Localized Concentration of Semi-Classical States for Nonlinear Dirac Equations
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| 003 | CHVBK | ||
| 005 | 20210128100707.0 | ||
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| 008 | 210128e20150501xx s 000 0 eng | ||
| 024 | 7 | 0 | |a 10.1007/s00205-014-0811-4 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s00205-014-0811-4 | ||
| 245 | 0 | 0 | |a Localized Concentration of Semi-Classical States for Nonlinear Dirac Equations |h [Elektronische Daten] |c [Yanheng Ding, Tian Xu] |
| 520 | 3 | |a The present paper studies concentration phenomena of the semiclassical approximation of a massive Dirac equation with general nonlinear self-coupling: $$\begin{array}{ll}-i \hbar \alpha \cdot \nabla w+a \beta w + V (x) w = g (|w|) w.\end{array}$$ - i ħ α · ∇ w + a β 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 $${\Lambda_j \subset \mathbb{R}^3}$$ Λ j ⊂ R 3 such that $${-a < \min_{\Lambda_j} V=V(x_j) < \min_{\partial \Lambda_j}V}$$ - a < min Λ j V = V ( x j ) < min ∂ Λ j V , $${x_j\in\Lambda_j}$$ x j ∈ Λ j , then the k-families of solutions $${w_\hbar^j}$$ w ħ j concentrate around x j as $${\hbar\to 0}$$ ħ → 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. | |
| 540 | |a Springer-Verlag Berlin Heidelberg, 2014 | ||
| 700 | 1 | |a Ding |D Yanheng |u Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China |4 aut | |
| 700 | 1 | |a Xu |D Tian |u Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China |4 aut | |
| 773 | 0 | |t Archive for Rational Mechanics and Analysis |d Springer Berlin Heidelberg |g 216/2(2015-05-01), 415-447 |x 0003-9527 |q 216:2<415 |1 2015 |2 216 |o 205 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00205-014-0811-4 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00205-014-0811-4 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Ding |D Yanheng |u Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Xu |D Tian |u Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Archive for Rational Mechanics and Analysis |d Springer Berlin Heidelberg |g 216/2(2015-05-01), 415-447 |x 0003-9527 |q 216:2<415 |1 2015 |2 216 |o 205 | ||