Entire Solutions of the Lane-Emden Equation in Heisenberg Groups

We consider the equation \(-\Delta_{\mathbb{H}^n}u=u^p\) in \(\mathbb{H}^n\). Using bifurcation theory, we give the existence of an entire solution when \(p>\frac{Q+2}{Q-2}\). By applying integration by parts, we derive the Liouville theorem for \(p\in(1,p_c(Q))\), where \[p_c(Q)=\frac{(Q-2)^2-4Q+8\sqrt{Q-1}}{(Q-2)(Q-10)}.\] Finally, we describe the asymptotic behavior of the solution outside a compact set with zero boundary conditions.