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Positive solutions of Yamabe-type equations on the Heisenberg group. (English) Zbl 0948.35027
Let $${\mathbb H}^n$$ be the Heisenberg group of real dimension $$2n+1.$$ As it is well-known, $${\mathbb H}^n={\mathbb C}^n\times{\mathbb R}$$ is a manifold, and the group structure is given by $$(z,t)\cdot(z',t')=(z+z',t+t'+2\operatorname {Im}\langle z,z'\rangle_{{\mathbb C}^n})$$, where $$\langle\cdot,\cdot\rangle_{{\mathbb C}^n}$$ denotes the usual Hermitian structure on $${\mathbb C}^n.$$ Let us consider the Kohn-Spencer Laplacian defined by $$\Delta_{{\mathbb H}^n}:=\sum_{j=1}^n(X_j^2+Y_j^2),$$ where $X_j=2 \operatorname {Re} {{\partial}\over{\partial z_j}}+2 \operatorname {Im}z_j{{\partial}\over{\partial t}},\quad Y_j=2 \operatorname {Im} {{\partial}\over{\partial z_j}}-2 \operatorname {Re}z_j{{\partial}\over{\partial t}}, j=1,\ldots,n,$ that, along with $$\partial/\partial t,$$ span the Lie algebra of left-invariant vector fields on $${\mathbb H}^n.$$ In this paper, the authors consider the equation $\Delta_{{\mathbb H}^n}u+a(z,t)u-b(z,t)|u|^{\sigma -1}u=0,$ where $$\sigma >1.$$ Their aim is to give conditions on the coefficients $$a(z,t)$$ and $$b(z,t)$$ in order for the above equation to have, or not to have, a positive solution on $${\mathbb H}^n.$$ They recall how the above equation is related to the classical one relative to the CR-Yamabe problem, pointing out that one of the difficulties in the case they treat is due to the non-compactness of the manifold they work on. They prove a number of results about existence and non-existence of a positive solution $$u$$ on $${\mathbb H}^n$$ based on growth assumptions of the coefficients $$a$$ and $$b$$ of the kind $\psi(z,t)a_1(d(z,t))\leq a(z,t)\qquad \text{(and/or)}\qquad a(z,t)\leq\psi(z,t)a_2(d(z,t)),$ $\psi(z,t)b_1(d(z,t))\leq b(z,t)\qquad \text{(and/or)}\qquad b(z,t)\leq\psi(z,t)b_2(d(z,t)),$ where $$d(z,t)=(|z|^4+t^2)^{1/4},$$ $$\psi(z,t):=|z|^2/d(z,t)^2$$ ($$(z,t)\not=(0,0)$$), and where $$a_i,b_i$$ are suitable nonnegative continuous functions on $$[0,+\infty).$$

##### MSC:
 35H10 Hypoelliptic equations 22E25 Nilpotent and solvable Lie groups 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
##### Keywords:
Yamabe problem; subelliptic Laplacean
Full Text:
##### References:
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