There are several issues in which libertarians disagree. One of the most noticeable of these is the issue of antitrust laws. Anarcho-capitalists believe in absolutely no government intervention in the economy. They believe that 100% free markets always function well. Other groups of libertarian thought would agree that minimal intervention is necessary, such as a social safety net, and many believe in some sort of antitrust laws as well.
Anarcho-capitalists, including many prominent libertarians such as Peter Schiff and Walter Block, say that the monopoly issue is not a problem in free markets. Walter Block argued in a speech in Auburn, AL, that companies who are under performing will die out naturally, leaving the best companies to take the market. He said that it is possible for one company to take the market, which is actually fine if they are maintaining good products and decent prices. If they are not producing a good product or charging a fair price, people are free to start new companies that charge less and produce more, an therefore the consumer will choose that company, forcing the other company to drop prices and increase production. Peter Schiff echoes Block's sentiment, saying that if a monopoly drives out competitors by lowering prices and then jacks up prices, the market will one again be flooded with competitors.
The majority of libertarians seem to agree that antitrust laws could be abolished. However, there is a sizable amount that say that antitrust laws should be maintained to some degree. Libertarians that say this can range from moderate libertarians, all the way to minarchists who believe that government's only role in economics is antitrust. Jeffery Miron of CATO has stated on his blog that "reasonable people can make a case for some degree of antitrust enforcement." The argument here is generally that once a monopoly is formed, it is much more difficult to build competing businesses once the monopoly does increase prices. This is in part due to the size. The monopoly is obviously quite large, and a competitor would likely have to build from the ground up. This takes long amounts of time and requires large amounts of work. The argument is that it is nearly impossible to introduce competitors once a monopoly is born.
There has been much in-depth research done on monopolies since they were abolished in the United States by Theodore Roosevelt. According to Thomas DiLorenzo of Mises, there is no evidence that a "natural monopoly," or one that is not aided by the government, ever existed. He notes that economist and Dean of Undergraduate Students at my University of Illinois, Horace Gray, said that "the public interest would be best promoted by grants of special privilege to private persons and to corporations." Indeed, when utility companies were monopolized in the United States, he notes that the companies were subsidized, thereby giving certain companies an edge. It is important to note this favoritism to certain companies. Had the government not subsidized companies, would that not have increased competition? It is important to ask if those monopolies would not have occurred had the playing field been level.
Competition is, however, a key attribute in Austrian Economics. Competition helps drive down prices and produce better products. Competition is touted by free market-activists as a reason to have free markets, so wouldn't taking away competition with a monopoly take away a reason to have free markets at all? This is what may lead some libertarians to argue that government's only role in the economy should be antitrust laws. This was, the theory of competition can be upheld and therefore the free market can be validated. This is why even those who are as small government as minarchists will sometimes express support for antitrust laws.
Antitrust laws are a key argument among libertarians. While the majority of well-known libertarians fall on the side of abolishment of antitrust, there still remains a sizable group in favor of the laws. It is up to you to decide which side you fall on. In the end, however, let me reiterate that we are all on the same side right now as libertarians. If we ever get to the point where our two-party system is between one party who favors antitrust and another that does not, then we may split, but at this point we are all committed to spreading libertarianism.