论文标题
Yamabe型方程和Brézis-Nirenberg问题的气泡塔的尺寸$ N \ GE 7 $
Towers of bubbles for Yamabe-type equations and for the Brézis-Nirenberg problem in dimensions $n \ge 7$
论文作者
论文摘要
令$（m，g）$是尺寸$ n \ ge 7 $和正Yamabe类型的当地封闭的当地封闭式riemannian歧管。如果$ξ_0$表示质量函数的非分级关键点，我们证明了存在的存在，对于任何$ k \ ge 1 $和$ \ \ varepsilon> 0 $，是一个积极的爆破解决方案$ _ {\ u _ {\ varepsilon} $ _ { H \ big）证明方法结合了有限维度的降低与线性问题解决方案的尖锐分析。作为这种证明方法的另一种应用，我们为brézis-nirenberg问题构建签字吹式解决方案$ u _ {\ varepsilon} $ $ \triangle_ξu _ {\ varepsilon} - \ varepsilon} - \ varepsilon u _ {\ varepsilon u _ {\ varepsilon} = | u _ {\ varepsilon} |^{\ frac {\ frac {4} {n-2}} u _ {\ varepsilon} \ textrm {in}ω，\ quad u _ {\ quad u _ {\ varepsilon} = 0 \ Mathbb {r}^n $，$ n \ ge 7 $，看起来像$ k $阳性符号的叠加。
Let $(M,g)$ be a closed locally conformally flat Riemannian manifold of dimension $n \ge 7$ and of positive Yamabe type. If $ξ_0$ denotes a non-degenerate critical point of the mass function we prove the existence, for any $ k \ge 1$ and $\varepsilon >0$, of a positive blowing-up solution $u_{\varepsilon}$ of $$\triangle_g u_{\varepsilon} +\big( c_n S_g +\varepsilon h\big) u_{\varepsilon} = u_{\varepsilon}^{2^*-1},$$ that blows up like the superposition of $k$ positive bubbles concentrating at different speeds at $ξ_0$. The method of proof combines a finite-dimensional reduction with the sharp pointwise analysis of solutions of a linear problem. As another application of this method of proof we construct sign-changing blowing-up solutions $u_{\varepsilon}$ for the Brézis-Nirenberg problem $$ \triangle_ξ u_{\varepsilon} - \varepsilon u_{\varepsilon} = |u_{\varepsilon}|^{\frac{4}{n-2}} u_{\varepsilon} \textrm{ in } Ω, \quad u_{\varepsilon} = 0 \textrm{ on } \partial Ω$$ on a smooth bounded open set $Ω\subset \mathbb{R}^n$, $n \ge 7$, that look like the superposition of $k$ positive bubbles of alternating sign.
