Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem (-Delta)(s)u(s) = vertical bar u(s)vertical bar(2)(*-2)(s)u(s), u(s) is an element of D-0(s)(Omega), 2(s)* :=2N/N - 2s, where s is any positive number, Omega is either IRN or a smooth symmetric bounded domain, and D-0(s) (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 - epsilon. 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, then 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.