The triharmonic equation on the Heisenberg group

Abstract

Abstract Consider the equation { - Δ H 3 ⁢ u = f ⁢ ( ξ , u , ∇ H ⁡ u , ∇ H 2 ⁡ u , ∇ H 3 ⁡ u , ∇ H 4 ⁡ u , ∇ H 5 ⁡ u ) in ⁢ Ω , u > 0 in ⁢ Ω , u = ∂ ∂ ⁡ ν ⁢ ( ∇ H 2 ⁡ u ) = ∂ ∂ ⁡ ν ⁢ ( ∇ H 3 ⁡ u ) = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\Delta_{H}^{3}u&\displaystyle=f(\xi,u% ,\nabla_{H}u,\nabla_{H}^{2}u,\nabla_{H}^{3}u,\nabla_{H}^{4}u,\nabla_{H}^{5}u)&% &\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle>0&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=\dfrac{\partial}{\partial\nu}(\nabla_{H}^{2}u)=% \dfrac{\partial}{\partial\nu}(\nabla_{H}^{3}u)=0&&\displaystyle\phantom{}\text% {on }\partial\Omega,\end{aligned}\right. where Ω is a domain of the finite-dimensional space ℍ n {\mathbb{H}^{n}} and f is a positive and bounded function. We prove the existence of a solution for the above equation. In addition, we prove the uniqueness and the cylindrical symmetry of the solution.