Multiple bound state solutions for fractional Choquard equation with Hardy–Littlewood–Sobolev critical exponent

Abstract

In this paper, we study the nonlinear Choquard equation ε2s(−Δ)su+V(x)u=Iα*|u|2α,s*|u|2α,s*−2u,u∈Ds,2(RN), where s ∈ (0, 1), N ≥ 3, ɛ is the positive parameter, and 2α,s*=N+αN−2s is the critical exponent with respect to the Hardy–Littlewood–Sobolev inequality. V(x)∈LN2s(RN), where V(x) is assumed to be zero in some region of RN, which means that it is of the critical frequency case. In virtue of a global compactness result in fractional Sobolev space and Lusternik–Schnirelmann theory of critical points, we succeed in proving the multiplicity of bound state solutions.