).
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SY wrote the main manuscript text; CY provided the method of the fifth part; C-LT studied the feasibility and modified the paper format. All authors reviewed the manuscript.
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Yu, S., Yang, C. & Tang, CL. Normalized bound states for the Choquard equations in exterior domains. Z. Angew. Math. Phys. 75, 35 (2024). https://doi.org/10.1007/s00033-024-02188-w
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DOI: https://doi.org/10.1007/s00033-024-02188-w