The Lin-Ni's problem for mean convex domains by Olivier Druet

By Olivier Druet

The authors turn out a few subtle asymptotic estimates for optimistic blow-up recommendations 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 gentle bounded area of $\mathbb{R}^n$, $n\geq 3$. particularly, they exhibit that focus can happen merely on boundary issues with nonpositive suggest curvature whilst $n=3$ or $n\geq 7$. As an instantaneous final result, they turn out 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 by means of Wang-Wei-Yan convey that the sure at the power is an important

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45) s2i,α μj,α = O(1) when α → +∞. μi,α (μ2j,α + |xi,α − xj,α |2 ) We prove the claim. We first assume that μj,α = o(μi,α ) when α → +∞. We then get that s2i,α μj,α μj,α s2i,α =O · = o(1) 2 2 μi,α (μj,α + |xi,α − xj,α | ) μi,α μ2j,α when α → +∞. 45) holds in this case. 5. 45) and then the claim. 6. 6. Therefore, points (i) to (v) of the hypothesis of Proposition 6 are satisfied with μα := μi,α and rα := si,α . This ends Step 1. Then we can apply Proposition 6 with rα := si,α and μα := μi,α . 46) lim α→+∞ d(xi,α , ∂Ω) = +∞.

CONVERGENCE AT GENERAL SCALE 47 for all α ∈ N. 4). 11) = μi,α rα 2 + |x − θi,α |2 ) μi,α μα lim α→+∞ n−2 2 n−2 2 ⎟ ⎠ |x − θi |2−n + o(1) for all x ∈ BR (0) \ {θi } when α → +∞. Note that these quantities are well-defined due to point (iv) of the hypothesis of Proposition 6. 2: Let i ∈ I c such that |xi,α − xα | = +∞. rα Let α0 ∈ N be large enough such that |xi,α − xα | ≥ 2Rrα for all α ≥ α0 . Then lim α→+∞ ||xα − xi,α + rα x| − |xi,α − xα || ≤ rα |x| = O(rα ) = o(|xα − xi,α |) when α → +∞ and uniformly for all x ∈ BR (0).

3: We fix i ∈ I. In particular, limα→+∞ xi,α = x0 . We assume that xi,α ∈ ∂Ω for all α ∈ N. 24) n−2 μα 2 = μi,α rα2 μα (μ2i,α + |ϕ((x1,α , xα ) + rα x) − πϕ−1 (xi,α )|2 ) n−2 2 for all α ∈ N and all x ∈ BR (0). Here, note that since we work in a neighborhood of x0 , we use the maps ϕ, π defined above. We define ((xi,α )1 , xi,α ) := ϕ−1 (xi,α ) for all α ∈ N. 26) when α → +∞. In particular, |x1,α | = ρ. 27) ˜i,α (ϕ((x1,α , xα ) + rα x)) rαn−2 U n−2 2 = lim α→+∞ μα μi,α μα n−2 2 |x−σ(θ˜i )|2−n +o(1) when α → +∞ uniformly on compact subsets of Rn \ {σ(θ˜i )}.

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