Entire weak solutions for an anisotropic equation in the Heisenberg group
Abdolrahman Razani · Proceedings of the American Mathematical Society · 2023
Here, we consider an anisotropic equation \[ − Δ H n , p → u + a ( q ) | u | p – 2 u = λ w ( q ) | u | m − 2 u − h ( q ) | u | l − 2 u , -\Delta _{{\mathbb {H}^n},\overrightarrow {p}}u +a(q)|u|^{p^–2}u=\lambda w(q)|u|^{m-2}u-h(q)|u|^{l-2}u, \] in the Heisenberg group H n {\mathbb {H}^n} , where the operator Δ H n , p → \Delta _{{\mathbb {H}^n},\overrightarrow {p}} is the horizontal anisotropic p 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.