Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian

Abstract

<p style='text-indent:20px;'>In this paper, we investigate radial symmetry and monotonicity of positive solutions to a logarithmic Choquard equation involving a generalized nonlinear tempered fractional <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian operator by applying the direct method of moving planes. We first introduce a new kind of tempered fractional <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta-\lambda_{f})_{p}^{s} $\end{document}</tex-math></inline-formula> based on tempered fractional Laplacian <inline-formula><tex-math id="M5">\begin{document}$ (\Delta+\lambda)^{\beta/2} $\end{document}</tex-math></inline-formula>, which was originally defined in [<xref ref-type="bibr" rid="b3">3</xref>] by Deng et.al [Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul., 16(1)(2018), 125-149]. Then we discuss the decay of solutions at infinity and narrow region principle, which play a key role in obtaining the main result by the process of moving planes.</p>