Existence of radial solutions for a weighted p-biharmonic problem with navier boundary condition on the Heisenberg group

Abstract

Abstract The existence of at least one positive radial solution of the p-biharmonic problem Δ H n ( w ( ξ ) | Δ H n u | p − 2 Δ H n u ) + R ( ξ ) w ( ξ ) | u | p − 2 u = ∑ i = 1 m a i ( | ξ | H n ) | u | q i − 2 u − ∑ j = 1 k b j ( | ξ | H n ) | u | r j − 2 u , $$\begin{array}{}\displaystyle\Delta_{\mathbb{H}^n}\big(w(\xi)|\Delta_{\mathbb{H}^n}u|^{p-2}\Delta_{\mathbb{H}^n}u\big)+ R (\xi)w(\xi) |u|^{p-2} u=\sum_{i=1}^ma_i (|\xi|_{\mathbb{H}^n}) |u|^{q_i-2} u -\sum_{j=1}^k b_j (|\xi|_{\mathbb{H}^n}) |u|^{r_j-2}u,\end{array}$$ with Navier boundary condition on a Korányi ball is proved, where w ∈ As is a Muckenhoupt weight function and Δ H n , p 2 $\begin{array}{}\displaystyle\Delta^2_{\mathbb{H}^n, p}\end{array}$ is the Heisenberg p-biharmonic operator.