Let be a smooth bounded simply connected domain in and let , where is the Minkowski space with the metric . Then the Lagrangean of the Maxwell-Chern-Simons-Higgs (MCSH) model is given by
where are constants, , and is the neutral scalar field which is introduced to attain self-duality.
The MCSH model was introduced as a unified system of the Abelian-Higgs (AH) model (Landau-Ginzburg model) and the Chern-Simons-Higgs (CSH) model [C. Lee, K. Lee and H. Min, Self-dual Maxwell-Chern-Simons solitons, Phys. Lett. B 252, No. 1, 79–83 (1990; Zbl 1079.58009)]. If , there exist global minimizers of the static energy functionals of the AH, CSH and MCSH models and they are achieved as solutions of self-dual equations. A static energy functional of the MCSH model reduces static energy functionals of the AH and CSH models by suitable choices of and . The solutions of the energy minimizing self-dual equation of the MCSH model converge to the solutions of the energy minimizing equations of the AH and CSH models [D. Chae, O. Yu. Imanuvilov, J. Funct. Anal. 196, No. 1, 87–118 (2002; Zbl 1079.58009); D. Chae and N. Kim, J. Differ. Equations 134, No. 1, 154–182 (1997; Zbl 0869.35094)]. In this paper, these results are extended to nonself-dual cases. If the gauge field is absent in the model, the results are similar to the self-adjoint case (§2. Th. 2.4 and 2.6). But if the gauge field is present, the tangential current of the CSH limit may be different from the original one (§3. Th. 3.5 and 3.7). The authors mention this phenomenon is closely related with the energy loss in the gauge potential. The asymptotic behavior of solutions to the Euler Lagrange equations for the static energy functional of the static MCSH model when is also studied (§4. Th. 4.8).
To study AH and CSH limits, the function spaces in the case and in the case are introduced. Here, ,
and , where
It is noted that the norm is equivalent to , because is simply connected. The subspace ,
is also introduced. Then it is shown that the MCSH energy functionals achieve their minimum on and on (Lemma 2.1 and Th. 3.3). For the minimizer in the case , the estimate is shown (Th. 2.3). An estimate on the minimum value is also given (Lemma 2.5). By using these estimates, the Maxwell limit and Chern-Simons limit are computed. In the case , estimates of minimum value is done by using the properties of (Lemma. 3.4 and 3.6). Maxwell limit and Chern-Simons limit are also computed by using these estimates. Discussions on the asymptotics for minimizers in §4 are done following F. Bethuel, H. Brézis and F. Hélein, Calc. Var. Partial Differ. Equ. 1, No. 2, 123–148 (1993; Zbl 0834.35014)].