Nonlinear Liouville theorems for some critical problems on \(H\)-type groups.
Let \(G\) be a stratified group. Using variational techniques, one is led to characterize the energy levels of the variational solutions of the following semilinear Dirichlet problem with critical growth \[ -\Delta_G u= u^{(Q+ 2)/(Q-2)},\quad u\in S^1_0(\Omega),\quad u> 0,\tag{\(*\)} \] when \(\Omega\) is a half-space of \(G\) or the whose space. Here, \(\Delta_G\) is a sub-Laplacian on \(G\), \(Q\) denotes the homogeneous dimension of \(G\) and \(S^1_0(\Omega)\) is the appropriate subelliptic Sobolev space. The authors prove the following results: 1. Let \(G\) be a \(H\)-type group and let \(\Omega\) be any half-space of \(G\). Then the Dirichlet problem \((*)\) has no solution. 2. Let \(G\) be a step-two stratified group. Let \(\Omega\) be any half-space of \(G\) whose boundary is parallel to the center of \(G\). Then the Dirichlet problem \((*)\) has no solution. The core of the proofs consists in finding suitable asymptotic estimates for the second-layer derivatives of the solutions of \((*)\), which will allow to apply some general Rellich- and Pokhozhaev-type identities.
scientific article; zbMATH DE number 2062726
scientific article