On a class of Kirchhoff type problems with singular exponential nonlinearity

IF 0.7 Q2 MATHEMATICS Tamkang Journal of Mathematics Pub Date : 2023-04-26 DOI:10.5556/j.tkjm.55.2024.5097
Mebarka Sattaf, Brahim Khaldi
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Abstract

We study the following singular Kirchhoff type problem \[\left( P\right) \left\{ \begin{array} [c]{c} -m\left({\displaystyle\int\limits_{\Omega}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u=h\left( u\right) \frac{e^{\alpha u^{2}}}{\left\vert x\right\vert ^{\beta}}\text{ \ \ \ in} \Omega,\\ u=0 \text{on}\; \partial\Omega \end{array} \right. \] where $\Omega\subset\mathbb{R}^{2}$ is a bounded domain with smooth boundary and $0\in\Omega,$ $\beta\in\left[ 0,2\right)$, $\alpha>0$ and $m$ is a continuous function on $\mathbb{R}^{+}.$ Here, $h$ is a suitable preturbation of $e^{\alpha u^{2}}$ as $u\rightarrow\infty.$ In this paper, we prove the existence of solutions of $(P)$. Our tools are Trudinger-Moser inequality with a singular weight and the mountain pass theorem
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一类奇异指数非线性Kirchhoff型问题
我们研究如下奇异Kirchhoff型问题 \[\left( P\right) \left\{ \begin{array} [c]{c} -m\left({\displaystyle\int\limits_{\Omega}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u=h\left( u\right) \frac{e^{\alpha u^{2}}}{\left\vert x\right\vert ^{\beta}}\text{ \ \ \ in} \Omega,\\ u=0 \text{on}\; \partial\Omega \end{array} \right. \] 在哪里 $\Omega\subset\mathbb{R}^{2}$ 有界域是否具有光滑边界和 $0\in\Omega,$ $\beta\in\left[ 0,2\right)$, $\alpha>0$ 和 $m$ 是连续函数吗 $\mathbb{R}^{+}.$ 这里, $h$ 合适的预扰是 $e^{\alpha u^{2}}$ as $u\rightarrow\infty.$ 的解的存在性 $(P)$. 我们的工具是奇异权的Trudinger-Moser不等式和山口定理
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来源期刊
CiteScore
1.50
自引率
0.00%
发文量
11
期刊介绍: To promote research interactions between local and overseas researchers, the Department has been publishing an international mathematics journal, the Tamkang Journal of Mathematics. The journal started as a biannual journal in 1970 and is devoted to high-quality original research papers in pure and applied mathematics. In 1985 it has become a quarterly journal. The four issues are out for distribution at the end of March, June, September and December. The articles published in Tamkang Journal of Mathematics cover diverse mathematical disciplines. Submission of papers comes from all over the world. All articles are subjected to peer review from an international pool of referees.
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