Existence results for some problems on Riemannian manifolds

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\ge 3$) Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{g}w+\alpha (\sigma )w=\mu K(\sigma )w^{\frac{d+2}{d-2}}+\lambda (w^{r-1}+f(w)),\sigma \in \mathcal{M}& \\ & \\ w\in H_{\alpha }^2(\mathcal{M}),w>0in\mathcal{M}& \end{array}$$ where, as usual, ${\mathrm{\Delta }}_g$ denotes the Laplace-Beltrami operator on $(\mathcal{M},g)$, $\alpha, K:$\mathcal{M}$\to$\mathbb{R}$$arepositive(essentially)boundedfunctions,$r\in(0,1)$,and$f:[0,+\infty)\to[0,+\infty)$ is a subcritical continuous function. Restricting ourselves to the unit sphere ${\mathbb{S}}^d$ via the stereographic projection, we also solve some parametrized Emden-Fowler equations in the Euclidean case.