Multiple solutions to a magnetic nonlinear Choquard equation. (English) Zbl 1247.35141

The authors prove the existence of multiple complex valued solutions of the stationary magnetic nonlinear Choquard equation \[ \begin{split} (-i \nabla + A(x))^2 u + V(x)u = \left(\frac{1}{|x|^\alpha} \star |u|^p \right) |u|^{p-2} u, \\ u \in L^2( \mathbb{R}^N,\mathbb{C}), \enspace \nabla u + iA(x)u \in L^2(\mathbb{R}^N,\mathbb{C}^N),\end{split} \] where \(A: \mathbb{R}^N \rightarrow \mathbb{R}^N\) is a real-valued vector potential, \(V: \mathbb{R}^N \rightarrow \mathbb{R}\) is a real-valued scalar potential, \(N \geq 3\), \(\alpha \in (0,N)\), and \( 2 - (\alpha / N) < p < (2N - \alpha) / (N -2)\), for the case that both the vector and the scalar potential satisfy the symmetry conditions \(A(gx) = gA(x)\), \(V(gx) = V(x)\) for all \(g \in G, x \in \mathbb{R}^N\), where \(G\) is a closed subgroup of the group \(O(N)\) of linear isometries of \( \mathbb{R}^N\). Precisely, one finds solutions that satisfy the symmetry condition \(u(gx) = \tau(g) u(x)\), where \(\tau: G \rightarrow S^1\) is a known group holomorphism into the unit complex numbers \(S^1\). The main results are formulated in two theorems whose proofs are based on variational methods. For the lack of compactness problem they refer to P.-L. Lions [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109–145 (1984; Zbl 0541.49009)].
This equation is a generalization of the stationary Choquard equation \[ -\Delta u + u = \left(\frac{1}{|x|} \star |u|^2 \right) u, \enspace u \in H^1(\mathbb{R}^3) \] that comes up in the Hartree-Fock theory of plasma. Existence and uniqueness results for this equation had been already established by E. H. Lieb [Studies Appl. Math. 57, 93–105 (1977; Zbl 0369.35022)] and P.-L. Lions [Nonlinear Anal., Theory Methods Appl. 4, 1063–1072 (1980; Zbl 0453.47042)].


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q40 PDEs in connection with quantum mechanics
35J20 Variational methods for second-order elliptic equations
35B06 Symmetries, invariants, etc. in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI arXiv


[1] Ackermann N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004) · Zbl 1059.35037
[2] Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) · Zbl 0273.49063
[3] Cingolani S., Clapp M.: Intertwining semiclassical bound states to a nonlinear magnetic equation. Nonlinearity 22, 2309–2331 (2009) · Zbl 1173.35678
[4] Cingolani, S., Clapp, M., Secchi, S.: Intertwining semiclassical solutions to a Schrödinger-Newton system, preprint · Zbl 1260.35198
[5] Cingolani S., Secchi S., Squassina M.: Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. Roy. Soc. Edinburgh 140 A, 973–1009 (2010) · Zbl 1215.35146
[6] Clapp M., Puppe D.: Critical point theory with symmetries. J. Reine Angew. Math. 418, 1–29 (1991) · Zbl 0722.58011
[7] tom Dieck T.: Transformation Groups. Walter de Gruyter, Berlin-New York (1987) · Zbl 0611.57002
[8] Esteban M.J., Lions P.L.: Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. In: (eds) Partial Differential Equations and the Calculus of Variations, Vol. 1, pp. 401–449. Progr. Nonlinear Differential Equations Appl., 1, Birkhäuser Boston, Boston, MA (1989)
[9] Fadell E., Husseini S., Rabinowitz P.H.: Borsuk-Ulam theorems for $${\(\backslash\)mathbb{S}\^{1} }$$ -actions and applications. Trans. Am. Math. Soc. 274, 345–359 (1982) · Zbl 0506.58010
[10] Fröhlich, J., Lenzmann, E.: Mean-field limit of quantum Bose gases and nonlinear Hartree equation. In: Séminaire: Équations aux Dérivées Partielles 2003–2004, Exp. No. XIX, pp. 26. École Polytech., Palaiseau
[11] Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in $${\(\backslash\)mathbb{R}\^{N} }$$ , Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7A. pp. 369–402. Academic Press, New York–London
[12] Ginibre J., Velo G.: On a class of nonlinear Schrödinger equations with nonlocal interaction. Math. Z. 170, 109–136 (1980) · Zbl 0415.35065
[13] Lieb E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977) · Zbl 0369.35022
[14] Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Math. vol. 14. American Mathematical Society, Providence RI (1997)
[15] Lieb E.H., Simon B.: The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys. 53, 185–194 (1977)
[16] Lions P.-L.: The Choquard equation and related questions. Nonlinear Anal. TMA 4, 1063–1073 (1980) · Zbl 0453.47042
[17] Lions, P.-L.: The concentration-compacteness principle in the calculus of variations. The locally compact case. Ann. Inst. Henry Poincaré, Analyse Non Linéaire vol. 1. pp. 109–145 and 223–283 (1984)
[18] Lions, P.-L.: Symmetries and the concentration-compacteness method. In: Nonlinear Variational Problems, pp. 47–56. Pitman, London, (1985)
[19] Ma L., Zhao L.: Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195, 455–467 (2010) · Zbl 1185.35260
[20] Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger-Newton equations. Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity vol. 15. pp. 2733–2742 (1998) · Zbl 0936.83037
[21] Moroz I.M., Tod P.: An analytical approach to the Schrödinger-Newton equations. Nonlinearity 12, 201–216 (1999) · Zbl 0942.35077
[22] Nolasco M.: Breathing modes for the Schrödinger–Poisson system with a multiple–well external potential. Commun. Pure Appl. Anal. 9, 1411–1419 (2010) · Zbl 1202.35304
[23] Palais R.: The principle of symmetric criticallity. Comm. Math. Phys. 69, 19–30 (1979) · Zbl 0417.58007
[24] Penrose R.: On gravity’s role in quantum state reduction. Gen. Rel. Grav. 28, 581–600 (1996) · Zbl 0855.53046
[25] Penrose R.: Quantum computation, entanglement and state reduction. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356, 1927–1939 (1998) · Zbl 1152.81659
[26] Penrose R.: The Road to Reality. A Complete Guide to the Laws of the Universe. Alfred A. Knopf Inc., New York (2005) · Zbl 1188.00007
[27] Struwe M.: Variational Methods. Springer, Berlin-Heidelberg (1996)
[28] Secchi S.: A note on Schrödinger–Newton systems with decaying electric potential. Nonlinear Anal. 72, 3842–3856 (2010) · Zbl 1187.35254
[29] Tod P.: The ground state energy of the Schrödinger-Newton equation. Phys. Lett. A 280, 173–176 (2001) · Zbl 0984.81024
[30] Wei J., Winter M.: Strongly interacting bumps for the Schrödinger–Newton equation. J. Math. Phys. 50, 012905 (2009) · Zbl 1189.81061
[31] Willem M.: Minimax theorems. PNLDE vol. 24. Birkhäuser, Boston-Basel-Berlin (1996) · Zbl 0856.49001
[32] Zhang Z., Küpper T., Hu A., Xia H.: Existence of a nontrivial solution for Choquard’s equation. Acta Math. Sci. Ser. B Engl. Ed. 26, 460–468 (2006) · Zbl 1152.35379
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.