Second Yamabe constant on Riemannian products

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 limt→+∞Y2(M×N,[g+th])=22m+nY(M×Rn,[g+ge]). If n≥2, we show the existence of nodal solutions of the Yamabe equation on (M×N,g+th) (provided t large enough). When sg is constant, we prove that limt→+∞YN2(M×N,g+th)=22m+nYRn(M×Rn,g+ge). Also we study the second Yamabe invariant and the second N-Yamabe invariant.