The existence of ground state normalized solution for Kirchhoff equation

Abstract

In this paper, we study the existence of ground state solution for the following Kirchhoff equation with combined nonlinearities − ( a + b ∫ R N | ∇u | 2 ) Δu = λu + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass ∫ R N u 2 = c 2 , where a > 0 , b > 0 , c > 0 , μ < 0 , 2 < q ≤ 2 + 8 N < p < 2 ∗ . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution ( u ~ , λ c ) ∈ S c × R , where S c = { u ∈ H 1 ( R N ) : ∫ R N u 2 = c 2 } .