Solutions of non-periodic super-quadratic Dirac equations

This paper is concerned with solutions to the Dirac equation: −i∑αk∂ku+aβu+M(x)u=Ru(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 ΦM such that the minimax value cM based on the linking structure of ΦM satisfies , where is the least energy of the “limit equation”. Thus we can show the (C)c-condition holds true for all and consequently obtain one least energy solution to the Dirac equation.