Permission problem installing bitcore
I solved this problem using NodeJS v4. I did a clean install again and ran these commands:

NOTE: My user is 'test' so my home is located at /home/test. Replace 'test' with your current user.


Install NVM:

$ curl -o- https://raw.githubusercontent.com/creationix/nvm/v0.33.0/install.sh |  bash


After that, close and reopen your terminal and test whether NVM was correctly installed:

$ nvm --version

Install NodeJS v4:

$ nvm install v4


Check whether NodeJS was correctly instaled:

$ node -v

Install ZeroMQ and Tools:

$ sudo apt-get install -y libzmq3-dev build-essential

Install Bitcore. Today (December 2017), the lastest version is 4.1.1. This version uses two different versions of bitcore-lib and cause a conflict when you run bitcored. So, install bitcore v4.1.0:

$ npm install -g bitcore@4.1.0

Done!! 


With these steps, I could use bitcore however they doesn't solve my original problem which is, install bitcore using NodeJS v8.9.3.