## Infinitely many bound state solutions of Choquard equations with potentials.(English)Zbl 1401.35055

Summary: In this paper, we consider the following Choquard equation $\begin{cases} - \Delta u+a(x)u=(I_\alpha *|u|^p)|u|^{p-2}u, &\quad x\in {\mathbb {R}}^N,\\ u(x)\rightarrow 0,&\quad |x|\rightarrow +\infty , \end{cases} \tag{CH}$ where $$N\geq 3, I_\alpha$$ is a Riesz potential, $$\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-2}$$ and $$a(x)$$ is a given nonnegative potential function. Under some assumptions of asymptotic properties on $$a(x)$$ at infinity and according to a concentration compactness argument, we obtain infinitely many solutions of (CH), whose energy can be arbitrarily large.

### MSC:

 35J20 Variational methods for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations
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### References:

 [1] Ackermann, N., On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248, 423-443, (2004) · Zbl 1059.35037 [2] Alves, CO; Nobrega, A.; Yang, M-B, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var., 55, 48, (2016) · Zbl 1347.35097 [3] Cerami, G.; Devillanova, G.; Solimini, S., Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differ. Equ., 23, 139-168, (2005) · Zbl 1078.35113 [4] Chang, K.-C.: Infinite Dimensional Morse Theory and Multiple Solution Problem. Birkhäuser, Boston (1993) [5] Cingolani, S.; Clapp, M.; Secchi, S., Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63, 233-248, (2012) · Zbl 1247.35141 [6] Clapp, M.; Salazar, D., Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407, 1-15, (2013) · Zbl 1310.35114 [7] Clapp, M.; Salazar, D., Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407, 1-15, (2013) · Zbl 1310.35114 [8] Devillanova, G.; Solimini, S., Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differ. Equ., 7, 1257-1280, (2002) · Zbl 1208.35048 [9] Ghimenti, M.; Schaftingen, J., Nodal solutions for the Choquard equation, J. Funct. Anal., 271, 107-135, (2016) · Zbl 1345.35046 [10] Ghimenti, M.; Schaftingen, J., Nodal solutions for the Choquard equation, J. Funct. Anal., 271, 107-135, (2016) · Zbl 1345.35046 [11] Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93-105 (1976/1977) · Zbl 0369.35022 [12] Lions, P.L.: The concentration-compactness method in the calculus of variations. The locally compact case, parts 1 and 2. Ann. Inst. H. Poincaré Anual. Non Linéair 1, 109-145, 223-283 (1984) [13] Lions, PL, The Choquard equation and related questions, Nonlinear Anal., 4, 1063-1072, (1980) · Zbl 0453.47042 [14] Ma, L.; Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rat. Mech. Anal., 195, 455-467, (2010) · Zbl 1185.35260 [15] Moroz, V., Van Schaftingen, J.: Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent. Commun. Contemp. Math. 17(5), 155005,12 (2015) · Zbl 1326.35109 [16] Moroz, V.; Schaftingen, J., Groundstates of nonlinear Choquard equations: existenc, qualitative properties and decay asymptotics, J. Funct. Anal., 265, 153-184, (2013) · Zbl 1285.35048 [17] Moroz, V.; Schaftingen, J., Existence of groundstates for a class of nonlinear Choquard equations, Trans. Am. Math. Soc., 367, 6557-6579, (2015) · Zbl 1325.35052 [18] Moroz, V.; Schaftingen, J., Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52, 199-235, (2015) · Zbl 1309.35029 [19] Moroz, IM; Penrose, R.; Tod, P., Spherically-symmetric solutions of the Schrödinger-Newton equations, Class. Quantum Grav., 15, 2733-2742, (1998) · Zbl 0936.83037 [20] Pekar, S.: Untersuchung über die Elektronentheorie der Kristalle. Akademie Verlag, Berlin (1954) · Zbl 0058.45503 [21] Penrose, R., On gravity’s role in quantum state reduction, Gen. Rel. Gravit., 28, 581-600, (1996) · Zbl 0855.53046 [22] Secchi, S., A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72, 3842-3856, (2010) · Zbl 1187.35254 [23] Struwe, M.: Variational Methods. Applications to Nonlinear Partial Differential equations and Hamiltonian Systems. Springer, Berlin (1996) · Zbl 0864.49001 [24] Wei, J-C; Winter, M., Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 012905, 22p, (2009) · Zbl 1189.81061 [25] Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996) · Zbl 0856.49001 [26] Zhang, H.; Xu, J-X; Zhang, F-B, Existence and multiplicity of solutions for a generalized Choquard equation, Comput. Math. Appl, 73, 1803-1814, (2017) · Zbl 1375.35134
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