We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical problem \begin{ equation*} (-\Delta)^su_s=|u_s|^{ 2^\star_s-2}u_s, \quad u_s\in D^s_0(\Omega),\quad 2^\star_s:=\frac{ 2N}{ N-2s}, \end{ equation*} where $s$ is any positive number, $\Omega$ is either $\mathbb{ R}^N$ or a smooth symmetric bounded domain, and $D^s_0(\Omega)$ is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign changing. We show that, up to a subsequence, a l.e.s.s. $u_s$ converges to a l.e.s.s. $u_{ t}$ as $s$ goes to any $t>0$. In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order $t-\varepsilon$. A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If $t$ is an integer, these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any $s>1$.
Categories: math.AP