{"title":"On a class of Kirchhoff type problems with singular exponential nonlinearity","authors":"Mebarka Sattaf, Brahim Khaldi","doi":"10.5556/j.tkjm.55.2024.5097","DOIUrl":null,"url":null,"abstract":"We study the following singular Kirchhoff type problem \n\\[\\left( P\\right) \\left\\{ \n\\begin{array} [c]{c} \n-m\\left({\\displaystyle\\int\\limits_{\\Omega}}\\left\\vert \\nabla u\\right\\vert ^{2}dx\\right) \\Delta u=h\\left( u\\right) \n\\frac{e^{\\alpha u^{2}}}{\\left\\vert x\\right\\vert ^{\\beta}}\\text{ \\ \\ \\ in} \\Omega,\\\\ \nu=0 \\text{on}\\; \\partial\\Omega \n\\end{array} \\right. \n\\] \nwhere $\\Omega\\subset\\mathbb{R}^{2}$ 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 \non $\\mathbb{R}^{+}.$ 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 \n$(P)$. Our tools are Trudinger-Moser inequality with a singular weight and the mountain pass theorem","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":"88 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tamkang Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5556/j.tkjm.55.2024.5097","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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
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.