It depends on the type of product and the type of stores.
If the stores sell thousands of different products, such as department and discount stores or grocery stores, they often try to match or beat the price of the competition for individual products. For example, Walmart stores check out the local competition regularly and lower prices on individual products so that the competition doesn't have lower prices. Another way stores can respond is to have a policy of matching advertised prices; if a customer shows a local store's advertised price, the store will match it.
Sometimes, stores don't try to match individual prices, but instead advertise themselves as having lower overall prices. This happens often in the grocery business. They count on customers wanting to do their shopping in one location instead of visiting several stores to get the best price on each individual product. They know that many customers don't have the time to research the prices on each product on their shopping lists and make several stops to do their shopping.
In short, there are different responses depending on the store policy.
It is different if the product is something that makes up a large share of a store's sales. In this case, stores that have the best financial resources, can control costs, and are run most efficiently can afford to sell at a lower price and run the competition out of business. Other than that, non-price competition can become part of the store's strategy. This could include such things as: Offering a better service plan; a better location; a more friendly atmosphere; better product expertise; easy credit terms; additional products available in the store.