Existence results for some problems on Riemannian manifolds

Abstract

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 $$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, ${\Delta }_g$ denotes the Laplace-Beltrami operator on $(\mathcal{M},g)$, $\alpha ,K:\mathcal{M}\to \mathbb{R}$ are positive (essentially) bounded functions, $r\in (0,1)$, 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.