Existence and concentration of solutions for the nonlinear Kirchhoff type equations with steep well potential

Abstract

Abstract In this paper, we study the following nonlinear problem of Kirchhoff type: { − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u + λ V ( x ) u = | u | p − 2 u , in R 3 , u ∈ H 1 ( R 3 ) , $$ \textstyle\begin{cases} - ( a + b\int_{{\mathbb{R}^{3}}} \vert {\nabla u} \vert ^{2} ) \Delta u + \lambda V(x)u = {{ \vert u \vert } ^{p - 2}}u,\quad \text{in }{\mathbb{R}^{3}}, \\ {u \in{H^{1}}({\mathbb{R}^{3}}),} \end{cases} $$ where the parameter λ > 0 $\lambda > 0$ and 4 ≤ p 0 $a, b>0$ . By variational methods, the results of the existence of nontrivial solutions and the concentration phenomena of the solutions as λ → + ∞ $\lambda \to + \infty$ are obtained. It is worth pointing out that, for the case p ∈ ( 4 , 6 ) $p\in(4,6)$ , the potential V is permitted to be sign-changing.

Keywords

• Kirchhoff equation
• variational methods
• concentration