تقرير
Asymptotic profiles of ground state solutions for Choquard equations with a general local perturbation
| العنوان: | Asymptotic profiles of ground state solutions for Choquard equations with a general local perturbation |
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| المؤلفون: | Ma, Shiwang, Moroz, Vitaly |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs |
| الوصف: | In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation $$ -\Delta u+\varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ g(u), \quad {\rm in} \ \mathbb R^N, \eqno(P_\varepsilon) $$ where $N\ge 3$ is an integer, $p=\frac{N+\alpha}{N}$, or $\frac{N+\alpha}{N-2}$, $I_\alpha$ is the Riesz potential and $\varepsilon>0$ is a parameter. Under some mild conditions on $g(u)$, we show that as $\varepsilon\to \infty$, after {\em a suitable rescaling} the ground state solutions of $(P_\varepsilon)$ converge to a particular solution of some limit equations, and establish a sharp asymptotic characterisation of such a rescaling, which depend in a non-trivial way on the asymptotic behavior of the function $g(s)$ at infinity and the space dimension $N$. Based on this study, we also present some results on the existence and asymptotic behaviors of positive normalized solutions of $(P_\varepsilon)$ with the normalization constraint $\int_{\mathbb R^N}|u|^2=a^2$. Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as $a\to 0$ and $a\to \infty$. Comment: 41 pages, 0 figure. arXiv admin note: substantial text overlap with arXiv:2302.13727 |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.02877 |
| رقم الأكسشن: | edsarx.2405.02877 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |