https://www.jeremydaly.com/aurora-serverless-the-good-the-bad-and-the-scalable/Max Connections
A major limitation of relational databases in serverless architectures is the maximum number of concurrent connections allowed by the database engine. While FaaS services like Lambda may scale infinitely (in theory anyway), massive spikes in volume can quickly saturate the number of available connections to the underlying database. There are ways to manage connections in serverless environments (also see Managing MySQL at Serverless Scale), but even with Aurora Serverless, this still appears to be a possible limiting factor.
AWS uses the following formula for generating the max_connections value for Aurora instances:
log( ( <Instance Memory> * 1073741824) / 8187281408 ) * 1000 = <Default Max Connections>
A db.r4.xlarge instance with 30.5 GB of memory for example would have a default max_connections value of 2,000.
log( (30.5 * 1073741824) / 8187281408 ) * 1000 = 2000