## 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)].

### MSC:

 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

### Citations:

Zbl 0541.49009; Zbl 0369.35022; Zbl 0453.47042
Full Text:

### References:

 [1] Ackermann N.: On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248, 423–443 (2004) · Zbl 1059.35037 · doi:10.1007/s00209-004-0663-y [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 · doi:10.1016/0022-1236(73)90051-7 [3] Cingolani S., Clapp M.: Intertwining semiclassical bound states to a nonlinear magnetic equation. Nonlinearity 22, 2309–2331 (2009) · Zbl 1173.35678 · doi:10.1088/0951-7715/22/9/013 [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 · doi:10.1017/S0308210509000584 [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) · doi:10.1007/BF01214768 [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) · doi:10.1007/BF01609845 [16] Lions P.-L.: The Choquard equation and related questions. Nonlinear Anal. TMA 4, 1063–1073 (1980) · Zbl 0453.47042 · doi:10.1016/0362-546X(80)90016-4 [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 · doi:10.1007/s00205-008-0208-3 [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 · doi:10.1088/0951-7715/12/2/002 [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 · doi:10.3934/cpaa.2010.9.1411 [23] Palais R.: The principle of symmetric criticallity. Comm. Math. Phys. 69, 19–30 (1979) · Zbl 0417.58007 · doi:10.1007/BF01941322 [24] Penrose R.: On gravity’s role in quantum state reduction. Gen. Rel. Grav. 28, 581–600 (1996) · Zbl 0855.53046 · doi:10.1007/BF02105068 [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 · doi:10.1098/rsta.1998.0256 [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 · doi:10.1016/j.na.2010.01.021 [29] Tod P.: The ground state energy of the Schrödinger-Newton equation. Phys. Lett. A 280, 173–176 (2001) · Zbl 0984.81024 · doi:10.1016/S0375-9601(01)00059-7 [30] Wei J., Winter M.: Strongly interacting bumps for the Schrödinger–Newton equation. J. Math. Phys. 50, 012905 (2009) · Zbl 1189.81061 · doi:10.1063/1.3060169 [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
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