First thing would be to understand how require
works in NodeJS. I do recommend you go through this article if you're interested in knowing more about it.
Now, once you have required your connection, you have it for good and it won't be required again. This matches what you're looking for as you don't want to overwhelm your database by creating a new connection every time.
But, there is a problem...
Lambda Cold Starts
Whenever you invoke a Lambda function for the first time, it will spin up a container with your function inside it and keep it alive for approximately 5 mins. It's very likely (although not guaranteed) that you will hit the same container every time as long as you are making 1 request at a time. But what happens if you have 2 requests at the same time? Then another container will be spun up in parallel with the previous, already warmed up container. You have just created another connection on your database and now you have 2 containers. Now, guess what happens if you have 3 concurrent requests? Yes! One more container, which equals one more DB connection.
As long as there are new requests to your Lambda functions, by default, they will scale out to meet demand (you can configure it in the console to limit the execution to as many concurrent executions as you want - respecting your Account limits)
You cannot safely make sure you have a fixed amount of connections to your Database by simply requiring your code upon a Function's invocation. The good thing is that this is not your fault. This is just how Lambda functions behave.
...one other approach is
to cache the data you want in a real caching system, like ElasticCache, for example. You could then have one Lambda function be triggered by a CloudWatch Event that runs in a certain frequency of time. This function would then query your DB and store the results in your external cache. This way you make sure your DB connection is only opened by one Lambda at a time, because it will respect the CloudWatch Event, which turns out to run only once per trigger.
EDIT: after the OP sent a link in the comment sections, I have decided to add a few more info to clarify what the mentioned article wants to say
From the article:
"Simple. You ARE able to store variables outside the scope of our
handler function. This means that you are able to create your DB
connection pool outside of the handler function, which can then be
shared with each future invocation of that function. This allows for
pooling to occur."
And this is exactly what you're doing. And this works! But the problem is if you have N connections (Lambda Requests) at the same time. If you don't set any limits, by default, up to 1000 Lambda functions can be spun up concurrently. Now, if you then make another 1000 requests simultaneously in the next 5 minutes, it's very likely you won't be opening any new connections, because they have already been opened on previous invocations and the containers are still alive.