  We study a nonlinear boundary value problem driven by the $p$-Laplacian plus
an indefinite potential with Robin boundary condition. The reaction term is a
Carath\'eodory function which is asymptotically resonant at $\pm\infty$ with
respect to a nonprincipal Ljusternik-Schnirelmann eigenvalue. Using variational
methods, together with Morse theory and truncation-perturbation techniques, we
show that the problem has at least three nontrivial smooth solutions, two of
which have a fixed sign.
