Semilinear elliptic problems on unbounded subsets of the Heisenberg group (Q2707301)

The author studies the existence of solutions for a Dirichlet problem of the equation NEWLINE\[NEWLINE-\Delta_H u=f(u)\tag{1}NEWLINE\]NEWLINE on a (generally unbounded) domain of \(H^N\), where \(H^N\) be the space \(\mathbb{R}^N \times\mathbb{R}^N \times\mathbb{R}\) equipped with group operation NEWLINE\[NEWLINE\eta=(\alpha, \beta,\tau),\quad \eta\cdot \eta'=\bigl( \alpha +\alpha', \beta+\beta', \tau+\tau'+ 2(\alpha\beta'- \beta\alpha') \bigr)\tag{2}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\Delta_H= \sum^N_{i=1} \partial^2_{x_i} +\partial^2_{y_i} +4y_i \partial_{x_i} \partial_t- 4x_i\partial y_i\partial_t +4(x_i^2+y_i^2) \partial_t^2. \tag{3}NEWLINE\]NEWLINE The author proves the existence of (1). To this end he uses an abstract version of concentration compactness approach taking into account noncompactness cases.