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 generalise and sharpen several recent results in the literature regardingthe existence and complete classification of the isolated singularities for abroad class of nonlinear elliptic equations of the form \begin{equation} -{\rmdiv}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\quad \text{in} B_1\setminus\{0\}, \end{equation} where $B_r$ denotes the open ball withradius $r>0$ centred at zero in $\mathbb{R}^N$ $(N\geq 2)$. We assume that$\mathcal{A} \in C^1(0,1]$, $b\in C(\bar{B_1}\setminus\{0\})$ and $h\inC[0,\infty)$ are positive functions associated with regularly varying functionsof index $\vartheta$, $\sigma$ and $q$ at $0$, $0$ and $\infty$ respectively,satisfying $q>p-1>0$ and $\vartheta-\sigma<p<N+\vartheta$. We prove that thecondition $b(x) \,h(\Phi)\not \in L^1(B_{1/2})$ is sharp for the removabilityof all singularities at zero for the positive solutions of our problem, where$\Phi$ denotes the "fundamental solution" of $-{\rm div}\,(\mathcal A(|x|)\,|\nabla u|^{p-2} \nabla u)=\delta_0$ (the Dirac mass at zero) in $B_1$, subjectto $\Phi|_{\partial B_1}=0$. If $b(x) \,h(\Phi)\in L^1(B_{1/2})$, we show thatany non-removable singularity at zero for a positive solution to our equationis either weak (i.e., $\lim_{|x|\to 0} u(x)/\Phi(|x|)\in (0,\infty)$) or strong($ \lim_{|x|\to 0} u(x)/\Phi(|x|)=\infty$). The main difficulty and novelty ofthis paper, for which we develop new techniques, come from the explicitasymptotic behaviour of the strong singularity solutions in the critical case,which had previously remained open even for $\mathcal{A}=1$. We also study theexistence and uniqueness of the positive solution to our problem with aprescribed admissible behaviour at zero and a Dirichlet condition on $\partialB_1$.

Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification