Abstract
Abstract We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem ( - Δ ) s u s = | u s | 2 s ⋆ - 2 u s , u s ∈ D 0 s ( Ω ) , 2 s ⋆ := 2 N N - 2 s , (-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^% {s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s}, where s is any positive number, Ω is either ℝ N {\mathbb{R}^{N}} or a smooth symmetric bounded domain, and D 0 s ( Ω ) {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 {u_{s}} converges to a l.e.s.s. u t {u_{t}} as s goes to any t > 0 {t>0} . In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order t - ε {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, 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 {s>1} .
Highlights
In this paper we study existence and convergence properties of solutions to pure critical problems such as (−∆)sus = |us|2⋆s−2us, us ∈ D0s(Ω), 2⋆ := 2N, s N − 2s (1.1)where N ≥ 1, s > 0, N > 2s, (−∆)s is the fractional Laplacian, Ω is either RN or a smooth bounded domain of RN, and D0s(Ω) is the homogeneous Sobolev space, namely, the closure of Cc∞(Ω) with respect to the Gagliardo seminorm · s, given by (2.2) below
The fractional Laplacian plays an important role in the study of anomalous and nonlocal diffusion, which appears for instance in continuum mechanics, graph theory, and ecology, see [16] and the references therein
It is not true that Dt(RN ) ⊂ Ds(RN ) for t > s, as it happens in bounded domains, and it is not trivial to find a suitable norm to describe the convergence properties of solutions; for instance, a characterization such as (1.6) is not possible in RN since us might not belong to Dt(RN ) for t = s
Summary
It is not true that Dt(RN ) ⊂ Ds(RN ) for t > s, as it happens in bounded domains, and it is not trivial to find a suitable norm to describe the convergence properties of solutions; for instance, a characterization such as (1.6) is not possible in RN since us might not belong to Dt(RN ) for t = s This is not a problem of local smoothness, but rather an incompatibility with the decay at infinity. For s ∈ (0, 1), this existence result was proved in the recent paper [46], for s = 1 it is shown in [18], and for s = 2 it is a particular case of [20, Theorem 1.1] All these papers follow a strategy based on a symmetric-concentration compactness argument, but at a technical level they have important differences and none of them can be extended to guarantee existence of solutions in the whole higher-order range s ∈ (1, ∞).
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