Existence of multi-bump solutions for a class of Kirchhoff type problems in $\mathbb {R}^3$R3

Abstract

Using variational methods, we establish existence of multi-bump solutions for a class of Kirchhoff type problems \documentclass[12pt]{minimal}\begin{document}$-(a+b\int _{\mathbb {R}^3}|\nabla u|^2dx)\Delta u + \lambda V(x)u = f(u)$\end{document}−(a+b∫R3|∇u|2dx)Δu+λV(x)u=f(u), where f is a continuous function with subcritical growth, V(x) is a critical frequency in the sense that \documentclass[12pt]{minimal}\begin{document}$\inf _{x\in \mathbb {R}^3}V(x)=0$\end{document}infx∈R3V(x)=0. We show that if the zero set of V(x) has several isolated connected components Ω1, …, Ωk such that the interior of Ωi is not empty and ∂Ωi is smooth, then for λ > 0 large there exists, for any non-empty subset J ⊂ {1, …, k}, a bump solution is trapped in a neighborhood of ∪j ∈ JΩj.