Semiclassical States for Fractional Choquard Equations With a Potential Well

In this paper, we studied the existence and concentration behavior of positive solutions for a fractional Choquard equation involving a small parameter and a nonlocal nonlinearity. Under suitable assumptions on the potential function and the nonlinear term, we established two main results. First, for sufficiently small parameters, the equation admits multiple positive solutions, with the number of solutions depending on the topological structure of the potential's minimum set. Moreover, these solutions concentrate near this set as the parameter tends to zero. Second, we proved the existence of a single-peak solution whose peak converges to the potential's minimum set, and its rescaled version approaches a least energy solution of the associated limiting problem.