Asymptotic Analysis of Multiple Solutions for Perturbed Choquard Equations

Abstract

In this paper, we study the following Choquard equations with small perturbation f$$-\Delta u + V(x)u = (I_\alpha * |u|^p)|u|^{p-2}u+f(x), x\in \mathbb{R}^N.$$ where N ≥ 3 and Iα denotes the Riesz potential. As is known that the above equation has a ground state uα and a bound state vα by fibering maps (see [22] or [23]), our aim is to show that for fixed $$p \in (1,\frac{N}{N-2})$$, uα and vα converge to a ground state and a bound state of the limiting local problem respectively, as α → 0.