Multi-bump solutions for a Kirchhoff-type problem

Abstract

Abstract In this paper, we study the existence of solutions for the Kirchhoff problem M ( ∫ ℝ 3 | ∇ u | 2 d x + ∫ ℝ 3 ( λ a ( x ) + 1 ) u 2 d x ) ( - Δ u + ( λ a ( x ) + 1 ) u ) = f ( u ) $M\Biggl (\int _{\mathbb {R}^{3}}|\nabla u|^{2}\, dx + \int _{\mathbb {R}^{3}} (\lambda a(x)+1)u^{2}\, dx\Biggl ) (- \Delta u + (\lambda a(x)+1)u) = f(u)$ in ℝ3, u ∈ H 1(ℝ3) Assuming that the nonnegative function a(x) has a potential well with int (a -1({0})) consisting of k disjoint components Ω1,Ω2,...,Ω k and the nonlinearity f(t) has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by using variational methods.