Weighted Sobolev Space and Hyperbolic Laplacian Equations I

TL;DR

This paper develops a framework using weighted Sobolev spaces to analyze hyperbolic Laplacian equations, establishing key properties and proving the existence of solutions with variational methods.

Contribution

It introduces a weighted Sobolev space approach for hyperbolic Laplacian equations, proving compactness, maximum principle, and existence of nonradial solutions.

Findings

Established compact embedding theorem for weighted Sobolev space.

Proved maximum principle for the operator involved.

Proved existence of nonradial solutions using variational methods.

Abstract

In this paper, the following problem in the hyperbolic space $\mathbb{B}^N$ will be considered \begin{equation*} -\Delta_{\mathbb{B}^N} u=f(x,u), \mathrm{in} \ \mathbb{B}^N.\eqno{(1)} \end{equation*} where, $\Delta_{\mathbb{B}^N}$ 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 $f(x,u)=|u|^{2^*-2} u+\lambda u^{q-2}u$, $\lambda>0$, $1<q<2^{\ast}$, 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.