## Localized concentration of semi-classical states for nonlinear Dirac equations.(English)Zbl 1309.35102

Summary: The present paper studies concentration phenomena of the semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: $-i \hbar \alpha \cdot \nabla w+a \beta w + V (x) w = g (| w| ) w.$ Compared with some existing issues, the most interesting results obtained here are twofold: the solutions concentrating around local minima of the external potential; and the nonlinearities assumed to be either super-linear or asymptotically linear at the infinity. As a consequence one sees that, if there are $$k$$ bounded domains $$\Lambda_j \subset \mathbb{R}^3$$ such that $$-a < \min_{\Lambda_j} V=V(x_j) < \min_{\partial \Lambda_j}V$$, $$x_j\in\Lambda_j$$, then the $$k$$-families of solutions $$w_\hbar^j$$ concentrate around $$x_j$$ as $$\hbar\to 0$$, respectively. The proof relies on variational arguments: the solutions are found as critical points of an energy functional. The Dirac operator has a continuous spectrum which is not bounded from below and above, hence the energy functional is strongly indefinite. A penalization technique is developed here to obtain the desired solutions.

### MSC:

 35Q41 Time-dependent Schrödinger equations and Dirac equations 81V10 Electromagnetic interaction; quantum electrodynamics 35P15 Estimates of eigenvalues in context of PDEs 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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### References:

 [1] Ackermann, N., A nonlinear superposition principle and multibump solution of periodic Schrödinger equations, J. Funct. Anal., 234, 423-443, (2006) · Zbl 1126.35057 [2] Ambrosetti, A.; Badiale, M.; Cignolani, S., Semi-classical states of nonlinear shrödinger equations. arch, Rational Mech. Anal., 140, 285-300, (1997) · Zbl 0896.35042 [3] Ambrosetti, A.; Felli, V.; Malchiodi, A., Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7, 117-144, (2005) · Zbl 1064.35175 [4] Byeon, J.; Jeanjean, L., Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal., 185, 185-200, (2007) · Zbl 1132.35078 [5] Byeon, J.; Wang, Z.Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 165, 295-316, (2002) · Zbl 1022.35064 [6] Dautray R., Lions J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 3. Springer, Berlin (1990) · Zbl 0683.35001 [7] Del Pino, M.; Felmer, P., Local mountain passes for semilinear ellipitc problems in unbounded domains, Calc. Var. Partial Differential Equations, 4, 121-137, (1996) · Zbl 0844.35032 [8] Del Pino, M., Felmer, P.: Multi-peak bound states for nonlinear Schrödinger equations, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, vol. 15. No. 2. Elsevier, Masson, pp. 127-149, (1998) · Zbl 0524.35002 [9] Ding Y.H.: Variational Methods for Strongly Indefinite Problems, Interdiscip. Math. Sci., vol. 7. World Scientific Publ, Singapore (2007) · Zbl 1133.49001 [10] Ding, Y.H., Semi-classical ground states concentrating on the nonlinear potentical for a Dirac equation, J. Differ. Equ., 249, 1015-1034, (2010) · Zbl 1193.35161 [11] Ding, Y.H., Lee, C., Ruf, B.: On semiclassical states of a nonlinear Dirac equation. Proc. R. Soc. Edinburgh Sect. A Math. 143.04, 765-790 (2013) · Zbl 1304.35590 [12] Ding, Y.H.; Liu, Xiaoying, Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differ. Equ., 252, 4962-4987, (2012) · Zbl 1236.35133 [13] Ding, Y.H.; Ruf, B., Existence and concentration of semi-classical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44, 3755-3785, (2012) · Zbl 1259.35171 [14] Ding, Y.H.; Wei, J.C., Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20, 1007-1032, (2008) · Zbl 1170.35082 [15] Ding, Y.H.; Wei, J.C.; Xu, T., Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system, J. Math. Phys., 54, 061505, (2013) · Zbl 1282.81073 [16] Ding, Y.H.; Xu, T., On semi-classical limits of ground states of a nonlinear Maxwell-Dirac system, Calc. Var. Partial Differ. Equ., 51, 17-44, (2014) · Zbl 1297.35197 [17] Ding, Y.H.; Xu, T., On the concentration of semi-classical states for a nonlinear Dirac-Klein-Gordon system, J. Differ. Equ., 256, 1264-1294, (2014) · Zbl 1283.35099 [18] Esteban, M.J.; Séré, E., Stationary states of the nonlinear Dirac equation: a variational approach, Commun. Math. Phys., 171, 323-350, (1995) · Zbl 0843.35114 [19] Finkelstein, R.; LeLevier, R.; Ruderman, M., Nonlinear spinor fields, Phys. Rev., 83, 326-332, (1951) · Zbl 0043.21603 [20] Finkelstein, R.; Fronsdal, C.; Kaus, P., Nonlinear spinor field, Phys. Rev., 103, 1571-1579, (1956) · Zbl 0073.44705 [21] Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69, 397-408, (1986) · Zbl 0613.35076 [22] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998) · Zbl 0361.35003 [23] Ivanenko, D.D., Notes to the theory of interaction via particles, Zh.Éksp. Teor. Fiz., 8, 260-266, (1938) · Zbl 0021.27604 [24] Jeanjean, L.; Tanaka, K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differ. Equ., 21, 287-318, (2004) · Zbl 1060.35012 [25] Lions, P.L.: The concentration-compactness principle in the calculus of variations: the locally compact case, Part II. AIP Anal. non linéaire1, 223-283 · Zbl 0704.49004 [26] Miguel, R.; Hugo, T., Solutions with multiple spike patterns for an elliptic system, Calc. Var. Partial Differ. Equ., 31, 1-25, (2008) · Zbl 1143.35027 [27] Miguel, R.; Yang, J.F., Spike-layered solutions for an elliptic system with Neumann boundary conditions, Trans. Am. Math. Soc., 357, 3265-3284, (2005) · Zbl 1136.35046 [28] Oh, Y.G., Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class ($$V$$)_{$$a$$}, Commun. Partial Differ. Equ., 13, 1499-1519, (1988) · Zbl 0702.35228 [29] Oh, Y.G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys., 131, 223-253, (1990) · Zbl 0753.35097 [30] Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Amer. Math. Soc., Providence, 1986 · Zbl 0609.58002 [31] Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 270-291, (1992) · Zbl 0763.35087 [32] Stein E.M.: Singular integrals and differentiability properties of functions, vol. 2. Princeton University Press, Princeton (1970) · Zbl 0207.13501 [33] Simon, B., Schrödinger semigroups, Bull. Am. Math. Soc. (N.S.), 7, 447-526, (1982) · Zbl 0524.35002 [34] Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153, 229-244, (1993) · Zbl 0795.35118
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