论文标题
加权Sobolev空间和双曲线拉普拉斯方程I
Weighted Sobolev Space and Hyperbolic Laplacian Equations I
论文作者
论文摘要
在本文中，双曲空间中的以下问题$ \ mathbb {b}^n $将被视为\ begin {equination*}-Δ_ {\ mathbb {b}^n} u = f（x，x，u），\ mathrm {in}在其中，$δ_ {\ mathbb {b}^n} $表示$ \ mathbb {b}^n $上的laplace beltrami操作员。并且可以将此问题转换为以下欧几里得问题\ begin {equination*} \ begin {case} - \ propatoTorname {div}（k（x）\ nabla u）= 4 k（x）^{\ frac {\ frac {n} n} \\ u（0）= 0，＆\ mathrm {on} \ \ partial \ partbb {b}^n，\ end {cases} \ eqno {（2）} \ end end {equication {qore {qore {qore*}，其中，$ k（x）：$ k（x）：= 1/\ weft（= 1/\ weft（1- | x | x | x |^2 \ right）^$ nocy n of inse^$ nose^$ nose^$ n of inse^$ n of inse^$ and。可以通过研究问题的存在来获得（2）。我们将配备问题（2）的加权Sobolev空间，并证明了加权Sobolev空间的紧凑型定理和浓度紧凑性原理。我们将证明，最大原则保留了操作员$ - \ operatorname {div}（k（x）\ nabla u）$。 当$ f（x，u）= | u |^{2^*-2} u+λu^{q-2} u $，$λ> 0 $，$ 1 <q <q <q <2^{\ ast} $，使用变分的方法，紧凑的嵌入theorem，ly be e gradial of corge of corge（2）。
In this paper, the following problem in the hyperbolic space $\mathbb{B}^N$ will be considered \begin{equation*} -Δ_{\mathbb{B}^N} u=f(x,u), \mathrm{in} \ \mathbb{B}^N.\eqno{(1)} \end{equation*} where, $Δ_{\mathbb{B}^N}$ denotes the Laplace Beltrami operator on $\mathbb{B}^N$. And this problem can be converted into the following Euclidean problem \begin{equation*} \begin{cases} -\operatorname{div}(K(x) \nabla u)=4 K(x)^{\frac{N}{N-2}}f(x,u), &\mathrm{in} \ \mathbb{B}^N, \\ u(0)=0, &\mathrm{on}\ \partial\mathbb{B}^N, \end{cases}\eqno{(2)} \end{equation*} where, $K(x):=1/\left(1-|x|^2\right)^{N-2}.$ Then, the existence of solution of problem (1) can be obtained by studying the existence of solution of problem (2). We will equip problem (2) with a weighted Sobolev space and prove the compact embedding theorem and the concentration compactness principle for the weighted Sobolev space. And we will prove that the maximum principle holds for the operator $-\operatorname{div}(K(x) \nabla u)$. When $f(x,u)=|u|^{2^*-2} u+λu^{q-2}u$, $λ>0$, $1<q<2^{\ast}$, using the variational method, the compact embedding theorem, the concentration compactness principle and the maximum principle, the existence of nonradial solutions of problem (2) will be obtained.
