非周期Dirac 方程的稳态解
摘要
This paper is concerned with solutions to the Dirac equation: -<italic>i</italic>∑<italic>α<sub>k</sub>∂<sub>k</sub>u</italic>+<italic>αβu</italic>+ <italic>M</italic>(<italic>x</italic>)<italic>u</italic> = <italic>g</italic>(<italic>x</italic>, |<italic>u</italic>|)<italic>u</italic> . Here <italic>M</italic>(<italic>x</italic>) is a general potential and <italic>g</italic>(<italic>x</italic>, |<italic>u</italic>|)<italic>u</italic> is super linear in <italic>u</italic> at infinity. We use variational methods to study this problem. By virtue of some auxiliary system related to the "limit equation" of the Dirac equation, we constructed linking levels of the variational functional Φ<sub><italic>M</italic></sub> such that the minimax value <italic>C</italic><sub><italic>M</italic></sub> based on the linking structure of Φ<sub><italic>M</italic></sub> satisfies 0 < <italic>C</italic><sub><italic>M</italic></sub> < , where is the least energy of the "limit equation". Thus we can show the (<italic>C</italic>)<sub><italic>c</italic></sub>-condition holds true for all <italic>c</italic> < and consequently, we obtain one least energy solution of the Dirac equation.