Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation

Abstract

We study the semi-classical limit of the least energy solutions to the nonlinear Dirac equation − i ε ∑ k = 1 3 α k ∂ k u + a β u = P ( x ) | u | p − 2 u for x ∈ R 3 . 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 R 3 , 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 .