Semiclassical states for nonlinear Dirac equations with singular potentials

Abstract

In this paper, we study the following nonlinear Dirac equationwhere p ∈ (2, 3), a > 0 is a constant, α = (α 1 , α 2 , α 3 ), α 1 , α 2 , α 3 and β are 4×4 Pauli-Dirac matrices. Our investigation focuses on the case in which V (x) may attain ±a at somewhere or at infinity and K (x) may approach ±∞ as x accumulating at some points and as |x| → ∞. This is a case with the potentials being singular, which has not been studied before as all the works in the literature require lim sup |x|→∞ |V (x)| < a and K ∈ L ∞ (R 3 , R + ). Under some mild assumptions on V and K , for ε > 0 small, we construct localized bound state solutions which concentrate around the singular set of K . Mathematics Subject Classification 35Q40 • 49J35 1 Introduction and main resultsIn this paper, we consider the existence and concentration behavior of semiclassical states of the following stationary Dirac equationCommunicated by M. del Pino.