Normalized solutions to the mass supercritical Kirchhoff-type equation with non-trapping potential

This paper is concerned with the existence of solutions to the Kirchhoff-type equation −a+b∫R3|∇u|2Δu+(V+λ)u=|u|p−2u+μ|u|q−2uinR3 under the normalized constraint ∫R3u2=ρ2, where a, b, ρ > 0, 14/3 < q < p ⩽ 6, μ > 0 is a constant, and λ∈R appears as a Lagrange multiplier. Under an explicit assumption on V, we can prove the existence of positive ground state solutions to the above equation. A new concentration compactness type result is established to recover compactness in the Sobolev critical case.