Abstract

We determine the exact behavior at the singularity of solutions to semilinear subelliptic problems of the type $$-\Delta _{\mathbb {G}}u -\mu \dfrac{\psi ^2}{d^2} u =f(\xi ,u)$$ in $$\Omega $$ , $$u=0$$ on $$\partial \Omega $$ , where $$\Delta _{\mathbb {G}}$$ is a sub-Laplacian on a Carnot group $$\mathbb {G}$$ of homogeneous dimension Q, $$\Omega $$ is an open subset of $$\mathbb {G}$$ , $$0\in \Omega $$ , d is the gauge norm on $$\mathbb {G}$$ , $$\psi :=|\nabla _{\mathbb {G}}d|$$ , where $$\nabla _{\mathbb {G}}$$ is the horizontal gradient associated with $$\Delta _{\mathbb {G}}$$ , f has at most critical growth and $$0\le \mu < \overline{\mu }$$ , where $$\overline{\mu }=\left( \frac{Q-2}{2} \right) ^2$$ is the best Hardy constant on $$\mathbb {G}$$ .

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