Multiplicity of solutions for a critical nonlinear Schrödinger–Kirchhoff-type equation

Abstract

In this paper, we study the following critical nonlinear Schrödinger–Kirchhoff equation: ($P$) { − ( a + b ∫ R N | ∇u | 2 d x ) Δu + V ( x ) u = P ( x ) | u | 2 ∗ − 2 u + μ | u | q − 2 u , in R N , u ∈ H 1 ( R N ) where a , b , μ > 0 , N ≥ 3 , max { 2 ∗ − 1 , 2 } < q < 2 ∗ = 2 N N − 2 , V ( x ) > 0 and P ( x ) ≥ 0 are two continuous functions. By using the variational method and truncation technique, we prove the multiplicity of solutions for Equation (P).