Fractional p-Kirchhoff system with sign changing nonlinearities

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• 作者：Pawan Kumar Mishra ; K. Sreenadh
• 关键词：Fractional p ; Laplacian system ; Kirchhoff type problem ; Critical exponent problem
• 刊名：Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：111
• 期：1
• 页码：281-296
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general; Applications of Mathematics; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Milan
• ISSN：1579-1505
• 卷排序：111

文摘

In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the following fractional p-Kirchhoff system $$\begin{aligned}&M\left( \displaystyle \int _{\mathbb {R}^{2n}}\frac{|u(x)-u(y)|^p}{\left| x-y\right| ^{n+ps}}dx\,dy\right) (-\Delta )^{s}_p u(x) \\&\quad = \lambda f(x)|u|^{q-2}u+ \frac{2\alpha }{\alpha +\beta }\left| u\right| ^{\alpha -2}u|v|^\beta \quad \text {in } \Omega ,\\&M\left( \displaystyle \int _{\mathbb {R}^{2n}}\frac{|v(x)-v(y)|^p}{\left| x-y\right| ^{n+ps}}dx\,dy\right) (-\Delta )^{s}_p v(x) \\&\quad =\mu g(x)|v|^{q-2}v+ \frac{2\beta }{\alpha +\beta }\left| u\right| ^{\alpha }|v|^{\beta -2}v\quad \text {in } \Omega ,\\&u = v = 0 \quad \text {in }\; \mathbb {R}^{n}{\setminus }\Omega , \end{aligned}$$where \((-\Delta )^{s}_p\) is the fractional p-Laplace operator, \(\Omega \) is a bounded domain in \(\mathbb {R}^n\) with smooth boundary, M is continuous function, \(\lambda , \mu \) are real parameters, \(f,g \in L^{\gamma }(\Omega )\) with \(\gamma =\frac{\alpha +\beta }{\alpha +\beta -q}\) are sign changing, \(ps<n<2ps\) and \(1<q<p\), \(2p<r\le p_s^*=\frac{np}{n-ps}\).KeywordsFractional p-Laplacian systemKirchhoff type problemCritical exponent problemMathematics Subject Classification35J5035R1135B33References1.Adriouch, K., El Hamidi, A.: The Nehari manifold for systems of nonlinear elliptic equations. 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Anal. 7, 383–405 (2008)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer-Verlag Italia 2016Authors and AffiliationsPawan Kumar Mishra1Email authorK. Sreenadh11.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia About this article CrossMark Publisher Name Springer Milan Print ISSN 1578-7303 Online ISSN 1579-1505 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; border-bottom: 0; padding: 8px 12px; position: relative; background-color: #f2f2f2; } .buybox__header .buybox__login { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; letter-spacing: .017em; 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