Existence of positive solutions for a class of singular elliptic problems with convection term and critical exponential growth

IF 1.7 4区 数学 Q1 Mathematics
Sami Baraket, Anis Ben Ghorbal, Giovany M. Figueiredo
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引用次数: 0

Abstract

This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by $$\begin{aligned} \textstyle\begin{cases} -\Delta u= \displaystyle \frac {\lambda _{0}}{u^{\beta _{0}}} + \Lambda _{0} |\nabla u|^{\gamma _{0}}+ \frac{f_{0}(u)}{|x|^{\alpha _{0}}}+ h_{0}(x), \ \ u>0 \ \ \text{in} \ \Omega , \\ u=0 \ \text{on} \ \ \partial \Omega , \end{cases}\displaystyle \end{aligned}$$ where $\Omega \subset \mathbb{R}^{2}$ is a bounded smooth domain, $0<\beta _{0}$ , $\gamma _{0} \leq 1$ , $\alpha _{0} \in [0,2)$ , $h_{0}(x)\geq 0$ , $h_{0}\neq 0$ , $h_{0}\in L^{\infty}(\Omega )$ , $0<\|h_{0}\|_{\infty} < \lambda _{0} < \Lambda _{0}$ , and $f_{0}$ are continuous functions. More precisely, $f_{0}$ has a critical exponential growth, that is, the nonlinearity behaves like $\exp (\overline{\Upsilon}s^{2})$ as $|s| \to \infty $ , for some $\overline{\Upsilon}>0$ .
一类具有对流项和临界指数增长的奇异椭圆问题正解的存在性
本文使用 Galerkin 方法研究了一类奇异椭圆问题的正解存在性,该问题由 $$\begin{aligned} 给出。\textstyle\begin{cases} -\Delta u= \displaystyle \frac {\lambda _{0}}{u^{\beta _{0}}}+ Lambda _{0}|\u|^{gamma _{0}}+ \frac{f_{0}(u)}{|x|^{alpha _{0}}+ h_{0}(x), \ u>0 \\text{in}\ u=0 (on)\ end{cases}\displaystyle \end{aligned}$ 其中 $\Omega \subset \mathbb{R}^{2}$ 是一个有界的光滑域,$00$ 。
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来源期刊
Boundary Value Problems
Boundary Value Problems MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.00
自引率
5.90%
发文量
83
审稿时长
4 months
期刊介绍: The main aim of Boundary Value Problems is to provide a forum to promote, encourage, and bring together various disciplines which use the theory, methods, and applications of boundary value problems. Boundary Value Problems will publish very high quality research articles on boundary value problems for ordinary, functional, difference, elliptic, parabolic, and hyperbolic differential equations. Articles on singular, free, and ill-posed boundary value problems, and other areas of abstract and concrete analysis are welcome. In addition to regular research articles, Boundary Value Problems will publish review articles.
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