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Let Ω be an open subset of the Euclidean space R. If the boundary of Ω is sufficiently regular and p < n, then the Sobolev embedding W (Ω) →֒ L(Ω) holds, where p ≤ q ≤ p := np n−p . Additionally, if the set Ω is bounded, then the Rellich– Kondrachov compactness theorem W (Ω) →֒→֒ L(Ω) is satisfied, where p ≤ q < p (see e.g. [1]). Furthermore, if Ω is not bounded, then compactness not necessarily holds. Indeed, by taking Ω = R, nonzero φ ∈ C 0 (R) such that supp φ ⊂ [− 1 2 , 1 2 ] and un(x) = φ(x− n), we obtain that un is bounded in W (R), but there does not exist a subsequence of un that converges in L (R). On the other hand, suppose that n ≥ 2 and define the subspace of W (R) consisting of radially symmetric functions (invariant under the action of ortogonal group O(n)), i.e., W 1,p r (R ) := { u ∈ W (R) | u is radially symmetric }

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