At a bankruptcy sale in Puerto Rico last week, auctioneers sold off the manufacturing equipment of Brooks Shoe Co., which little more than a year ago was making a profit as the third-largest running shoe company in the nation.

On the same day, in a handsomely-furnished office on Capitol Hill, Philip H. Knight, president of Nike Inc., talked expansively of his company's future, of its bold leap into the European shoe market and the start of production in four factories in China.

Nike exploded from $28.7 million in annual sales in 1977 to $457.7 million last year and recently scored first in the nation in Forbes magazine's annual survey of major corporations' return on investment. Already the biggest athletic footwear company in the United States, Nike has opened a factory in Ireland to challenge world leader Adidas in Europe.

"We think we can compete with Adidas," said Knight, who has brought in former Secretary of Transportation Neil Goldschmidt as Nike's international marketing director. "It's a natural market for us."

Nike and Brooks represent the extremes of the volatile athletic shoe industry, an industry that grew from nothing to megabucks in about five years and now is entering a new phase of expansion, competition and shakeout of the weaker entries.

The running shoe business is no longer of interest only to athletes and insiders of the jogging craze. With annual sales in the U.S. alone of well over $1 billion, it has attracted the attention of Wall Street analysts, federal regulators and the business press, who watched several recent developments that seem sure to restructure the industry:

* Wolverine World Wide Inc., a Michigan firm best known as the manufacturer of Hush Puppies, has acquired the Brooks name, trademarks and patents, and will begin delivery of its new Brooks line to retailers next month.

* Allied Corp., the giant chemical conglomerate, put its Converse shoe subsidiary up for sale. Converse, which produces 12 million pairs of sports shoes a year and is the leader in basketball footwear, is profitable, but Allied said it no longer wants to be in the consumer products business.

* Running shoes, which before the big jogging boom were designed by runners in a kind of cottage industry, are now being designed by engineers and aerospace technicians, and prices are rising accordingly. New Balance, of Allston, Mass., one of the smaller companies, recently introduced a model called the 990, at a suggested retail price of $99.50.

* Following Nike's lead, several smaller running shoe companies have begun marketing apparel. Prominent runners who are technically amateurs, such as Bill Rodgers and Frank Shorter, have developed their own lines of athletic gear.

* Nike, which has imported most of its shoes from Taiwan and South Korea, has begun production at four factories in China. Up to now these have been lower-priced shoes that did not bear the Nike swoosh-stripe trademark, but Knight said the quality is now high enough that the China-made shoes can be marketed under the Nike brand.

Beyond these shifts, the industry is going through a period of uncertainty as the Federal Trade Commission, which subpoenaed documents from nearly all companies late in 1980, conducts a wide-ranging inquiry into competitive practices in the industry.

The FTC has filed no official complaints and has not even opened a public dossier on the case. The FTC has confirmed that its Cleveland office is conducting "an investigation, the purpose of which is to determine whether any person may have violated any provision of the laws administered by the commission," but it has rejected Freedom of Information Act requests for further information.

Also clouding the picture are several pending lawsuits that show, if nothing else, that the fraternity of running does not extend to cooperative business relationships.

Nike sued the Rubber Manufacturers Association and seven other shoe companies, including Brooks, in an antitrust case, claiming that they had conspired to persuade the U.S. Customs service to impose additional duties on Nike's imported shoes.

Brooks countersued, charging Nike with illegally attempting to monopolize the industry. In that case, Brooks embarrassed Nike by showing that prominent athletes, including Mark Moseley of the Redskins and Mike Schmidt of the Philadelphia Phillies, had altered other makers' shoes to make them look like Nikes.

Judge Dudley Bonsal of U.S. District Court in New York barred Nike from "authorizing, permitting or causing" any professional athlete to "doctor" another maker's shoes with the Nike trademark. The clear implication was that the athletes were receiving compensation from Nike to wear its shoes but preferred shoes of other companies.

Runner's World magazine sued Nike for libel after Nike withdrew its advertising, claiming that the magazine slanted its annual shoe ratings survey to favor Brooks at Nike's expense.

With a circulation that peaked at more than 400,000, the magazine has influenced especially inexperienced runners in their shoe selection. The five-star or highest rating it gave the Brooks Vantage model in 1977 propelled that company from a small firm operating out of a warehouse in Hanover, Pa., into the third-largest domestic running shoe company.

