Johnson & Johnson (NYSE:JNJ) is stuck between a rock and a hard place. Instead of cowering in the corner being potentially smashed by competitors, it's decided to take on that rock.

Johnson & Johnson is suing the beloved American Red Cross -- our metaphorical rock -- for trademark infringement. The two institutions have both used the red cross on a white background as their trademark for more than a century. The shared symbol wasn't a problem with Johnson and Johnson, which owns the trademark, until the Red Cross started licensing it out to companies that make disaster preparedness supplies. If it lets the Red Cross continue to sell products using its trademark, Johnson & Johnson opens itself up to having other companies, such as competitors Pfizer (NYSE:PFE) and 3M (NYSE:MMM), use the symbol on their products. That would substantially lower the value of the brand it has built over the last 120 years.

Of course, Johnson & Johnson benefits more from the suit than just stopping potential trademark violations. The Red Cross has begun selling items such as first-aid kits and hand sanitizers at such large retail stores as Wal-Mart (NYSE:WMT) and Target (NYSE:TGT). Stopping the Red Cross from selling items would mean that it would no longer have to compete with the Red Cross' products for shelf space.

The Red Cross is fighting back and trying to make Johnson & Johnson look like a bully taking on a beloved charity. And they're at least partially right; Johnson & Johnson's suit asks that the charity do more than just stop using the symbol on products it sell. It wants the Red Cross to destroy its inventory, hand over any profits to Johnson and Johnson, and pay punitive damages, including legal fees. Maybe the lawyers are asking for that much as a starting place to bargain from, but it sounds like a potential public relations disaster to me.

It really is a no-win situation for Johnson and Johnson. The best it can do is try to settle the court case quickly and quietly. It doesn't really need any of the monetary damages it seeks in the lawsuit; they'll hardly put a dent in its cost structure tightening plan. Johnson & Johnson just needs to walk a fine line between having its brand eroded by the Red Cross and having its brand eroded by a public relations nightmare.

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Fool contributor Brian Orelli, Ph.D., doesn't own shares of any company mentioned in this article. The Fool has a disclosure policy.