Existence of multiple semiclassical solutions for a critical Choquard equation with indefinite potential

Abstract We study the existence and multiplicity of semiclassical states for the Choquard equation with critical growth − e 2 Δ u + V ( x ) u = ( ∫ R N G ( y , u ( y ) ) | x − y | μ d y ) g ( x , u ) in  R N , where N ≥ 3 , 0 μ min { 4 , N } , V ( x ) is sign changing and G is the primitive of g which is of critical growth due to the well known Hardy–Littlewood–Sobolev inequality. Under suitable assumptions on V and g , we prove the existence and multiplicity of semiclassical states by critical point theory.