Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation
Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation
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DOI：
10.1016/j.jde.2010.03.022
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发表时间：
2010-09
影响因子：
2.4
通讯作者：
Yanheng Ding
中科院分区：
文献类型：
--
作者：
Yanheng Ding
研究了x∈R3非线性Dirac方程最小能量解的半经典极限。由于狄拉克算子上下无界，其伴生能量泛函是强不定的，并且由于问题是在全局空间R3中考虑的，因此不满足Palais-Smale条件。在使用变分法时，出现了新的现象和数学兴趣。证明了该方程在ε>0小时具有最小能量解，并且这些解收敛于关联极限问题的最小能量解，并在一定意义上集中于非线性势P(x)在ε→0时的最大值。
We study the semi-classical limit of the least energy solutions to the nonlinear Dirac equation for x∈R3. Since the Dirac operator is unbounded from below and above, the associate energy functional is strongly indefinite, and since the problem is considered in the global space R3, the Palais–Smale condition is not satisfied. New phenomena and mathematical interests arise in the use of the calculus of variations. We prove that the equation has the least energy solutions for all ε>0 small, and additionally these solutions converge to the least energy solutions of the associate limit problem and concentrate to the maxima of the nonlinear potential P(x) in certain sense as ε→0.