Multiplicity of constant scalar curvature metrics on warped products

Let $(M^m,g_M)$ be a closed, connected manifold with positive scalar curvature and $(T^k,g)$ some flat $k$-Torus of unit volume. By a result of F. Dobarro and E. Lami Dozo, there exists a unique $f: M \rightarrow \mathbf{R}_{>0}$ such that the warped product $M\times_f T^k$ has constant scalar curvature and unit volume. We study the Yamabe equation on these spaces. We use techniques from bifurcation theory, along with spectral theory for warped products, to prove multiplicity results for the Yamabe problem.