Existence of semiclassical solutions for some critical Dirac equation

Abstract

In this paper, we study the following critical Dirac equation −iε∑k=13αk∂ku+aβu+V(x)u=P(x)f(|u|)u+Q(x)|u|u,x∈R3, where ɛ > 0 is a small parameter; a > 0 is a constant; α1, α2, α3, and β are 4 × 4 Pauli–Dirac matrices; and V, P, Q, and f are continuous but are not necessarily of class C1. We prove the existence and concentration of semiclassical solutions under suitable assumptions on the potentials V(x), P(x), and Q(x) by using variational methods. We also show the semiclassical solutions ωɛ with maximum points xɛ of |ωɛ| concentrating at a special set HP characterized by V(x), P(x), and Q(x) and for any sequence xε→x0∈HP,vε(x)≔ωε(εx+xε) converges in W1,q(R3,C4) for q ≥ 2 to a ground state solution u of −i∑k=13αk∂ku+aβu+V(x0)u=P(x0)f(|u|)u+Q(x0)|u|u,inR3. Finally, we estimate the exponential decay properties of solutions.