Existence of multiple semiclassical solutions for a critical Choquard equation with indefinite potential. (June 2020) - Gao, Fashun; Yang, Minbo; Zhou, Jiazheng
Abstract: We study the existence and multiplicity of semiclassical states for the Choquard equation with critical growth − ε 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.