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\le f\in C^{0,1}(\mathbb R^n)$. Given a domain $\Omega\subset \mathbbR^n$, we prove that any stable solution to the equation $-\Delta u=f(u)$ in$\Omega$ satisfies a BMO interior regularity when $n=10$, and an Morrey$M^{p_n,4+2/(p_n-2)}$ interior regularity when $n\ge 11$, where$$p_n=\frac{2(n-2\sqrt{n-1}-2)}{n-2\sqrt{n-1}-4}. $$  This result is optimal as hinted by earlier results, and answers an openquestion raised by Cabr\'e, Figalli, Ros-Oton and Serra. As an application, weshow a sharp Liouville property: Any stable solution $u \in C^2(\mathbb R^n)$to $-\Delta u=f(u)$ in $\mathbb R^n$ satisfying the growth condition, i.e.\$|u(x)|= o\left( \log|x| \right)$ as $|x|\to+\infty$ when $n=10$; or $|u(x)|=o\left( |x| ^{ -\frac n2+\sqrt{n-1}+2 }\right)$ as $x|\to+\infty$ when $n\ge11$, must be a constant. This extends the well-known Liouville property forradial stable solutions obtained by Villegas.

Optimal regularity & Liouville property for stable solutions to semilinear elliptic equations in $\mathbb R^n$ with $n\ge10$