Positive solutions of Yamabe-type equations with general potentials

Let $(M^n,g)$ be a smooth closed connected Riemannian manifold of dimension $n\ge3$, and let $ψ\in C^\infty(M,\mathbb R)$. We prove that the equation $$ -\frac{4(n-1)}{n-2}Δ_g u+ψu=u^{\frac{n+2}{n-2}} $$ has a smooth positive solution if and only if its linear part has positive first eigenvalue. In