The multiplicity of solutions for the critical problem involving the fracional p-Laplacian operator

IF 0.4 Q4 MATHEMATICS Boletim Sociedade Paranaense de Matematica Pub Date : 2022-12-27 DOI:10.5269/bspm.62706
Djamel Abid, K. Akrout, A. Ghanmi
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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.
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涉及分数阶p-拉普拉斯算子的临界问题解的多重性
本文讨论了下列临界分数阶拉普拉斯问题\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的多重解的存在性。\end{equation*}%其中$p>1$, $s\in (0,1)$, $\Omega \子集\mathbb{R}^{n}(n>ps) $是有界光滑域,$ \lambda $, $\mu $是正参数,函数$f,g:\overline{% \Omega}\乘以\ lbrack0,\ inty)\ lonightarrow [0,\ inty),$是连续的,对于第二个变量是可微的。我们的主要工具是基于变分方法结合经典的浓度紧致法。
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来源期刊
CiteScore
1.40
自引率
0.00%
发文量
140
审稿时长
25 weeks
期刊最新文献
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