Translator Disclaimer
March/April 2022 Bubbling solutions for a planar exponential nonlinear elliptic equation with a singular source
Jingyi Dong, Jiamei Hu, Yibin Zhang
Adv. Differential Equations 27(3/4): 147-192 (March/April 2022). DOI: 10.57262/ade027-0304-147

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following elliptic Dirichlet problem $$ \begin{cases} -\Delta\upsilon= e^{\upsilon}-s\phi_1-4\pi\alpha\delta_p-h(x)\,\,\,\, \,\textrm{in}\,\,\,\,\,\Omega,\\ \upsilon=0 \quad\quad\quad\quad\quad\quad \qquad\qquad\quad\quad\,\,\,\, \textrm{on}\,\ \,\partial\Omega, \end{cases} $$ where $s > 0$ is a large parameter, $h\in C^{0,\gamma}(\overline{\Omega})$, $p\in\Omega$, $\alpha\in(-1,+\infty)\setminus\mathbb{N}$, $\delta_p$ denotes the Dirac measure supported at point $p$ and $\phi_1$ is a positive first eigenfunction of the problem $-\Delta\phi=\lambda\phi$ under Dirichlet boundary condition in $\Omega$. If $p$ is a strict local maximum point of $\phi_1$, we show that such a problem has a family of solutions $\upsilon_s$ with arbitrary $m$ bubbles accumulating to $p$, and the quantity $$ \int_{\Omega}e^{\upsilon_s} \rightarrow8\pi(m+1+\alpha)\phi_1(p) \ \text{ as $s\rightarrow+\infty$.} $$

Citation

Download Citation

Jingyi Dong. Jiamei Hu. Yibin Zhang. "Bubbling solutions for a planar exponential nonlinear elliptic equation with a singular source." Adv. Differential Equations 27 (3/4) 147 - 192, March/April 2022. https://doi.org/10.57262/ade027-0304-147

Information

Published: March/April 2022
First available in Project Euclid: 7 February 2022

Digital Object Identifier: 10.57262/ade027-0304-147

Subjects:
Primary: 35B25 , 35B40 , 35J25

Rights: Copyright © 2022 Khayyam Publishing, Inc.

JOURNAL ARTICLE
46 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.27 • No. 3/4 • March/April 2022
Back to Top