Local behavior of solutions to subelliptic problems with Hardy potential on Carnot groups (Q723773)

The author considers the problem \[ -\Delta_{\mathbb G}u-\mu\frac{\psi^2}{d^2}u=f(\xi,u) \] in \(\Omega\), subject to the boundary condition \[ u=0 \] on \(\partial\Omega\). In the above equation, \(\Omega\) is an arbitrary open subset of \(\mathbb G\), where \(\mathbb G\) is a Carnot group of homogeneous dimension \(Q\geq 3\). In addition, \(\Delta_{\mathbb G}\) is the sub-Laplacian on \(\mathbb G\). The growth condition \[ \big| f(\xi,t)\big|\leq C\left(| t|+| t|^{2^\ast-1}\right), \] for all \(t\in\mathbb R\) is imposed, where \[ 2^\ast:=\frac{2Q}{Q-2}. \] The main result argues that whenever \(u\in S_0^1(\Omega)\) is a solution to the above problem under the above growth assumption, it follows that \(\big| u(\xi)\big|\) satisfies a specific quantitative bound.