Abstract
We study the saddle type nodal solutions for the Choquard equation
where \(N\ge 3\), \(I_\alpha \) is the Riesz potential of order \(\alpha \in (0, N)\) and \(\frac{N+\alpha }{N-2}\) refers to the upper critical exponent with respect to the Hardy–Littlewood–Sobolev inequality. By introducing the symmetric groups of Coxeter, we give a unified framework to construct saddle solutions with prescribed symmetric nodal structures. To overcome the difficulties arising from the lack of compactness, we give a fine analyzes on the profile decompositions of the symmetric Palais–Smale sequence. These results further feature the nonlocal nature of the Choquard equation, in contrast, the counterpart Yamabe equation \(-\Delta u=|u|^{\frac{4}{N-2}}u\) can not permit such type of nodal solutions.
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Ackermann, N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004)
Alves, C.O., Nóbrega, A.B., Yang, M.: Multi-bump solutions for Choquard equation with deepening potential well. Calc. Var. Partial Differ. Equ. 55, Article ID, 28 pages (2016)
Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)
Battaglia, L., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations in the plane. Adv. Nonlinear Stud. 17, 581–594 (2017)
Benci, V., Cerami, G.: Existence of positive solutions of the equation \(-\Delta u+a(x)u=u^{(N+2)/(N-2)}\) in \({\mathbb{R}}^N\). J. Funct. Anal. 88, 90–117 (1990)
Cassani, D., Van Schaftingen, J., Zhang, J.: Groundstates for Choquard type equations with Hardy–Littlewood–Sobolev lower critical exponent. Proc. R. Soc. Edinburgh Sect. A 150, 1377–1400 (2020)
Clapp, M., Salazar, D.: Positive and sign changing solutions to a nonlinear Choquard equation. J. Math. Anal. Appl. 407, 1–15 (2013)
Clapp, M.: Entire nodal solutions to the pure critical exponent problem arising from concentration. J. Differ. Equ. 261, 3042–3060 (2016)
Coxeter, H.S.M.: The complete enumeration of finite groups of the form \(r_i^2=(r_ir_j)^{k_{ij}}=1\). J. London Math. Soc. 10, 21–25 (1935)
Davis, M.W.: The Geometry and Topology of Coxeter Groups, vol. 32. Princeton University Press, Princeton (2008)
del Pino, M., Musso, M., Pacard, F., Pistoia, A.: Large energy entire solutions for the Yamabe equation. J. Differ. Equ. 251, 2568–2597 (2011)
Ding, W.: On a conformally invariant elliptic equation on \({{\mathbb{R}}}^N\). Commun. Math. Phys. 107, 331–335 (1986)
Diósi, L.: Gravitation and quantum-mechanical localization of macro-objects. Phys. Lett. A 105, 199–202 (1984)
Esteban, M.J., Lions, P.-L.: Existence and nonexistence results for semilinear elliptic problems in unbounded domains. Proc. R. Soc. Edinburgh Sect. A 93, 1–14 (1982)
Gao, F., Yang, M.: On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents. J. Math. Anal. Appl. 448, 1006–1041 (2017)
Gao, F., Yang, M.: The Brezis–Nirenberg type critical problem for the nonlinear Choquard equation. Sci. China Math. 61, 1219–1242 (2018)
Ghimenti, M., Van Schaftingen, J.: Nodal solutions for the Choquard equation. J. Funct. Anal. 271, 107–135 (2016)
Ghimenti, M., Moroz, V., Van Schaftingen, J.: Least action nodal solutions for the quadratic Choquard equation. Proc. Am. Math. Soc. 145, 737–747 (2017)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin (2001)
Guo, L., Hu, T., Peng, S., Shuai, W.: Existence and uniqueness of solutions for Choquard equation involving Hardy-Littlewood-Sobolev critical exponent. Calc. Var. Partial Differ. Equ. 58, Article ID 128, 34 pages (2019)
Jones, K.R.W.: Newtonian quantum gravity. Aust. J. Phys. 48, 1055–1082 (1995)
Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1976/77)
Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence (1997)
Lions, P.-L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109–145 (1984)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana 1, 145–201 (1985)
Liu, X., Ma, S., Zhang, X.: Infinitely many bound state solutions of Choquard equations with potentials. Z. Angew. Math. Phys. 69, Article ID 118, 29 pages (2018)
Ma, L., Zhao, L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010)
Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger–Newton equations. Class. Quantum Gravity 15, 2733–2742 (1998)
Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)
Moroz, V., Van Schaftingen, J.: Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367, 6557–6579 (2015)
Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent. Commun. Contemp. Math. 17, Article ID 1550005, 12 pages (2015)
Moroz, V., Van Schaftingen, J.: A guide to the Choquard equation. J. Fixed Point Theory Appl. 19, 773–813 (2017)
Obata, M.: Conformal changes of Riemannian metrics on a Euclidean sphere. In: Differential Geometry (in Honor of Kentaro Yano), Kinokuniya, Tokyo, pp. 347–353, (1972)
Pekar, S.: Untersuchungen über die Elektronentheorie der Kristalle. Akademie-Verlag, Berlin (1954)
Seok, J.: Nonlinear Choquard equations: doubly critical case. Appl. Math. Lett. 76, 148–156 (2018)
Struwe, M.: Variational Methods. Springer-Verlag, Berlin (1996)
Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)
Thomas, A.: Geometric and topological aspects of Coxeter groups and buildings, Vorlesungsmanuskript (2017)
Tintarev, C.: Concentration analysis and cocompactness, Trends Mathematics. Concentration analysis and applications to PDE, Birkhäuser/Springer, Basel, pp. 117–141 (2013)
Van Schaftingen, J., Xia, J.: Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent. J. Math. Anal. Appl. 464, 1184–1202 (2018)
Wang, Z.-Q., Xia, J.: Saddle solutions for the Choquard equation II. Nonlinear Anal. 201, Artice ID 112053, 25 pages (2020)
Wei, J., Winter, M.: Strongly interacting bumps for the Schrödinger–Newton equations. J. Math. Phys. 50, Article ID 012905, 22 pages (2009)
Weth, T.: Energy bounds for entire nodal solutions of autonomous superlinear equations. Calc. Var. Partial Differ. Equ. 27, 421–437 (2006)
Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser Boston Inc., Boston (1996)
Wu, Q., Qin, D., Chen, J.: Ground states and non-existence results for Choquard type equations with lower critical exponent and indefinite potentials. Nonlinear Anal. 197, Article ID 111863, 20 pages (2020)
Xia, J., Wang, Z.-Q.: Saddle solutions for the Choquard equation. Calc. Var. Partial Differ. Equ. 58, Article ID 85, 30 pages (2019)
Acknowledgements
The authors are grateful to Professor Zhi-Qiang Wang for his useful and illuminating discussions which motivate the paper. This paper is partially supported by NSFC 11771324,11811540026, 11901456 and 11901582. J. Xia is also supported by the Natural Science Foundation of Shaanxi Province (Grant No. 2020JQ–120).
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Communicated by P. H. Rabinowitz.
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