By Ndiaye C.B.

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Ph,u ∈ M4δ Bpi,u (2r) ⊂ M2δ and l points q1,u , . . , ql,u ∈ ∂M , Bq+i,u (2r) ⊂ ∂M × [0, 2δ ] such that e4u dVg < ; (106) M4δ \∪h i=1 Bpi,u (r) and ∂M ×[0, δ4 ]\∪li=1 Bq+i,u (r) e4u dVg < . Proof. Suppose that by contradiction the statement is not true. Then there exists > 0, r > 0, δ > 0 and a sequence (un ) ∈ H∂n such that M e4un dVg = 1, II(un ) → −∞ as n → +∞ and such that Eiter 1) e4un dVg < and k˜ tuples of points p1 , . . , pk ∈ M4δ and Bp (2r) ⊂ M2δ ,we have M \M4δ i e4u dVg < (107) M4δ ∩(∪h i=1 Bpi,u (r)) f dVg − ; M4δ Or 2) Mδ e4un dVg < and ∀k tuples of points q1 , .

Then there exists > 0, r > 0, δ > 0 and a sequence (un ) ∈ H∂n such that M e4un dVg = 1, II(un ) → −∞ as n → +∞ and such that Eiter 1) e4un dVg < and k˜ tuples of points p1 , . . , pk ∈ M4δ and Bp (2r) ⊂ M2δ ,we have M \M4δ i e4u dVg < (107) M4δ ∩(∪h i=1 Bpi,u (r)) f dVg − ; M4δ Or 2) Mδ e4un dVg < and ∀k tuples of points q1 , . . , qk ∈ ∂M we have 4 ∂M ×[0, δ4 ∩(∪li=1 Bq+i,u (r)) e4u dVg < f dVg − . ∂M ×[0, δ4 ] Or 3) M \M4δ e4un dVg ≥ , Mδ e4un dVg ≥ and ∀(h, l) ∈ N∗ × N∗ , 2h + l ≤ k, for every h tuples of points 4 p1 , .

F dVg > ε; Bp˜ (r) h and (104) Bq+ (r) 1 H f dVg > ε, . . , Bq+ (r) f dVg > ε; Bq+i (2r) ∩ Bq+j (2r) = ∅ for i = j. 1. It characterizes some functions in for which the value of II is large negative. 1, and for k ≥ 1 given by (6), the following property holds. For any > 0, and r > 0 (all small) there exists large positive L = L( , r) such that for any u ∈ H ∂ with II(u) ≤ −L, M e4u dVg = 1 the following holds, ∀δ > 0 (small) there the following holds ∂n 36 1) If that M \M4δ ∈ M4δ Bpi,u (2r) ⊂ M2δ such then we have there exists k˜ points p1,u , .

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Conformal metrics with constant q-curvature for manifolds with boundary by Ndiaye C.B.


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