As a “dog”, your company is in the low growth and low relative market share versus your competitors. Without knowing the specifics, there are a few ways to play it from there.
The first question to ask is are you the “natural owner” of the business. Sometimes you just have to grind it out and do the best you can as there is, as you say, no one to buy the business. Grinding it out might include bulking up. There are examples of competitive businesses, consolidating later in their industry life cycle and the last few standing gain pricing power and make good money. Tough to pull off, but it can happen.
Other than that, you could seek to reinvent your business… who would have known that a DVD delivery rental business could become the Netflix of today! Or Apple could transform from a niche PC manufacturer into the mobile telephony and services company it is today. Again, this can be difficult also, but it is the essence of strategy… to find a way to continually renew a business to remain relevant and profitable over the long term.
Finally, I wouldn’t assume that you can’t sell. Sometimes it is a matter of looking for the right kind of buyer – this can even be only from a financial perspective. As a more mature business, while you lack growth you may have revenue (albeit declining) and profit. Another business which has strong growth prospects, but lacks weight of revenue and profitability might combine with your business to help transform it to become a “Star”!
Don’t give up… sometimes “old Dogs have new tricks”!