Multi-bump solutions for a Kirchhoff-type problem

Abstract

In this paper, we study the existence of solutions for the Kirchhoff problem M(∫ℝ3|∇u|2dx+∫ℝ3(λa(x)+1)u2dx)(-Δu+(λa(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 ∈ H1(ℝ3) Assuming that the nonnegative function a(x) has a potential well with int (a -1({0})) consisting of k disjoint components Ω1,Ω2,...,Ωk 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.

Keywords

• kirchhoff problem
• multi-bump solution
• variational methods
• 35j65
• 34b15