Existence and Concentration of Semiclassical Solutions for Dirac Equations with Critical Nonlinearities

Abstract

We study the semiclassical ground states of the Dirac equation with critical nonlinearity: $-i\hbar\alpha\cdot\nabla w + a\beta w +V(x)w= W(x)(g(|w|)+|w|)w$ for $x\in\mathbb{R}^3$. The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. We develop an argument to establish the existence of least energy solutions for $\hbar$ small. We also describe the concentration phenomena of the solutions as $\hbar\to 0$.