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 $(d \geq 3)$ 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.

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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