Normalized solutions to the Sobolev critical Kirchhoff-type equation with non-trapping potential

Abstract

The paper is concerned about the existence of solutions with prescribed L 2 -norm to the following Kirchhoff-type equation − ( a + b ∫ R 3 | ∇u | 2 ) Δu + ( V + λ ) u = | u | p − 2 u + μ | u | q − 2 u in R 3 , where a , b > 0 , 2 < q < 14 / 3 < p ⩽ 6 or 14 / 3 < q < p ⩽ 6 , μ > 0 . Noting that 14/3 is the mass critical exponent, a Pohozaev constraint method is adopted in two cases. In the mass mixed critical case, i.e., 2 < q < 10 / 3 , 14 / 3 < p ⩽ 6 , we get a normalized solution to above equation with small enough μ by Ekeland's variational principle. In the mass supercritical case, i.e., 14 / 3 < q < p ⩽ 6 , we obtain a positive ground state normalized solution, and energy comparison argument is used in the Sobolev critical case.