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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $0<\alpha<2$ be any real number. In this paper, we investigate thefollowing semilinear equations involving the fractional Laplacian\begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on$\mathbb{R}^n$ and $\mathbb{R}^n_+$.  Applying a direct method of moving planes for the fractional Laplacian, weprove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and$\mathbb{R}^n_+$ under mild conditions on $f$.

Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on $\mathbb{R}^n$ and $\mathbb{R}^n_+$