Existence and convergence of solutions to fractional pure critical exponent problems

TL;DR

This paper investigates the existence and convergence of symmetric solutions to fractional critical exponent problems, analyzing their behavior as the fractional parameter varies, and establishing nonexistence results in certain domains.

Contribution

It provides a comprehensive analysis of the convergence of least-energy symmetric solutions across different fractional orders and domains, including nonlocal-to-local transition insights.

Findings

Solutions converge to a limit as the fractional parameter varies.

Convergence in bounded domains characterized by fractional norms.

Nonexistence of nontrivial nonnegative solutions in a ball for s>1.

Abstract

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 $s>1$.