By Pomponio A., Secchi S.

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Now, by the definition of tn , we have that E˜n (tn un ) ≥ inf sup E˜n (tu) =: bn . 3 with obvious modifications, that lim ε−N n bn = c 0 , n→∞ and this, together with (40), (42) and (43), easily implies the validity of Claim (39). 5. Set vn : x → un (εn x). We claim that sup Iz0 (tvn ) ≤ c0 + o(1). (44) t>0 To see this, we recall that {vn } is bounded in the H 1 norm. Since J(εn x)∇vn | ∇vn + V (εn x)|vn |2 = ε−1 n Ω g(εn x, vn )vn ≤ ε−1 n Ω f (vn )vn , ε−1 n Ω similar arguments as those above show that |vn |p+1 ≥ σ > 0.

Wang, B. Zeng, On concentration of positive bound states of nonlinear Schr¨odinger equations with competing potential functions, SIAM J. Math. , 28, (1997), 633–655. [31] M. Willem. Minimax theorems. Birkh¨auser, Boston, 1996.

1 n Ω But from assumption (f3) we see that F (u) ≥ Cuθ , so that tθ−2 n |vn |θ ≤ C. ε−1 n Ω This and (45) imply that {tn } is bounded. Therefore from (39) we deduce |tn vn |2 = 0. lim n→∞ ˜ RN \ε−1 n Λ 32 (46) From the properties of z0 we easily get (here we use (J1) to get rid of the ˜ contribution of J both inside and outside Λ) εn c0 + o(1) ≥ ε−N n E (tn un ) ≥ Iz0 (tn vn ) − t2n 2 V (z0 )|vn |2 . ˜ RN \ε−1 n Λ and so we get (44). If we set wn = tn vn , we see that {wn } is a minimizing sequence for Iz0 constrained to its Nehary manifold Nz0 .