Contemporary Mathematics. Fundamental DirectionsContemporary Mathematics. Fundamental DirectionsСовременная математика. Фундаментальные направления2413-36392949-0618Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University)4390810.22363/2413-3639-2025-71-1-85-95TZCNJNArticlesСтатьиResearch ArticleApplications of the s-harmonic extension to the study of singularities of Emden’s equationsПрименение s-гармонического расширения к изучению особенностей уравнений ЭмденаVeronLaurentВеронЛ.veronl@univ-tours.frUniversité de Tours15122025711Nonlocal and nonlinear problemsНелокальные и нелинейные задачи859521042025Copyright ©; 2025, V´eron L.Copyright ©; 2025, Верон Л.2025V´eron L.Верон Л.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/CMFD/article/view/43908

We use the Caffarelli–Silvestre extension to \( \mathrm{R}_+\times\mathrm{R}^N \) to study the isolated singularities of functions satisfying the semilinear fractional equation \( (-\Delta)^sv+\epsilon v^p=0 \) in a punctured domain of \( \mathrm{R}^N \) where \(\epsilon=\pm 1\), \(0<s<1\) and \(p>1\). We emphasise the obtention of a priori estimates and analyse the set of self-similar solutions. We provide a complete description of the possible behaviour of solutions near a singularity.

Мы используем расширение Каффарелли—Сильвестра на \( \mathrm{R}_+\times\mathrm{R}^N \) для изучения изолированных особенностей функций, удовлетворяющих дробно-полулинейному уравнению \( (-\Delta)^sv+\epsilon v^p=0 \) в проколотой области \( \mathrm{R}^N, \) где \(\epsilon=\pm 1,\) \(0<s<1\) и \(p>1.\) Мы получаем априорные оценки и анализируем множество самоподобных решений. Мы даем полное описание возможного поведения решений вблизи особенности.

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