论文标题
多临界椭圆问题的积极解决方案
Positive solutions to multi-critical elliptic problems
论文作者
论文摘要
在本文中，我们研究了以下多种解决方案的存在，以下到以下多临界椭圆问题\ begin {equination} \ label {eq：0.1} \ left \ left \ {\ oken {aligned}-ΔU＆=λ| +\ sum_ {i = 1}^k（| x |^{ - （n-α_i）}*| u |^{2^*_ i}）| U |^{2^*_ I-2} u \ quad {\ quad {\ quad {\ rm in}与有限域$ω\ subset \ mathbb {r}^n，\，n \ geq 4 $，其中$λ> 0 $，$ 2^*_ i = _ i = \ frac {n+α_i} $ n $ n-α_i} { i = 1,2，\ cdot \ cdot \ cdot，k $是关键的强硬木材 - 索伯夫指数，$ 2 <p <p <22^*_ {min} $，带有$ 2^*_ {min} = \ min} = \ min \ min \ {2^*_ I，我们表明有$λ^*> 0 $，这样，如果$ 0<λ<λ^*$问题\ eqref {eq：0.1}至少具有$cat_Ω（ω）$阳性解决方案。我们还研究了\ eqref {eq：0.1}的极限问题的解决方案的存在和唯一性。
In this paper, we investigate the existence of multiple solutions to the following multi-critical elliptic problem \begin{equation}\label{eq:0.1} \left\{\begin{aligned} -Δu & =λ|u|^{p-2}u +\sum_{i=1}^k(|x|^{-(N-α_i)}*|u|^{2^*_i})|u|^{2^*_i-2}u\quad {\rm in}\quad Ω,\\ &u\in H^1_0(Ω)\\ \end{aligned}\right. \end{equation} in connection with the topology of the bounded domain $Ω\subset \mathbb{R}^N, \,N\geq 4$, where $λ>0$, $2^*_i=\frac{N+α_i}{N-2}$ with $N-4<α_i<N,\ \ i=1,2,\cdot\cdot\cdot, k$ are critical Hardy-Littlewood-Sobolev exponents and $2<p<22^*_{min}$ with $2^*_{min}=\min\{2^*_i, \ i=1,2,\cdot\cdot\cdot, k\}$. We show that there is $λ^*>0$ such that if $0<λ<λ^*$ problem \eqref{eq:0.1} possesses at least $cat_Ω(Ω)$ positive solutions. We also study the existence and uniqueness of solutions for the limit problem of \eqref{eq:0.1}.
