2026/07/23 by Qiaoqiao Hua, Jungang Li, Chunxia Tao
#math.AP #math.DG
Let M=ℍn×\mathbbSm, where n≥ 2, m≥ 1, and N=n+m. Let Pk be the order-2k GJMS operator, with 1≤ k<N/2, and assume that Λ0=infσL2(M)(Pk)>0. We studyPkU-λU=|U|q-2U, q=(2N)/(N-2k), 0<λ≤Λ0,and attainment of the associated critical quotient Sλ,k(M). Let SN,k be the Euclidean best Sobolev constant. For 0<λ<Λ0, the inequality Sλ,k(M)<SN,k implies attainment and a nontrivial weak solution. Localized Euclidean extremals establish this inequality when N≥4k, or when 2k+2≤ N<4k and λ>Λloc, where Λloc is explicit. If N≥2k+2 and SΛ0,k(M)<SN,k, attainment also holds at λ=Λ0 in the threshold form completion. At the threshold, L2-coercivity fails precisely on the constant spherical eigenspace. We combine cocompactness for its hyperbolic coefficient with a profile decomposition relative to the critical transformations preserving A=ℝn-1×0. Under the threshold hypotheses above, the strict Euclidean inequality excludes concentration escaping A from normalized minimizing sequences. If rj(J) denotes the remainder after the first J extracted profiles, thenlimJ→∞\limsupj→∞|rj(J)|Lq(ℝN)=0,which yields compactness modulo the axis-preserving transformations.