Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group (Q1575059)

Denoting by \(\Delta_{\mathbb{H}}\) the Laplacian operator on the \((2N+1)\)-dimensional Heisenberg group \(\mathbb{H}^N\), we prove some nonexistence results for solutions of inequalities of the three types \[ \begin{aligned} \Delta_{\mathbb{H}}(au)+|u|^p &\leq 0,\\ \partial_tu- \Delta_{\mathbb{H}}(au)-|u|^p &\leq 0,\\ \partial_{tt}u- \Delta_{\mathbb{H}}(au)-|u|^p &\leq 0,\end{aligned} \] in \(\mathbb{H}^N\) and \(\mathbb{H}^N\times \mathbb{R}_+\), with \(a\in L^\infty\), when \(1< p\leq p_0\), where \(p_0\) depends on \(N\) and the type of the equation.