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

Garofalo, Nicola; Vassilev, Dimiter

Duke Math. J. 106, No. 3, 411-448 (2001).

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 \(V_1\).

Reviewer: Alberto Parmeggiani (Bologna)

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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