2024/12/01 by Cheikh-Ali, Hussein, Premoselli, Bruno · 3 citations
#35B33 #35B40 #35B44 #35B45 #35J57 #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.00817
Let Ω be a bounded, smooth connected open domain in ℝn with n≥ 3. We investigate in this paper compactness properties for the set of sign-changing solutions v ∈ H10(Ω) of -Δv+h v =|v|2^*-2v \hbox in Ω, v = 0 \hbox on ∂ Ω where h∈ C1(Ω) and 2^*:=2n/(n-2). Our main result establishes that the set of sign-changing solutions of (*) at the lowest sign-changing energy level is unconditionally compact in C2(Ω) when 3 ≤ n ≤ 5, and is compact in C2(Ω) when n ≥ 7 provided h never vanishes in Ω. In dimensions n ≥ 7 our results apply when h >0 in Ω and thus complement the compactness result of Devillanova-Solimini, Adv. Diff. Eqs. 7 (2002). Our proof is based on a new, global pointwise description of blowing-up sequences of solutions of (*) that holds up to the boundary. We also prove more general compactness results under perturbations of h.