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 −ε2Δu+V(x)u=(∫RNG(y,u(y))|x−y|μdy)g(x,u)in RN,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.