Concentration phenomenon of solutions for a class of Kirchhoff-type equations with critical growth

In this paper, we study the following Kirchhoff-type equations with critical growth -(epsilon(2) a + epsilon b integral(R3) vertical bar del u vertical bar(2) dx) Delta u + V(x)u = P(x)f(u) + Q(x)vertical bar u vertical bar(4)u, x is an element of R-3, where epsilon > 0, a > 0, b > 0 and f is a continuous superlinear but subcritical nonlinearity. Under suitable assumptions on the potentials V(x), P(x) and Q(x), we obtain the existence and concentration of positive solutions and prove that the semiclassical solutions omega(epsilon) with maximum points x(epsilon) concentrating at a special set S-p characterized by V(x), P(x) and Q(x). Furthermore, for any sequence x(epsilon) -> x(0) is an element of S-p, v(epsilon) (x) := omega(epsilon) (epsilon x + x(epsilon)) converges in H-1 (R-3) to a ground state solution v of -(a + b integral(R3) vertical bar del v vertical bar(2)dx) Delta v + V(x(0))v = P(x(0)) f (v) + Q(x(0))vertical bar v vertical bar(4)v, x is an element of R-3. (C) 2020 Elsevier Inc. All rights reserved.