Solutions of non-periodic super-quadratic Dirac equations

This paper is concerned with solutions to the Dirac equation: − i ∑ α k ∂ k u + a β u + M ( x ) u = R u ( x , u ) . Here M ( x ) is a general potential and R ( x , u ) is a self-coupling which is super-quadratic in u 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 construct linking levels of the variational functional Φ M such that the minimax value c M based on the linking structure of Φ M satisfies 0 < c M < C ˆ , where C ˆ is the least energy of the “limit equation”. Thus we can show the ( C ) c -condition holds true for all c < C ˆ and consequently obtain one least energy solution to the Dirac equation.