Concentrating Bound States for Kirchhoff Type Problems in ℝ3 Involving Critical Sobolev Exponents

Abstract We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, where ε is a small positive parameter and a, b > 0 are constants, f ∈ C 1 (ℝ + ,ℝ) is subcritical, V : ℝ 3 → ℝ is a locally Hölder continuous function. We first prove that for ε 0 > 0 sufficiently small, the above problem has a weak solution u ε with exponential decay at infinity. Moreover, u ε concentrates around a local minimum point of V in Λ as ε → 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum.