In this paper, We studied a class of fractional $p$-Laplacian equations with nonsmooth potential. By nonsmooth analysis, nonsmooth critical point theory, and truncation techniques, we established two multiplicity theorems: One guaranteeing the existence of at least four solutions, and the other ensuring the existence of at least two solutions. In our approach, we addressed the challenges posed by the nonsmoothness of the potential, and the results extended the nonlocal problems with discontinuous nonlinearities.