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Existence of multiple positive solutions for Choquard equation with perturbation. (English) Zbl 1375.35220

Summary: This paper is concerned with the following Choquard equation with perturbation: \(- \Delta u + V(x) u = (1 / | x |^\alpha \ast | u |^p) | u |^{p - 2} u + g(x)\), \(u \in H^1(\mathbb{R}^N)\), where \(N \geq 3\), \(\alpha \in(0, N)\), and \(2 -(\alpha / N) < p <(2 N - \alpha) /(N - 2)\). This kind of equations is well known as the Choquard or nonlinear Schrödinger-Newton equation. Under some assumptions for the functions \(V(x)\), we prove the existence of multiple positive solutions of the equation. Moreover, we also show that these results still hold for more generalized Choquard equation with perturbation.

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
35B09 Positive solutions to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35R09 Integro-partial differential equations
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[1] Pekar, S., Untersuchungen über die Elektronentheorie der Kristalle, (1954), Berlin, Germany: Akademie, Berlin, Germany · Zbl 0058.45503
[2] Lieb, E. H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Applied Mathematics, 57, 2, 93-105, (1976/1977) · Zbl 0369.35022
[3] Jones, K. R. W., Newtonian quantum gravity, Australian Journal of Physics, 48, 6, 1055-1081, (1995)
[4] Moroz, I. M.; Penrose, R.; Tod, P., Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical and Quantum Gravity, 15, 9, 2733-2742, (1998) · Zbl 0936.83037
[5] Lions, P.-L., The Choquard equation and related questions, Nonlinear Analysis, 4, 6, 1063-1072, (1980) · Zbl 0453.47042
[6] Ackermann, N., On a periodic Schrödinger equation with nonlocal superlinear part, Mathematische Zeitschrift, 248, 2, 423-443, (2004) · Zbl 1059.35037
[7] Wei, J.-C.; Winter, M., Strongly interacting bumps for the Schrödinger–Newton equations, Journal of Mathematical Physics, 50, 1, (2009) · Zbl 1189.81061
[8] Ma, L.; Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Archive for Rational Mechanics and Analysis, 195, 2, 455-467, (2010) · Zbl 1185.35260
[9] Clapp, M.; Salazar, D., Positive and sign changing solutions to a nonlinear Choquard equation, Journal of Mathematical Analysis and Applications, 407, 1, 1-15, (2013) · Zbl 1310.35114
[10] Moroz, V.; Van Schaftingen, J., Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, Journal of Functional Analysis, 265, 2, 153-184, (2013) · Zbl 1285.35048
[11] Moroz, V.; Van Schaftingen, J., Existence of groundstates for a class of nonlinear choquard equations, Transactions of the American Mathematical Society, 367, 9, 6557-6579, (2015) · Zbl 1325.35052
[12] Moroz, V.; Van Schaftingen, J., Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, Journal of Differential Equations, 254, 8, 3089-3145, (2013) · Zbl 1266.35083
[13] Cingolani, S.; Secchi, S., Ground states for the pseudo-relativistic Hartree equation with external potential, Proceedings of the Royal Society of Edinburgh Section A—Mathematics, 145, 1, 73-90, (2015) · Zbl 1320.35300
[14] Cingolani, S.; Clapp, M.; Secchi, S., Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete and Continuous Dynamical Systems S, 6, 4, 891-908, (2013) · Zbl 1260.35198
[15] Cingolani, S.; Secchi, S.; Squassina, M., Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140, 5, 973-1009, (2010) · Zbl 1215.35146
[16] Cingolani, S.; Secchi, S., Semiclassical analysis for pseudo-relativistic Hartree equations, Journal of Differential Equations, 258, 12, 4156-4179, (2015) · Zbl 1319.35204
[17] Moroz, V.; van Schaftingen, J., Semi-classical states for the Choquard equation, Calculus of Variations and Partial Differential Equations, 52, 1-2, 199-235, (2015) · Zbl 1309.35029
[18] Bartsch, T.; Wang, Z. Q., Existence and multiplicity results for some superlinear elliptic problems on \(\mathbb{R}^N\), Communications in Partial Differential Equations, 20, 9-10, 1725-1741, (1995) · Zbl 0837.35043
[19] Bartsch, T.; Pankov, A.; Wang, Z.-Q., Nonlinear Schrödinger equations with steep potential well, Communications in Contemporary Mathematics, 3, 4, 549-569, (2001) · Zbl 1076.35037
[20] Lieb, E.-H.; Loss, M., Analysis. Analysis, Graduate Studies in Mathematics, 14, (2001), Providence, RI, USA: American Mathematical Society, Providence, RI, USA
[21] Tarantello, G., On nonhomogeneous elliptic equations involving critical Sobolev exponent, Annales de l’Institut Henri Poincaré Analyse Non Linéaire, 9, 3, 281-304, (1992) · Zbl 0785.35046
[22] Chen, C.-Y.; Kuo, Y.-C.; Wu, T.-F., The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, Journal of Differential Equations, 250, 4, 1876-1908, (2011) · Zbl 1214.35077
[23] Wu, T.-F., Multiple positive solutions for a class of concave-convex elliptic problems in \(\mathbb{R}^N\) involving sign-changing weight, Journal of Functional Analysis, 258, 1, 99-131, (2010) · Zbl 1182.35119
[24] Qi, Z. X.; Zhang, Z. T., Existence of multiple solutions to a class of nonlinear Schrödinger system with external sources terms, Journal of Mathematical Analysis and Applications, 420, 2, 972-986, (2014) · Zbl 1300.35133
[25] Chang, K. C., Methods in Nonlinear Analysis. Methods in Nonlinear Analysis, Springer Monographs in Mathematics, (2005), Berlin, Germany: Springer, Berlin, Germany · Zbl 1081.47001
[26] Willem, M., Minimax Theorems. Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, (1996), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA
[27] Riesz, M., L’intégrale de Riemann-Liouville et le probléme de Cauchy, Acta Mathematica, 81, 1, 1-222, (1949) · Zbl 0033.27601
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