English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We introduce a double iterative scheme and local variational method to solvethe Yamabe-type equation $ - \frac{4(n - 1)}{n - 2}\Delta_{g} u + (S_{g} +\beta ) u = \lambda u^{\frac{n + 2}{n - 2}} $ for some constant $ \beta\leqslant 0 $, locally on Riemannian domain $ (\Omega, g) $ with trivialDirichlet condition and globally on closed manifolds $ (M, g) $; the dimensionsof $ \Omega $ and $ M $ are at least 3. In contrast to the traditional globalvariational method, these Yamabe-type equations on closed manifolds areanalyzed by local analysis and monotone iteration scheme. In particular, we donot need to use the Weyl tensor. In particular, the sign of the firsteigenvalue $ \eta_{1} $ of conformal Laplacian $ \Box_{g} u : = - \frac{4(n -1)}{n - 2}\Delta_{g} u + S_{g} u $ plays the central role. When letting $ \beta\rightarrow 0 $ from the left, we reproof the classical Yamabe problem as anatural consequence of the Yamabe-type equations. The results are classified bythe sign of $ \eta_{1} $.

Solving the Yamabe-Type Equations on Closed Manifolds by Iteration Schemes