Existence of Multiple Positive Solutions for Choquard Equation with Perturbation

This paper is concerned with the following Choquard equation with perturbation: -Delta u + V(x)u = (1/vertical bar x vertical bar(alpha) * vertical bar u vertical bar(p))vertical bar u vertical bar(p-2) u + g(x), u is an element of H-1 (R-N), where N >= 3, alpha is an element of (0, N), and 2 - (alpha/N) < p < (2N - alpha)/(N - 2). This kind of equations is well known as the Choquard or nonlinear Schrodinger-Newton equation. Under some assumptions for the functions V(x), we prove the existence of multiple positive solutions of the equation. Moreover, we also show that these results still hold for more generalized Choquard equation with perturbation.