A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group
The author considers a Dirichlet problem in the Heisenberg group \(\mathbb H^n\) that extends to the context of the Heisenberg group a result by \textit{M. J. Esteban} and \textit{P. L. Lions} related to the classical Laplacian in \(\mathbb R^n\) [Proc. R. Soc. Edinb., Sect. A 93, 1--14 (1982; Zbl 0506.35035)]. Before giving the statement of the main theorem, it will be convenient to express some background in the Heisenberg group setting. The Heisenberg group \(\mathbb H^n =\{\xi=(x,y,t)\}\) is the Lie group \((\mathbb R^{2n +1},\circ)\) with the composition law \[ \xi\circ\xi' = (x +x',y + y', t + t' + 2 (x'\cdot y- x\cdot y')) \] where \(\cdot\) denotes the inner product in \(\mathbb R^n.\) The Kohn-Laplacian is the operator on \(\mathbb H^n\) \[ \Delta_{\mathbb H^n} = \sum_{j=1}^{n} (\partial_{x_j} +2y_j\partial_t)^2 + (\partial_{y_j}-2x_j\partial_t)^2. \] The principal result is the following: Theorem 1. Let \(\Pi\) be an arbitrary half-space of \(\mathbb H^n\). Then the Dirichlet problem \[ -\Delta_{\mathbb H^n} u = \begin{cases} u^{(Q+2)/(Q-2)} & \text{ in } \Pi \\ u\in S_0^1(\Pi) &{} \end{cases} \] has no nontrivial nonnegative weak solutions. Here \(Q= 2n +2\) is the homogeneous dimension of \(\mathbb H^n.\) It is remarked that \((Q+2)/(Q-2)\) is the critical exponent for \(\Delta_{\mathbb H^n},\) as well as \((n+2)/(n-2)\) is the corresponding critical exponent of some important semilinear Poisson equations in \(\mathbb R^n \), e.g. the conformal scalar curvature equation in the neigbourhood of isolated singularities in the standard Euclidean ball. There is a Sobolev-type inequality and a notion of weak solution of the above Dirichlet problem on the Heisenberg group. The reader is referred to the article for further details. The techniques to prove the theorem, according to the author, is based on a systematic use of cylindrically symmetric barrier and make uses of the Rellich-Pokhozhaev type identity proved by \textit{N. Garofalo} and \textit{E. Lanconelli} in [Indiana Univ. Math. J. 41, No. 1, 71--98 (1992; Zbl 0793.35037)].
scientific article; zbMATH DE number 1310756