{"title":"The multiplicity of solutions for the critical problem involving the fracional p-Laplacian operator","authors":"Djamel Abid, K. Akrout, A. Ghanmi","doi":"10.5269/bspm.62706","DOIUrl":null,"url":null,"abstract":"This paper deals with the existence of multiple solutions \nfor the following critical fractional $p$-Laplacian problem \n\\begin{equation*} \n\\left\\{ \n\\begin{array}{l} \n\\mathbf{(-}\\Delta \\mathbf{)}_{p}^{s}u(x)=\\lambda \\left\\vert u\\right\\vert \n^{p-2}u+f(x,u)+\\mu g(x,u)\\ \\text{in }\\Omega ,u>0, \\\\ \n\\\\ \nu=0\\text{ on}\\ \\mathbb{R}^{n}\\setminus \\Omega ,% \n\\end{array}% \n\\right. \n\\end{equation*}% \nwhere $p>1$, $s\\in (0,1)$, $\\Omega \\subset \\mathbb{R}^{n}(n>ps),$ be a bounded smooth domain, $\\lambda $, $\\mu $ are positive parameters and the functions $f,g:\\overline{% \n\\Omega }\\times \\lbrack 0,\\infty )\\longrightarrow [0,\\infty),$ are continuous and differentiable with respect to the second variable. Our main tools are based on variational methods combined with a classical concentration \ncompacteness method.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Boletim Sociedade Paranaense de Matematica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5269/bspm.62706","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper deals with the existence of multiple solutions
for the following critical fractional $p$-Laplacian problem
\begin{equation*}
\left\{
\begin{array}{l}
\mathbf{(-}\Delta \mathbf{)}_{p}^{s}u(x)=\lambda \left\vert u\right\vert
^{p-2}u+f(x,u)+\mu g(x,u)\ \text{in }\Omega ,u>0, \\
\\
u=0\text{ on}\ \mathbb{R}^{n}\setminus \Omega ,%
\end{array}%
\right.
\end{equation*}%
where $p>1$, $s\in (0,1)$, $\Omega \subset \mathbb{R}^{n}(n>ps),$ be a bounded smooth domain, $\lambda $, $\mu $ are positive parameters and the functions $f,g:\overline{%
\Omega }\times \lbrack 0,\infty )\longrightarrow [0,\infty),$ are continuous and differentiable with respect to the second variable. Our main tools are based on variational methods combined with a classical concentration
compacteness method.