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

Abstract

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