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 ∈ C1(ℝ+,ℝ) is subcritical, V : ℝ3 → ℝ is a locally Hölder continuous function. We first prove that for ε0 > 0 sufficiently small, the above problem has a weak solution uε with exponential decay at infinity. Moreover, uε 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.
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