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By Olivier Druet
The authors turn out a few sophisticated asymptotic estimates for optimistic blow-up options 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 exhibit that focus can happen simply on boundary issues with nonpositive suggest curvature whilst $n=3$ or $n\geq 7$. As a right away outcome, they end up the validity of the Lin-Ni's conjecture in measurement $n=3$ and $n\geq 7$ for suggest convex domain names and with bounded power. contemporary examples by means of Wang-Wei-Yan exhibit that the certain at the strength is an important situation
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The Lin-Ni's problem for mean convex domains
The authors turn out a few sophisticated asymptotic estimates for confident blow-up options 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 delicate bounded area of $\mathbb{R}^n$, $n\geq 3$. specifically, they express that focus can take place in basic terms on boundary issues with nonpositive suggest curvature whilst $n=3$ or $n\geq 7$.
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N } and assume that lim α→+∞ d(xi,α , ∂Ω) = ρ ∈ [0, +∞). si,α ´ ERIC ´ OLIVIER DRUET, FRED ROBERT, JUNCHENG WEI 52 In particular limα→+∞ xi,α = x0 ∈ ∂Ω. We let ϕ be a chart around x0 as in Lemma 2. For x ∈ s−1 i,α (Ω − xi,α ), we define v˜i,α (x) := sn−2 i,α n−2 2 u ˜α ◦ ϕ(ϕ−1 (xi,α ) + si,α x). , N }/ xj,α − xi,α = O(si,α ) and μj,α = o(si,α ) when α → +∞} and ϕ−1 (xj,α ) − ϕ−1 (xi,α ) for all j ∈ Ii . 33) We define σ(x1 , x ) := (2ρ − x1 , x ) for all (x1 , x ) ∈ Rn . 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 }).
3: we assume that d(xi,α , ∂Ω) = ρ ≥ 0. 54) lim In this case, the proof of Proposition 8 goes basically as the proof of Proposition 7. We stress here on the differences. 35) holds. We define δ := 1 min{|θ˜j |/ j ∈ Ii and θ˜j = 0}. 1. 2, we have that v˜i (x) = for all x ∈ B2δ (0). λi + ψ˜i (x) |x|n−2 ´ ERIC ´ OLIVIER DRUET, FRED ROBERT, JUNCHENG WEI 56 We claim that ψ˜i (0) > 0. 55) We prove the claim. 55) holds if K > 0. Assume that K = 0. 1: we assume that si,α := d(xi,α , ∂Ω) for all α ∈ N. 54) that that ρ = 1 > 0 and then σ(θ˜i ) = σ(0) = (2ρ, 0) = 0 and then ψ˜i (0) ≥ λi |σ(θ˜i )|2−n = λi (2ρ)2−n > 0.
3: From now on, we assume that Rκ+1,α (xα ) → +∞ as α → +∞ . 51) Rκ+1,α (xα ) for large α’s (the argument goes by contradiction). 52) |xα − x|2−n uα (x)2 ≤ n(n − 2)C0 −1 κ Ω + C0 Ui,α (x)2 − −1 dx i=1 |xα − x|2−n uα (x) dx + o (¯ uα ) . 33) that κ |xα − x| 2−n uα (x) dx ≤ A4 Ω n−2 |xα − x| μi,α2 =O |xα − x|2−n |xi,α − x|2 + μ2i,α i=κ+1 κ n−2 μi,α2 2 |xi,α − xα | + μ2i,α 1− n 2 Ω + O (¯ uα ) i=1 n−2 N |xi,α − xα |2 + μ2i,α 2 + O νκ,α 1− n 2 i=κ+1 κ =O n−2 Ui,α (xα ) i=1 1− n 2 N n−2 2 |xα − x|2−n dx + A4 νκ,α Ω 2 |xi,α − x| + μ2i,α dx Ω i=1 + A4 u ¯α 2−n 2−n 2 + O (¯ uα ) + O νκ,α Rκ+1,α (xα ) .