Concentrating Bound States for Kirchhoff Type Problems in ℝ 3 Involving Critical Sobolev Exponents - He, Yi; Li, Gongbao; Peng, Shuangjie (Advanced nonlinear studies, 2014)
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.