Semiclassical states for Dirac-Klein-Gordon system with critical growth

In this paper, we study the following critical Dirac-Klein-Gordon system in R-3: {i epsilon Sigma(3)(k=1) alpha(k)partial derivative(k)u - alpha beta u + V(x)u - lambda phi beta u = P(x) f (vertical bar u vertical bar)u + Q(x) vertical bar u vertical bar u, -epsilon(2)Delta phi + M phi + 4 pi lambda(beta u) . u, where epsilon > 0 is a small parameter, alpha > 0 is a constant. We prove the existence and concentration of solutions under suitable assumptions on the potential V(x), P(x) and Q(x). We also show the semiclassical solutions w E with maximum points w(epsilon) concentrating at a special set H-P characterized by V(x), P(x) and Q(x), and for any sequence x(epsilon) -> x(0 )is an element of H-p, v(epsilon) (x) := w(epsilon) (epsilon x + x(epsilon)) converges in H-1 (R-3 , C-4) to a least energy solution u of {i Sigma(3)(k=1) alpha(k)partial derivative(k)u - alpha beta u + V(x(0))u - lambda phi beta u = P(x(0)) f (vertical bar u vertical bar)u + Q(x(0)) vertical bar u vertical bar u, -Delta phi + M phi + 4 pi lambda(beta u) . u. (C) 2020 Elsevier Inc. All rights reserved.