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Schr\”{o}dinger-Kirchhoff equations of fractional $p$-Laplacian involving logarithmic and critical nonlinearity
  • Huilin Lv,
  • Shenzhou Zheng
Huilin Lv
Beijing Jiaotong University
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Shenzhou Zheng
Beijing Jiaotong University

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Abstract

For $00$, we consider a class of Schr\”{o}dinger-Kirchhoff equations of fractional $p$-Laplacian involving logarithmic nonlinearity and critical exponential growth: \begin{equation*} m\left( \left\Vert u\right\Vert ^{p}\right) \left( \left( -\Delta \right) _{p}^{s}u+\lambda V\left( x\right) \left\vert u\right\vert ^{p-2}u\right) =\lambda h\left( x\right) \left\vert u\right\vert ^{\theta p-2}u\ln \left\vert u\right\vert +\sigma \left\vert u\right\vert ^{p_{s}^{\ast }-2}u\quad \mbox{in}\ \mathbb{R}^{n} \end{equation*}% with \begin{equation*} \left\Vert u\right\Vert =\left( \int_{\mathbb{R}^{2n}}\frac{\left\vert u\left( x\right) -u\left( y\right) \right\vert ^{p}}{\left\vert x-y\right\vert ^{n+sp}}dxdy+\lambda \int_{\mathbb{R}^{n}}V\left( x\right) \left\vert u\right\vert ^{p}dx\right) ^{1/p}, \end{equation*}% where $p_{s}^{\ast }=np/\left( n-sp\right) $, $1\leq \theta