the above inequality and using (6.1) we get lim sup k→∞ Fk(vk, U \ Ju) α̂E(u,U ′ \ Ju) = α̂ ˆ U ′ W (x, e(u)) dx , (6.4) for all α̂ > α. Finally gathering together (6.2)–(6.4), we obtain lim sup k→∞ Fk(uk) [...] upper bound are summarized in remark 6.2 for the readers convenience. The proof of theorem 2.1 is divided into three main steps contained respectively in § 4, 5 and 6. As a consequence of theorem 2.1 and [...] https://doi.org/10.1017/prm.2023.51 28 R. Marziani and F. Solombrino 6. Upper bound In this section we prove the upper bound. Proposition 6.1. Let Fk and F be as in (2.5) and (2.10) respectively. Then for …