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

We consider the following critical elliptic system: \begin{equation*}\begin{cases} -\Delta u_i=\mu_i u_i^{3}+\beta u_i^{ } \sum\limits_{j\neq i}u_j^{2} \quad \hbox{in}\ \Omega_\varepsilon \\ u_i=0 \hbox{ on }\partial\Omega_\varepsilon , \qquad u_i>0 \hbox{ in } \Omega_\varepsilon\end{cases}\qquad i=1,\ldots, m, \end{equation*} in a domain$\Omega_\varepsilon \subset \mathbb{R}^4$ with a small shrinking hole$B_\varepsilon(\xi_0)$. For $\mu_i>0$, $\beta<0$, and $\varepsilon>0$ small, weprove the existence of a non-synchronized solution which looks like a fountainof positive bubbles, i.e. each component $u_i$ exhibits a towering blow-uparound $\xi_0$ as $\varepsilon \to 0$. The proof is based on theLjapunov-Schmidt reduction method, and the velocity of concentration of eachlayer within a given tower is chosen in such a way that the interaction betweenbubbles of different components balance the interaction of the first bubble ofeach component with the boundary of the domain, and in addition is dominantwhen compared with the interaction of two consecutive bubbles of the samecomponent.

A fountain of positive Bubbles on a Coron's Problem for a Competitive Weakly Coupled Gradient System