Boundary Value Problems (Sep 2017)
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.
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