Existence of positive radial solution for Neumann problem on the Heisenberg group

Abstract

Abstract The existence of at least one positive radial solution of the Neumann problem − Δ H n u + R ( ξ ) u = a ( | ξ | H n ) | u | p − 2 u − b ( | ξ | H n ) | u | q − 2 u , $$ -\Delta _{\mathbb{H}^{n}} u+R(\xi ) u=a \bigl( \vert \xi \vert _{\mathbb{H}^{n}} \bigr) \vert u \vert ^{p-2} u - b\bigl( \vert \xi \vert _{\mathbb{H}^{n}}\bigr) \vert u \vert ^{q-2}u, $$ is proved on the Heisenberg group H n $\mathbb{H}^{n}$ , via the variational principle, where a ( | ξ | H n ) $a(|\xi |_{\mathbb{H}^{n}})$ , b ( | ξ | H n ) $b(|\xi |_{\mathbb{H}^{n}})$ are nonnegative radial functions and R ( ξ ) $R(\xi )$ satisfies suitable conditions.

Keywords

• Heisenberg group
• Variational principle
• Supercritical Neumann problem
• Mountain Pass Geometry theorem
• Palais–Smale compactness condition