تقرير
Semilinear elliptic Schr\'odinger equations involving singular potentials and source terms
| العنوان: | Semilinear elliptic Schr\'odinger equations involving singular potentials and source terms |
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| المؤلفون: | Gkikas, Konstantinos T., Nguyen, Phuoc-Tai |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, 35J10, 35J25, 35J61, 35J75 |
| الوصف: | Let $\Omega \subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in $\Omega \setminus \Sigma$, where $d_\Sigma(x) = \mathrm{dist}(x,\Sigma)$ and $\mu$ is a parameter. We study the boundary value problem (P) $-L_\mu u = g(u) + \tau$ in $\Omega \setminus \Sigma$ with condition $u=\nu$ on $\partial \Omega \cup \Sigma$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function and $\tau$ and $\nu$ are positive measures. The interplay between the inverse-square potential $d_\Sigma^{-2}$, the nature of the source term $g(u)$ and the measure data $\tau,\nu$ yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases. Comment: We have modified statement (iii) of Theorem 1.3 and its proof. We also removed Theorem 1.4 in the former version. arXiv admin note: text overlap with arXiv:2203.01266 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2203.01328 |
| رقم الأكسشن: | edsarx.2203.01328 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |