Liouville-type results for elliptic equations with advection and potential terms on the Heisenberg group

Abstract

<p style='text-indent:20px;'>We investigate nonlinear elliptic equations of the form</p><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 \xi\in \mathbb{H}^n, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \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}^n\to \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}^n\to (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&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ u&gt;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.</p>