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# Communications in Analysis and Geometry

## Volume 28 (2020)

### Number 3

### Existence results for some problems on Riemannian manifolds

Pages: 677 – 706

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n3.a6

#### Authors

#### 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 Yamabe-type problem\[\begin{cases}-\Delta_g w + \alpha(\sigma) w = \mu K(\sigma) w^{\frac{d+2}{d-2}} +\lambda \left( w^{r-1} + f(w)\right), \quad \sigma\in\mathcal{M} \\w\in H^2_\alpha(\mathcal{M}), \quad w \gt 0 \; \textrm{in} \; \mathcal{M} \: \textrm{,}\end{cases}\]here, 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 furthermore solve some parametrized Emden–Fowler equations in the Euclidean case.

The manuscript was realized under the auspices of the Italian MIUR project *Variational methods, with applications to problems in mathematical physics and geometry* (2015KB9WPT 009). The second author was supported by the Slovenian Research Agency grants P1-0292, N1-0114, N1-0083, N1-0064, and J1-8131. The third author was partially supported by the INdAM-GNAMPA Project 2017 *Metodi variazionali per fenomeni non-locali*.

Received 7 September 2017

Accepted 25 January 2018

Published 6 July 2020