The role of the boundary in the existence of blow-up solutions for a doubly critical elliptic problem Cruz-Blázquez, Sergio Nonlinear elliptic equations Critical Sobolev exponents Blow-up solutions S.C. acknowledges financial support from the Spanish Ministry of Universities and Next Generation EU funds, through a Margarita Salas grant from the University of Granada, by the FEDER-MINECO Grant PID2021-122122NB-I00 and by J. Andalucia (FQM-116). He has also been partially supported by the INAM-GNAMPA project Fenomeni di blow-up per equazioni nonlineari, code CUP_E55F22000270001. In this paper we consider a doubly critical nonlinear elliptic problem with Neumann boundary conditions. The existence of blow-up solutions for this problem is related to the blow-up analysis of the classical geometric problem of prescribing negative scalar curvature K = −1 on a domain of Rn and mean curvature H = D(n(n − 1))−1/2, for some constant D > 1, on its boundary, via a conformal change of the metric. Assuming that n ≥ 6 and D > √(n + 1)/(n − 1), we establish the existence of a positive solution which concentrates around an elliptic boundary point which is a nondegenerate critical point of the original mean curvature. 2025-04-04T12:38:51Z 2025-04-04T12:38:51Z 2024-07-03 journal article Published version: Blázquez, S.C. The role of the boundary in the existence of blow-up solutions for a doubly critical elliptic problem. Nonlinear Differ. Equ. Appl. 32, 40 (2025). https://doi.org/10.1007/s00030-025-01042-w https://hdl.handle.net/10481/103463 10.1007/s00030-025-01042-w eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional Springer Nature