Design a Python web scraper using BeautifulSoup library that extracts product information (name, price, and description) from an e-commerce website's search results page. Save the extracted data into a JSON file.

First, you need to install the required libraries. You can do this by running:

```bash
pip install beautifulsoup4 requests
```

Now, let's create a Python script that will scrape product information from an e-commerce website using BeautifulSoup.

```python
import json
import requests
from bs4 import BeautifulSoup

# Replace with the URL of the search results page you want to scrape
url = 'https://www.example.com/search?q=product'

response = requests.get(url)
html_content = response.text

soup = BeautifulSoup(html_content, 'html.parser')

# Find the container that holds the product information.
# Replace the CSS selector with the appropriate one for your target website.
products_container = soup.select_one('.products-container')

# Find all individual products within the container.
# Replace the CSS selector with the appropriate one for your target website.
products = products_container.select('.product-item')

product_list = []

for product in products:
    # Extract product name, price, and description.
    # Replace the CSS selectors with the appropriate ones for your target website.
    name = product.select_one('.product-name').text.strip()
    price = product.select_one('.product-price').text.strip()
    description = product.select_one('.product-description').text.strip()

    product_info = {
        'name': name,
        'price': price,
        'description': description
    }

    product_list.append(product_info)

# Save the extracted data into a JSON file.
with open('products.json', 'w') as f:
    json.dump(product_list, f, indent=2)

print("Product information has been saved to products.json")
```

Replace the example URL and CSS selectors with the appropriate values for the e-commerce website you want to scrape. Run the script, and it will save the extracted product information into a JSON file named `products.json`.