Rockefeller believed his success was the idea of " survival of the fittest " ( social darwinism) how did this idea him? what problems came from it?

Social Darwinism is how we’ve named the application of Charles Darwin's scientific theories of evolution and natural selection to contemporary social development.

The term “Social Darwinism” raised up in the 1880s, and it gained popularity altogether with other sociological theories that empowered and granted the justification of verge ideas like imperialism, racism, and laissez-faire ("let do") to promote social and economic policies.

Thanks to the development of some of those ideas, extreme capitalism was also established as a norm for economic growth, which helped a great deal of businessmen, including of course John D. Rockefeller, a man in the oil trade, who benefited from it becoming one of the best businessmen, by rapidly raising a fortune from his “Standard Oil Company” in 1870, revolutionizing the petroleum industry; and by the late 1800s Rockefeller  had nearly a monopoly of the oil business in the U.S., such business practices at a point, led to the adoption of “Antitrust Laws”  

“Antitrust Laws” also referred to as "competition laws," a collection of federal and state government laws nursing the regulation of the conduct of business corporations, to promote a fair competition that benefits consumers, protecting them from predatory business practices, ensuring fair competition in open-market economies.

Rockefeller joint other Social Darwinism enthusiasts like Russell H. Conwell in the early 1900s believing that building a fortune was a profound Christian duty, to follow God.

Monopoly is described as the exclusive possession or control of the supply in a commodity or service; but further than that, Rockefeller was accused of unethical business practices like acquiring pipelines and forest areas to prevent the success of competitors.  

Some other problems came from those ideas too; at its worst, Social Darwinism was used as scientific justification for the Holocaust.  

Aforementioned, many Darwinism enthusiasts embraced laissez-faire capitalism and racism, demanding that the governments should not interfere in the “survival of the fittest” by helping the poor. This promoted nonsensical ideas such as, whimsy human ranking based on external traits making certain individuals superior to others.  

Thus, “The Nazis” claimed that the murder of Jews in World War II was an example of cleaning out the inferior genetics.

Avoiding the resurgence of such detrimental ideas from popularity again requires the federal governments to investigate and pursue companies, and organizations suspected of being in violation of those statutes to limit cartels and monopolies.

the exclusive possession or control of the supply or trade in a commodity or service.