Ground state sign-changing solutions for critical Choquard equations with steep well potential

Abstract

In this paper, we study sign-changing solution of the Choquard type equation − Δ u + ( λ V ( x ) + 1 ) u = ( I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ − 2 u + μ | u | p − 2 u in R N , where N ≥ 3 , α ∈ ( ( N − 4 ) + , N ) , I α is a Riesz potential, p ∈ [ 2 α ∗ , 2 N N − 2 ) , 2 α ∗ := N + α N − 2 is the upper critical exponent in terms of the Hardy–Littlewood–Sobolev inequality, μ > 0 , λ > 0 , V ∈ C ( R N , R ) is nonnegative and has a potential well. By combining the variational methods and sign-changing Nehari manifold, we prove the existence and some properties of ground state sign-changing solution for λ , μ large enough. Further, we verify the asymptotic behaviour of ground state sign-changing solutions as λ → + ∞ and μ → + ∞ , respectively.