Entire weak solutions for an anisotropic equation in the Heisenberg group

Abstract:

Here, we consider an anisotropic equation \[ -\Delta _{{\mathbb {H}^n},\overrightarrow {p}}u +a(q)|u|^{p^–2}u=\lambda w(q)|u|^{m-2}u-h(q)|u|^{l-2}u, \] in the Heisenberg group ${\mathbb {H}^n}$, where the operator $\Delta _{{\mathbb {H}^n},\overrightarrow {p}}$ is the horizontal anisotropic $p$-Laplacian on the Heisenberg group and is defined in the sequel. By the variational methods, we prove the existence of the entire weak solutions.