## Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type.(English)Zbl 1012.35014

Let $$G$$ be a stratified, nilpotent Lie group (i.e. a Carnot group). Let $${\mathbf g}$$ denote its Lie algebra. Let $$G$$ have step 2, that is $${\mathbf g}=V_1\oplus V_2,$$ where $$V_2=[V_1,V_1]$$ and $$[V_1,V_2]=\{0\}.$$ Endow $${\mathbf g}$$ with an inner product $$\langle\cdot,\cdot\rangle$$ with respect to which the above $$\oplus$$ is orthogonal. Let $$J: V_2\rightarrow\text{End}(V_1)$$ be the map defined by $$\langle J(v_2)v_1',v''_2\rangle=\langle v_2,[v_1',v''_2]\rangle,$$ $$v_2\in V_2,$$ $$v_1',v''_2\in V_1.$$ One says that $$G$$ is of Heisenberg type precisely when for every given $$v_2\in V_2$$ with $$\langle v_2,v_2\rangle=1,$$ the map $$J(v_2): V_1\rightarrow V_1$$ is orthogonal. In this paper, the authors study positive solutions to the CR Yamabe equation $L u=-u^{(Q+2)/(Q-2)},\quad u\in\overline{C_0^\infty(\Omega)}^{\|\cdot\|_{1,2}},\quad u\geq 0,$ on groups of Heisenberg type, where $$\Omega\subset G$$ is a domain of $$G,$$ $$L=-\sum_{j=1}^{m} X_j^*X_j$$ is a given sub-Laplacian on $$G$$, $$\{X_1,\ldots,X_m\}$$ being a basis of $$V_1,$$ $$Q=\dim V_1+2\dim V_2$$ is the homogeneous dimension of $$G$$, and finally where $$\|u\|_{1,2}= \|u\|_{L^{2Q/(Q-2)}(\Omega)}+\sum_{j=1}^{m}\|X_ju\|_{L^2(\Omega)}.$$ For the subclass of groups of Iwasawa type, they characterize those solutions that are invariant with respect to the action of the orthogonal group in $$V_1$$.

### MSC:

 35H20 Subelliptic equations 35A30 Geometric theory, characteristics, transformations in context of PDEs 43A80 Analysis on other specific Lie groups

### Keywords:

Carnot group; sub-Laplacian
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