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/

Zendy Plus

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

Zendy Tools

Zendy Open

The necessary and sufficient conditions for a regular positive entiresolution $u$ of the biharmonic equation: \begin{equation} \label{0.1} -\Delta^2u=u^{-p} \;\; \mbox{in $\R^N \; (N \geq 3)$}, \;\; p>1 \end{equation} to be aradially symmetric solution are obtained via the moving plane method (MPM) of asystem of equations. It is well-known that for any $a>0$, \eqref{0.1} admits aunique minimal positive entire radial solution ${\underline u}_a (r)$ and afamily of non-minimal positive entire radial solutions $u_a (r)$ such that $u_a(0)={\underline u}_a (0)=a$ and $u_a (r) \geq {\underline u}_a (r)$ for $r \in(0, \infty)$. Moreover, the asymptotic behaviors of ${\underline u}_a (r)$ and$u_a (r)$ at $r=\infty$ are also known. We will see in this paper that theasymptotic behaviors similar to those of ${\underline u}_a (r)$ and $u_a (r)$at $r=\infty$ can determine the radial symmetry of a general regular positiveentire solution $u$ of \eqref{0.1}. The precisely asymptotic behaviors of $u(x)$ and $-\Delta u (x)$ at $|x|=\infty$ need to be established such that themoving-plane procedure can be started. We provide the necessary and sufficientconditions not only for a regular positive entire solution $u$ of \eqref{0.1}to be the minimal entire radial solution, but also for $u$ to be a non-minimalentire radial solution.

Radial symmetry of positive entire solutions of a fourth order elliptic equation with a singular nonlinearity