Existence results for some problems on Riemannian manifolds

TL;DR

This paper establishes new existence results for Yamabe-type equations on compact Riemannian manifolds using variational methods, including solutions to singular problems and Euclidean Emden-Fowler equations.

Contribution

It introduces novel variational techniques to prove existence of solutions for Yamabe-type equations with subcritical perturbations on Riemannian manifolds.

Findings

Existence of solutions to singular Yamabe-type problems on compact manifolds.

Solutions to parametrized Emden-Fowler equations on the sphere.

Application of variational methods to subcritical perturbations.

Abstract

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) 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 $$ \left\lbrace \begin{array}{ll} -\Delta_g w + \alpha(\sigma)w = \mu K(\sigma) w^\frac{d+2}{d-2} +\lambda \left( w^{r-1} + f(w)\right), \quad \sigma\in\mathcal{M} &\\ &\\ w\in H^2_\alpha(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ 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.