Infinitely many localized semiclassical states for critical nonlinear Dirac equations

Abstract

In this paper, we are concerned with semiclassical states to the nonlinear Dirac equation with Sobolev critical exponent , where , 2 < q < 3, ϵ > 0 is a small parameter, a > 0 is a constant, α = (α 1, α 2, α 3), α j and β are 4 × 4 Pauli-Dirac matrices. We construct an infinite sequence of semiclassical states with higher energies concentrating around the local minimum points of the potential V. The problem is strongly indefinite and the solutions correspond to critical points of the underlying energy functional at energy levels where compactness condition breaks down. The proof relies on truncation techniques, blow-up arguments together with a local type Pohozaev identity.