Concentrating Bound States for Kirchhoff Type Problems in ℝ 3 Involving Critical Sobolev Exponents

• Published in: Advanced nonlinear studies Vol. 14; no. 2; pp. 483 - 510
• Main Authors: He, Yi, Li, Gongbao, Peng, Shuangjie
• Format: Journal Article
• Language: English
• Published: 01.05.2014
• ISSN: 1536-1365; 2169-0375
• DOI: 10.1515/ans-2014-0214

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.