Ground state of a magnetic nonlinear Choquard equation. (English) Zbl 1420.35344
Authors’ abstract: We consider the stationary magnetic nonlinear Choquard equation \[-(\nabla + i A(x))^2 u + V(x) u = \left(\frac{1}{| x |^\alpha} \ast F(| u |)\right) \frac{f(| u |)}{| u |} u,\] where \(A : \mathbb{R}^N \rightarrow \mathbb{R}^N\) is a vector potential, \(V\) is a scalar potential, \(f : \mathbb{R} \rightarrow \mathbb{R}\) and \(F\) is the primitive of \(f\). Under mild hypotheses, we prove the existence of a ground state solution for this problem. We also prove a simple multiplicity result by applying Ljusternik-Schnirelmann methods.
Reviewer: Alessandro Selvitella (Fort Wayne)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35J20 | Variational methods for second-order elliptic equations |
| 35A15 | Variational methods applied to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
References:
| [1] | Alves, C. O.; Gao, F.; Squassina, M.; Yang, M., Singularly perturbed critical Choquard equations, J. Differential Equations, 263, 7, 3943-3988 (2017) · Zbl 1378.35113 |
| [2] | Alves, C. O.; Nobrega, A.; Yang, M., Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55, 3, 48 (2016), 28 pp · Zbl 1347.35097 |
| [3] | Alves, C. O.; Yang, M., Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations, 257, 11, 4133-4164 (2014) · Zbl 1309.35036 |
| [4] | Alves, C. O.; Yang, M., Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh Sect. A, 146, 23-58 (2016) · Zbl 1366.35050 |
| [5] | H. Bueno, O. Miyagaki, G.A. Pereira, Remarks about a generalized pseudo-relativistic Hartree equation, submitted for publication, arXiv:1802.03963; H. Bueno, O. Miyagaki, G.A. Pereira, Remarks about a generalized pseudo-relativistic Hartree equation, submitted for publication, arXiv:1802.03963 · Zbl 1433.35033 |
| [6] | 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 |
| [7] | Cingolani, S.; Clapp, M.; Secchi, M., Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6, 4, 891-908 (2013) · Zbl 1260.35198 |
| [8] | Cingolani, S.; Secchi, S.; Squassina, M., Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh, 140 A, 973-1009 (2010) · Zbl 1215.35146 |
| [9] | Felmer, P.; Quaas, A.; Tan, J., Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142, 6, 1237-1262 (2012) · Zbl 1290.35308 |
| [10] | Furtado, M. A.; Maia, L. A.; Medeiros, E. S., Positive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential, Adv. Nonlinear Stud., 8, 353-373 (2008) · Zbl 1168.35433 |
| [11] | Gao, F.; Yang, M., The Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61, 1219-1242 (2018) · Zbl 1397.35087 |
| [12] | Gao, F.; Yang, M., A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., 20, 4, 1750037 (2018), 22 pp · Zbl 1391.35126 |
| [13] | Kavian, O., Introduction à la Théorie des Points Critiques (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0797.58005 |
| [14] | Lieb, E. H.; Loss, M., (Analysis. Analysis, Graduate Studies in Mathematics, vol. 14 (2001), American Mathematical Society: American Mathematical Society Providence, Rhode Island) · Zbl 0966.26002 |
| [15] | Moroz, V.; Van Schaftingen, J., Semi-classical states for the Choquard equations, Calc. Var. Partial Differential Equations, 52, 1, 199-235 (2015) · Zbl 1309.35029 |
| [16] | Moroz, V.; Van Schaftingen, J., A guide to the Choquard equation, J. Fixed Point Theory Appl., 19, 1, 773-813 (2017) · Zbl 1360.35252 |
| [17] | Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 2, 270-291 (1992) · Zbl 0763.35087 |
| [18] | Struwe, M., A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187, 4, 511-517 (1984) · Zbl 0535.35025 |
| [19] | Szulkin, A., Ljusternik-Schnirelmann theory on \(C^1\)-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5, 2, 119-139 (1988) · Zbl 0661.58009 |
| [20] | Szulkin, A.; Weth, T., The method of Nehari manifold, (Handbook of Nonconvex Analysis and Applications (2010), Int. Press: Int. Press Somerville, MA), 597-632 · Zbl 1218.58010 |
| [21] | Willem, M., Minimax Theorems (1996), Birkhäuser: Birkhäuser Boston · Zbl 0856.49001 |
| [22] | 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 |
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.
