Let (Mm,g) be a closed Riemannian manifold (m≥2) of positive scalar curvature and (Nn,h) any closed manifold. We study the asymptotic behaviour of the second Yamabe constant and the second N−Yamabe constant of (M×N,g+th) as t goes to +∞. We obtain that If n≥2, we show the existence of nodal solutions of the Yamabe equation on (M×N,g+th) (provided t large enough). When the scalar curvature of (M,g) is constant, we prove that . Also we study the second Yamabe invariant and the second N−Yamabe invariant.