Existence results for some problems on Riemannian manifolds

Molica Bisci, Giovanni (author), Repovš, Dušan (author), Vilasi, Luca (author)

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 Yamabe-type problem ▫$$\begin{cases} -\Delta_gw + \alpha(\sigma)w = \mu K(\sigma)w^{\frac{d+2}{d-2}} + \lambda (w^{r-1} + f(w)), \quad \sigma \in \mathcal{M} \\ w \in H^2_\alpha(\mathcal{M}), \quad w>0 \; \text{in} \; \mathcal{M}, \end{cases}$$▫ 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.

• Language: English
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2020
• Number of pages: Str. 677-706
• Numbering: Vol. 28, no. 3
• PID: 20.500.12556/RUL-118042
• UDC: 517.956
• ISSN on article: 1019-8385
• DOI: 10.4310/CAG.2020.v28.n3.a6
• COBISS.SI-ID: 22044675
• Publication date in RUL: 17.08.2020

Record is a part of a journal

• Title: Communications in analysis and geometry
• Shortened title: Commun. anal. geom.
• Publisher: International Press
• ISSN: 1019-8385
• COBISS.SI-ID: 3760729