تقرير
Semilinear elliptic Schr\'odinger equations with singular potentials and absorption terms
| العنوان: | Semilinear elliptic Schr\'odinger equations with singular potentials and absorption terms |
|---|---|
| المؤلفون: | Gkikas, Konstantinos T., Nguyen, Phuoc-Tai |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, 35J10, 35J25, 35J61, 35J75 |
| الوصف: | Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) 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 investigate 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 complex interplay between the competing effects of the inverse-square potential $d_\Sigma^{-2}$, the absorption term $g(u)$ and the measure data $\tau,\nu$ discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of $g$ for the existence of solutions. When $g$ is a power function, namely $g(u)=|u|^{p-1}u$ with $p>1$, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. $p$ is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. $p$ is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P). Comment: 40 pages |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1112/jlms.12844 |
| URL الوصول: | http://arxiv.org/abs/2203.01266 |
| رقم الأكسشن: | edsarx.2203.01266 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1112/jlms.12844 |
|---|