The critical Choquard equations with a Kirchhoff type perturbation in bounded domains

Abstract

This paper deals with the following critical Choquard equation with a Kirchhoff type perturbation in bounded domains, { − ( 1 + b ‖ u ‖ 2 ) Δu = ( ∫ Ω u 2 ( y ) | x − y | 4 d y ) u + λu in Ω , u = 0 on ∂Ω , where Ω ⊂ R N ( N ≥ 5 ) is a smooth bounded domain and ‖ ⋅ ‖ is the standard norm of H 0 1 ( Ω ) . Under the suitable assumptions on the constant b ≥ 0 , we prove the existence of solutions for 0 < λ ≤ λ 1 , where λ 1 > 0 is the first eigenvalue of − Δ on Ω. Moreover, we prove the multiplicity of solutions for λ > λ 1 and b>0 in suitable intervals.