Existence and concentration of solutions for a nonlinear Choquard equation. (English) Zbl 1322.35031
Summary: In this paper, we consider the nonlinear Choquard equation
\[
-\Delta u+(1+\mu g(x))u=(K_\alpha(x)\ast |u|^p)|u|^{p-2}u,\quad x\in\mathbb R^N,
\]
where \(N\geq 3\), \(\alpha\in(0, N)\), \(p\in(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2})\), \(\mu>0\) is a parameter, \(K_\alpha(x)\) is the Riesz potential and \(g(x)\) is a nonnegative continuous potential. Under some assumptions on \(g(x)\), we obtain the existence of ground state solutions and concentration results by using the critical point theory.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
Keywords:
nonlinear Choquard equation; critical point theory; ground state solution; concentration behaviorReferences:
| [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] | Brézis H., Lieb E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486-490 (1983) · Zbl 0526.46037 · doi:10.2307/2044999 |
| [3] | Bartsch T., Wang Z.Q.: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys. 51, 366-384 (2000) · Zbl 0972.35145 · doi:10.1007/PL00001511 |
| [4] | 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 · doi:10.1007/s00033-011-0166-8 |
| [5] | Cingolani S., Clapp M., Secchi S.: Intertwining semiclassical solutions to a Schrödinger-Newton system. Discret. Contin. Dyn. Syst. Ser. S 6, 891-908 (2013) · Zbl 1260.35198 |
| [6] | Cingolani S., Secchi S., Squassina M.: Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities. Proc. R. Soc. Edinburgh Sect. A 140, 973-1009 (2010) · Zbl 1215.35146 · doi:10.1017/S0308210509000584 |
| [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 · doi:10.1016/j.jmaa.2013.04.081 |
| [8] | Lieb, E.H., Loss, M.: Analysis, 2nd edn. Graduate Studies in Mathematics 14, AMS, USA (2001) · Zbl 1285.35048 |
| [9] | 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 |
| [10] | Lieb E.H., Simon B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys 53, 185-194 (1977) · doi:10.1007/BF01609845 |
| [11] | Lions P.L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063-1072 (1980) · Zbl 0453.47042 · doi:10.1016/0362-546X(80)90016-4 |
| [12] | Lions P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case. Part I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109-145 (1984) · Zbl 0541.49009 |
| [13] | Lü D.: A note on Kirchhoff-type equations with Hartree-type nonlinearities. Nonlinear Anal. 99, 35-48 (2014) · Zbl 1286.35108 · doi:10.1016/j.na.2013.12.022 |
| [14] | 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 |
| [15] | Moroz V., Van Schaftingen J.: Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal 265, 153-184 (2013) · Zbl 1285.35048 · doi:10.1016/j.jfa.2013.04.007 |
| [16] | 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 |
| [17] | Penrose R.: On gravity’s role in quantum state reduction. Gen. Relativ. Gravit. 28, 581-600 (1996) · Zbl 0855.53046 · doi:10.1007/BF02105068 |
| [18] | Wei J., Winter M.: Strongly interacting bumps for the Schrödinger-Newton equations. J. Math. Phys. 50, 012905 (2009) · Zbl 1189.81061 · doi:10.1063/1.3060169 |
| [19] | Willem, M.: Minimax theorems. In: Progress in nonlinear differential equations and their applications, vol. 24. Birkhäuser, Boston (1996) · Zbl 0856.49001 |
| [20] | Yang M., Wei Y.: Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities. J. Math. Anal. Appl. 403, 680-694 (2013) · Zbl 1294.35149 · doi:10.1016/j.jmaa.2013.02.062 |
| [21] | Zhang Z., Tassilo K., Hu A., Xia H.: Existence of a nontrivial solution for Choquard’s equation. Acta Math. Sci. 26B(3), 460-468 (2006) · Zbl 1152.35379 · doi:10.1016/S0252-9602(06)60070-2 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
