By Olivier Druet
The authors turn out a few subtle asymptotic estimates for confident blow-up suggestions to $\Delta u+\epsilon u=n(n-2)u^{\frac{n+2}{n-2}}$ on $\Omega$, $\partial_\nu u=0$ on $\partial\Omega$, $\Omega$ being a tender bounded area of $\mathbb{R}^n$, $n\geq 3$. particularly, they express that focus can take place simply on boundary issues with nonpositive suggest curvature while $n=3$ or $n\geq 7$. As an instantaneous outcome, they end up the validity of the Lin-Ni's conjecture in size $n=3$ and $n\geq 7$ for suggest convex domain names and with bounded power. fresh examples via Wang-Wei-Yan convey that the certain at the strength is an important situation
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This proves the claim. 2. 2: We assume that d(xα , ∂Ω) =ρ α→+∞ rα lim 1. CONVERGENCE AT GENERAL SCALE 49 with ρ ∈ [0, +∞). In particular, limα→+∞ xα = x0 ∈ ∂Ω. We consider the domain ˜α Ux0 , the extension g˜ of the Euclidean metric ξ, the chart ϕ and the extension u defined in Lemma 2. Let R > 0 and let α > 0 large enough such that BR (0) ⊂ rα−1 (ϕ−1 (Ux0 ) − ϕ−1 (xα )). Let us define (x1,α , xα ) := ϕ−1 (xα ) with x1,α ≤ 0 and xα ∈ Rn−1 . Therefore, as is easily checked, we have that for any x ∈ BR (0), ϕ(ϕ−1 (xα ) + rα x) ∈ Ω ⇔ x1 ≤ |x1,α | .
Then there exists v˜i ∈ C 2 (Rn \ {θ˜j , σ(θ˜j )/ j ∈ Ii }) such that 1 lim v˜i,α = v˜i in Cloc (Rn \ {θ˜j , σ(θ˜j )/ j ∈ Ii }). 35) λj |x − θ˜j |2−n + |x − σ(θ˜j )|2−n for all x ∈ Rn \{θj , σ(θ˜j )/j ∈ Ii }. 36) v˜i (x) := + ψ˜i (x) for all x ∈ B2δ (0) \ {0} with ψ˜i (0) > 0. |x|n−2 Proof of Propositions 7 and 8: We apply Proposition 6. 1: we claim that points (i) to (v) of Proposition 6 hold with μα := μi,α and rα := si,α for all α ∈ N. We prove the claim. 1 We claim that (i) holds. We prove this claim via two claims.
4) si,α = o(d(xi,α , ∂Ω)) when α → +∞. , N } such that μi,α = o(μj,α ) when α → +∞ and si,α = μi,α 2 (μ + |xi,α − xj,α |2 ) μj,α j,α 1 2 for all α ∈ N. Proof of Theorem 3: For x ∈ s−1 i,α (Ω − xi,α ), we define vi,α (x) := sn−2 i,α n−2 uα (xi,α + si,α x). 7) vi (x) = λi + ψi (x) for all x ∈ B2δ (0) \ {0} with ψi (0) > 0. |x|n−2 We prove the claim. 8) si,α for all α ∈ N. In particular, vi,α is well defined on B1/2 (0). d(xi,α ,∂Ω) si,α Assume that limα→+∞ quences of Proposition 7. 8) that ρ ≥ 1 and that si,α limα→+∞ xi,α = x0 ∈ ∂Ω.