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Existence of semiclassical solutions for some critical Dirac equation. (English) Zbl 1456.81166

Summary: In this paper, we study the following critical Dirac equation \(- i \varepsilon \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + V(x) u = P(x) f(| u |) u + Q(x) | u | u, x \in \mathbb{R}^3\), where \(\varepsilon > 0\) is a small parameter; \(a > 0\) is a constant; \( \alpha_1, \alpha_2, \alpha_3\), and \(\beta\) are \(4 \times 4\) Pauli-Dirac matrices; and \(V, P, Q\), and \(f\) are continuous but are not necessarily of class \(\mathcal{C}^1\). We prove the existence and concentration of semiclassical solutions under suitable assumptions on the potentials \(V(x), P(x)\), and \(Q(x)\) by using variational methods. We also show the semiclassical solutions \(\omega_\varepsilon\) with maximum points \(x_\varepsilon\) of |\( \omega_\varepsilon\)| concentrating at a special set \(\mathcal{H}_P\) characterized by \(V(x), P(x)\), and \(Q(x)\) and for any sequence \(x_\varepsilon \to x_0 \in \mathcal{H}_P, v_\varepsilon(x) := \omega_\varepsilon(\varepsilon x + x_\varepsilon)\) converges in \(W^{1, q}(\mathbb{R}^3, \mathbb{C}^4)\) for \(q \geq 2\) to a ground state solution \(u\) of \(- i \sum_{k = 1}^3 \alpha_k \partial_k u + a \beta u + V(x_0) u = P(x_0) f(| u |) u + Q(x_0) | u | u, \text{in} \mathbb{R}^3\). Finally, we estimate the exponential decay properties of solutions.
©2021 American Institute of Physics

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
15A66 Clifford algebras, spinors
35A15 Variational methods applied to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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[1] Finkelstein, R.; Fronsdal, C.; Kaus, P., Nonlinear spinor field, Phys. Rev., 103, 1571-1579 (1956) · Zbl 0073.44705
[2] Finkelstein, R.; LeLevier, R.; Ruderman, M., Nonlinear spinor fields, Phys. Rev., 83, 326-332 (1951) · Zbl 0043.21603
[3] Ivanenko, D., Notes to the theory of interaction via particles, Zh. Éksp. Teor. Fiz., 8, 260-266 (1938) · Zbl 0021.27604
[4] Ranada, A., On nonlinear classical Dirac fields and quantum physics, Old and New Questions in Physics, Cosmology, Philosophy, and Theoretical Biology, 363-376 (1983), Plenum: Plenum, New York
[5] Thaller, B., The Dirac Equation (1992), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0881.47021
[6] Ding, Y., Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differ. Equations, 249, 5, 1015-1034 (2010) · Zbl 1193.35161
[7] Ding, Y.; Liu, X., Semi-classical limits of ground states of a nonlinear Dirac equation, J. Differ. Equations, 252, 9, 4962-4987 (2012) · Zbl 1236.35133
[8] Rabinowitz, P. H., On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 2, 270-291 (1992) · Zbl 0763.35087
[9] Ding, Y.; Ruf, B., Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal., 44, 6, 3755-3785 (2012) · Zbl 1259.35171
[10] Ding, Y.; Xu, T., Localized concentration of semi-classical states for nonlinear Dirac equations, Arch. Ration. Mech. Anal., 216, 2, 415-447 (2015) · Zbl 1309.35102
[11] Byeon, J.; Wang, Z.-Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165, 4, 295-316 (2002) · Zbl 1022.35064
[12] Ding, Y.; Yu, Y., The concentration beavior of ground state solutions for nonlinear Dirac equation, Nonlinear Anal., 195, 111738 (2020) · Zbl 1435.35325
[13] Szulkin, A.; Weth, T., Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257, 12, 3802-3822 (2009) · Zbl 1178.35352
[14] Szulkin, A.; Weth, T., The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, 597-632 (2010), International Press: International Press, Somerville, MA · Zbl 1218.58010
[15] Wang, Z.; Zhang, X., An infinite sequence of localized semiclassical bound states for nonlinear Dirac equations, Calc. Var. Partial Differ. Equations, 57, 2, 56 (2018) · Zbl 1403.35258
[16] Ding, Y.; Wei, J., Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20, 8, 1007-1032 (2008) · Zbl 1170.35082
[17] Figueiredo, G. M.; Pimenta, M. T. O., Existence of ground state solutions to Dirac equations with vanishing potentials at infinity, J. Differ. Equations, 262, 1, 486-505 (2017) · Zbl 1352.35141
[18] Merle, F., Existence of stationary states for nonlinear Dirac equations, J. Differ. Equations, 74, 1, 50-68 (1988) · Zbl 0696.35154
[19] Zhang, J.; Zhang, W.; Zhao, F., Existence and exponential decay of ground-state solutions for a nonlinear Dirac equation, Z. Angew. Math. Phys., 69, 5, 116 (2018) · Zbl 1401.35070
[20] Zhao, F.; Ding, Y., Infinitely many solutions for a class of nonlinear Dirac equations without symmetry, Nonlinear Anal., 70, 2, 921-935 (2009) · Zbl 1152.35501
[21] Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153, 229-244 (1993) · Zbl 0795.35118
[22] Wang, X.; Zeng, B., On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28, 633-655 (1997) · Zbl 0879.35053
[23] Cingolani, S.; Lazzo, M., Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differ. Equations, 160, 118-138 (2000) · Zbl 0952.35043
[24] Ackermann, N., A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234, 2, 277-320 (2006) · Zbl 1126.35057
[25] Dautray, R.; Lions, J., Mathematical Analysis and Numerical Methods for Science and Technology (1990), Springer-Verlag: Springer-Verlag, Berlin
[26] Zhang, X., On the concentration of semiclassical states for nonlinear Dirac equations, Discrete Contin. Dyn. Syst., 38, 11, 5389-5413 (2018) · Zbl 1401.35256
[27] Bartsch, T.; Ding, Y., Solutions of nonlinear Dirac equations, J. Differ. Equations, 226, 1, 210-249 (2006) · Zbl 1126.49003
[28] Pankov, A., Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73, 259-287 (2005) · Zbl 1225.35222
[29] Lions, P. L., The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1, 4, 223-283 (1984) · Zbl 0704.49004
[30] Willem, M., Minimax Theorems (1996), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc., Boston, MA
[31] Ding, Y., Variational Methods for Strongly Indefinite Problems (2007), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ · Zbl 1133.49001
[32] Esteban, M. J.; Séré, E., Stationary states of the nonlinear Dirac equation: A variational approach, Commun. Math. Phys., 171, 2, 323-350 (1995) · Zbl 0843.35114
[33] Ding, Y.; Wei, J.; Xu, T., Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system, J. Math. Phys., 54, 6, 061505 (2013) · Zbl 1282.81073
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