Semiclassical states for coupled Schrödinger-Maxwell equations: concentration around a sphere. (English) Zbl 1074.81023
The author studies a coupled nonlinear Schroedinger-Maxwell system of equations. In this framework, he is concerned with the existence of semiclassical states and uses a perturbation scheme in a variational setting to study the concentration of the solutions.
Reviewer: Klaus Brod (Wiesbaden)
MSC:
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81Q15 | Perturbation theories for operators and differential equations in quantum theory |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
References:
| [1] | Agmon, S.Douglis, A.Nirenberg, L., Comm. Pure Appl. Math.12, 623 (1959). · Zbl 0093.10401 |
| [2] | Ambrosetti, A.Badiale, M., Ann. Inst. H. Poincaré Anal. Non Lineare15, 233 (1998). · Zbl 1004.37043 |
| [3] | Ambrosetti, A.Badiale, M., Proc. Royal Soc. EdinburghA128, 1131 (1998). |
| [4] | Ambrosetti, A.Badiale, M.Cingolani, S., Arch. Rat. Mech. Anal.140, 285 (1997). · Zbl 0896.35042 |
| [5] | Ambrosetti, A.Malchiodi, A.Ni, W.-M., Comm. Math. Phys.235, 427 (2003). · Zbl 1072.35019 |
| [6] | Badiale, M.D’Aprile, T., Nonlinear Anal.49, 947 (2002). |
| [7] | Bechouche, P.Mauser, N. J.Poupaud, F., Comm. Pure Appl. Math.54, 851 (2001). · Zbl 1031.81024 |
| [8] | Benci, V.Fortunato, D., Top. Math. Nonlinear Anal.11, 283 (1998). · Zbl 0926.35125 |
| [9] | Benci, V.et al., Math. Z.232, 73 (1999). |
| [10] | Beresticky, H.Lions, P. L., Arch. Rat. Mech. Anal.82, 313 (1983). · Zbl 0533.35029 |
| [11] | G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, preprint. · Zbl 1064.35180 |
| [12] | T. D’Aprile, Semiclassical states for the nonlinear Schrödinger equation with the electromagnetic field, preprint. · Zbl 1128.35382 |
| [13] | T. D’Aprile and D. Mugnai, Existence of solitary waves for the nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, preprint. |
| [14] | D’Avenia, P., Adv. Nonlinear Stud.2, 177 (2002). · Zbl 1007.35090 |
| [15] | Del Pino, M.Felmer, P., Calc. Var. PDE4, 121 (1996). |
| [16] | Del Pino, M.Felmer, P., J. Funct. Anal.149, 245 (1997). · Zbl 0887.35058 |
| [17] | Floer, A.Weinstein, A., J. Funct. Anal.69, 397 (1986). · Zbl 0613.35076 |
| [18] | Frölich, J.Tsai, T. P.Yau, H. T., Comm. Math. Phys.225, 223 (2002). · Zbl 1025.81015 |
| [19] | Gidas, B.Ni, W.-M.Nirenberg, L., Advances in Mathematics Supplementary Studies7-A, ed. Nachbin, A. () pp. 369-402. |
| [20] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 1983, Springer-Verlag · Zbl 0562.35001 |
| [21] | Grillakis, M.Shatah, J.Strauss, W., J. Funct. Anal.94, 308 (1990). · Zbl 0711.58013 |
| [22] | Kwong, M. K., Arch. Rat. Mech. Anal.105, 243 (1989). · Zbl 0676.35032 |
| [23] | Li, Y. Y., Adv. Diff. Eqn.2, 955 (1997). · Zbl 1023.35500 |
| [24] | Oh, Y.-G., Comm. Math. Phys.121, 11 (1989). · Zbl 0693.35132 |
| [25] | Wang, X., Comm. Math. Phys.153, 229 (1993). · Zbl 0795.35118 |
| [26] | Strauss, W. A., Comm. Math. Phys.55, 149 (1977). · Zbl 0356.35028 |
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.
