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- W4312490692 abstract "<p style='text-indent:20px;'>We investigate nonlinear elliptic equations of the form <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=FE1> begin{document}$ -Delta_{H} u(xi)+ A(xi) cdot nabla_{H} u(xi) = V(xi)f(u),quad xiin mathbb{H}^n, $end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id=M1>begin{document}$ mathbb{H}^n = (mathbb{R}^{2n+1},circ) $end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id=M2>begin{document}$ (2n+1) $end{document}</tex-math></inline-formula>-dimensional Heisenberg group, <inline-formula><tex-math id=M3>begin{document}$ Delta_{H} $end{document}</tex-math></inline-formula> is the Kohn-Laplacian operator, <inline-formula><tex-math id=M4>begin{document}$ nabla_{H} $end{document}</tex-math></inline-formula> is the Heisenberg gradient, <inline-formula><tex-math id=M5>begin{document}$ cdot $end{document}</tex-math></inline-formula> is the inner product in <inline-formula><tex-math id=M6>begin{document}$ mathbb{R}^{2n} $end{document}</tex-math></inline-formula>, the advection term <inline-formula><tex-math id=M7>begin{document}$ A: mathbb{H}^nto mathbb{R}^{2n} $end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id=M8>begin{document}$ C^1 $end{document}</tex-math></inline-formula> vector field satisfying a certain decay condition, the potential function <inline-formula><tex-math id=M9>begin{document}$ V: mathbb{H}^nto (0,infty) $end{document}</tex-math></inline-formula> is continuous, and the nonlinearity <inline-formula><tex-math id=M10>begin{document}$ f(u) $end{document}</tex-math></inline-formula> has the form <inline-formula><tex-math id=M11>begin{document}$ -u^{-p} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=M12>begin{document}$ p>0 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=M13>begin{document}$ u>0 $end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id=M14>begin{document}$ e^u $end{document}</tex-math></inline-formula>. Namely, we establish Liouville-type results for the class of stable solutions to the considered problems. Next, some special cases of the potential function <inline-formula><tex-math id=M15>begin{document}$ V $end{document}</tex-math></inline-formula> are discussed." @default.
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- W4312490692 date "2023-01-01" @default.
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- W4312490692 title "Liouville-type results for elliptic equations with advection and potential terms on the Heisenberg group" @default.
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- W4312490692 doi "https://doi.org/10.3934/dcdss.2022171" @default.
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