Solutions of super linear Dirac equations with general potentials

Abstract

This paper is concerned with solutions to the Dirac equation: −iΣα k ∂ k u + aβu + M(x)u = g(x, ‖u‖)u. Here M(x) is a general potential and g(x, ‖u‖) is a self-coupling which grows super-quadratically 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 constructed 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 < Ĉ, where Ĉ is the least energy of the limit equation. Thus we can show the (C) c -condition holds true for all c < Ĉ and consequently we obtain one solution of the Dirac equation.