Positive solutions of Yamabe-type equations on the Heisenberg group
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 \([0,+\infty).\)
scientific article; zbMATH DE number 1425077
scientific article