Multiple positive solutions for a nonlinear Choquard equation with nonhomogeneous

In this paper, we study the existence of multiple positive solutions for the following equation: −Δu+u = (Kα (x)∗ |u|p)|u|p−2u +λ f (x), x ∈ R , where N 3, α ∈ (0,N), p ∈ (1+ α/N,(N + α)/(N− 2)), Kα (x) is the Riesz potential, and f (x) ∈ H−1(RN) , f (x) 0 , f (x) ≡ 0. We prove that there exists a constant λ ∗ > 0 such that the equation above possesses at least two positive solutions for all λ ∈ (0,λ ∗) . Furthermore, we can obtain the existence of the ground state solution.