chore(docs): Add Neon's Postgres documents for edge runtime (#11129)

chore: Update PostgreSQL adapter in documentation examples

Co-authored-by: Nico Domino <[email protected]>

Author: k-taro56

docs/pages/getting-started/adapters/pg.mdx

@@ -51,6 +51,26 @@ export const { handlers, auth, signIn, signOut } = NextAuth({
 })
 ```
 
+If you are using [Neon](https://neon.tech)'s PostgreSQL like [Vercel Postgres](https://vercel.com/docs/postgres), you can use `@neondatabase/serverless` to work with edge runtime.
+
+```ts filename="./auth.ts"
+import NextAuth from "next-auth"
+import PostgresAdapter from "@auth/pg-adapter"
+import { Pool } from "@neondatabase/serverless"
+
+// *DO NOT* create a `Pool` here, outside the request handler.
+// Neon's Postgres cannot keep a pool alive between requests.
+
+export const { handlers, auth, signIn, signOut } = NextAuth(() => {
+  // Create a `Pool` inside the request handler.
+  const pool = new Pool({ connectionString: process.env.DATABASE_URL });
+  return {
+    adapter: PostgresAdapter(pool),
+    providers: [],
+  }
+})
+```
+
 </Code.Next>
 <Code.Svelte>
 
@@ -75,6 +95,26 @@ export const { handle, signIn, signOut } = SvelteKitAuth({
 })
 ```
 
+If you are using [Neon](https://neon.tech)'s PostgreSQL like [Vercel Postgres](https://vercel.com/docs/postgres), you can use `@neondatabase/serverless` to work with edge runtime.
+
+```ts filename="./src/auth.ts"
+import { SvelteKitAuth } from "@auth/sveltekit"
+import PostgresAdapter from "@auth/pg-adapter"
+import { Pool } from "@neondatabase/serverless"
+
+// *DO NOT* create a `Pool` here, outside the request handler.
+// Neon's Postgres cannot keep a pool alive between requests.
+
+export const { handle, signIn, signOut } = SvelteKitAuth(() => {
+  // Create a `Pool` inside the request handler.
+  const pool = new Pool({ connectionString: process.env.DATABASE_URL });
+  return {
+    adapter: PostgresAdapter(pool),
+    providers: [],
+  }
+})
+```
+
 </Code.Svelte>
 <Code.Express>