Given any closed Riemannian manifold (M, g) we use the Lyapunov–Schmidt finite-dimensional reduction method and the classical Morse and Lusternick–Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on (M, g). If (N, h) is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation on the Riemannian product \((M\times N , g + \varepsilon ^2 h )\), for \(\varepsilon >0\) small. For example, if M is a closed Riemann surface of genus \(\mathbf{g}\) and \((N,h) = (S^2 , g_0)\) is the round 2-sphere, we prove that for \(\varepsilon >0\) small enough and a generic metric g on M, the Yamabe equation on \((M\times S^2 , g + \varepsilon ^2 g_0 )\) has at least \(2 + 2 \mathbf{g}\) solutions.

给定任何封闭的黎曼流形（M，  g），我们使用Lyapunov–Schmidt有限维约简方法以及经典的Morse和Lusternick – Schnirelmann理论来证明（M，  g）上次临界Yamabe型方程正解的多重性结果。如果（Ñ，  ħ）是恒定的正标量曲率的闭合黎曼流形，我们得到用于在黎曼产品的山部方程多重结果\（（M \次N，G + \ varepsilon ^ 2小时）\） ，用于\（ \ varepsilon> 0 \）小。例如，如果M是\（\ mathbf {g} \）属的闭合Riemann曲面，则\（（N，h）=（S ^ 2，g_0）\）是圆形的2球，我们证明对于\（\ varepsilon> 0 \）足够小并且在M上有一个通用度量g，在\（（M×S ^ 2，g + \ varepsilon ^ 2 g_0）\）至少具有\（2 + 2 \ mathbf {g} \）个解。