Bounded Solutions for a Class of Semilinear Equations on the Heisenberg Group

Abstract

We establish existence and nonexistence results of nonnegative nontrivial bounded solutions of the semilinear equation (E):\({\Delta }_{\mathbb {H}} u=q\, u^{\gamma }\) in \({\mathbb {H}^{N}}\) where \({\Delta }_{\mathbb {H}}\) is the Kohn Laplacian on the Heisenberg group \({\mathbb {H}^{N}}\), q is a nonnegative locally bounded Borel function in \(\mathbb {H}^{N}\) and γ is a positive real number. In the particular case where \(q(\xi )=\overline {q}(\rho (\xi ))\) is ρ-radially symmetric, it is shown that (E) has a nonnegative nontrivial bounded solution if and only if $${\int}_{0}^{\infty} r\overline{q}(r)\,dr<\infty. $$