In 1978, Brooks reported a profit of $1 million on sales of $14 million, behind only Nike and Adidas in the United States. Sales more than doubled in 1979 and by 1980 the company had sales of $43 million and profits of $4.1 million.

The rapid growth in demand forced the company into a dramatic increase in production, and quality control problems developed almost immediately. Runners swore at the company as often as they swore by it.

Shoes were delivered to retailers without tongues or with the wrong number of eyelets. Nylon uppers frayed. As many as 30 percent of all shoes produced by Brooks, compared to less than one percent for Nike, were returned by retailers, causing a severe cash drain.

Brooks also strained its resources by an attempt to diversify that was apparently financed out of operating revenues. Brooks undertook a major expansion into tennis shoes and apparel, signing Jimmy Connors to a $125,000 a year promotion contract and buying the GUTS (General Universal Training Supplies) apparel line.

In addition, Brooks was plagued by a bitter feud between its two top officers, brothers-in-law Jay Goldenberg and Jerry Turner, which divided the loyalties of the company's managers.

"You either worked for Jay or you worked for Jerry, but you never saw the two of them talking. Each side thought the other side was incompetent," said one former employe. In fact, the two executives went for six months without saying anything more than hello to each other when passing in the hall, said the former employe.

Though sales continued to soar, profits peaked, then declined, and last year Brooks lost $1.4 million. For the first quarter of its 1982 fiscal year, Brooks sales, which had reached $53 million a year in 1981, were only $8.5 million, and the company showed a loss of $4 million.

In addition, the company was so heavily in debt after scrapping 50,000 pairs of defective shoes that in August the Puerto Rico Power and Energy Authority said Brooks' Puerto Rico plant owed an electric bill of $183,885.44 and had made no payments in eight months.

The plant is owned by the Puerto Rican government. Its equipment, which included modern shoe molding and sewing machinery, belonged to Brooks. It was all sold at auction last week, but the auctioneer declined to say how much money was raised.

Brooks executives refused to discuss the company's demise.

"They grew so quickly they just got out of control," said Thomas D. Gleason, president of Wolverine World Wide. To acquire the Brooks name and the right to manufacture shoes bearing Brooks trademarks, Wolverine World Wide paid $1.25 million, plus a percentage of sales for three years. Wolverine will collect Brooks' accounts receivable and take a percentage of the revenue.

There are no firm figures on market share of the overall sports shoe business. Nike's Knight calls the available numbers "pitiful." According to Sports Marketing and Retail Technology, an industry newsletter known as the SMART report, Nike has 48 percent to 60 percent of the market among "serious" runners, those who log 20 or more miles a week. Adidas has 20 percent to 25 percent and Brooks still has about 14 percent.

There is another whole market among joggers, school children, and others who wear running shoes because they are fashionable. Nike, unlike Brooks, has diversified its product line to capture some of that market, as well as the market for "court shoes," used in tennis or basketball.

As of 1981, according to Nike officials, 34 percent of the company's sales were running shoes, 38 percent court shoes, 15 percent children's footwear, 8 percent apparel and 4 percent field-sports and leisure shoes. The figures for running shoes were down sharply from the previous year, those for other products up correspondingly.

Knight said in an interview that the leisure-wear and sports-shoe business is still "on an upward trend" and will continue to grow because "it is health related." Nike intends to continue its corporate growth, he said, by expanding into new markets abroad, where it will take on entrenched competitors, and by continuing to diversify its line of domestic products.

Investors evidently share some of Knight's confidence about Nike's future. The company's stock, traded over the counter, was selling at about $28 a share last week, up from $19 last summer.

In going after the European market, Nike is taking on entrenched, dominant competiton, Adidas and Puma. Adidas especially is one of the first brand names known to -- and demanded by -- European children. Adidas' domination, however, is largely in soccer shoes, and Nike feels it can penetrate the growing European market for running shoes.

By establishing a factory in Ireland, Nike is able to ship shoes to Britain and the continent without paying import duties. The plant, which has a monthly capacity of 40,000 pairs of shoes, went into operation late last year.

Knight, co-founder of the company with Bill Bowerman, his former track coach at the University of Oregon, said in the firm's first annual report to stockholders that the formation of its Nike International subsidiary, headed by Goldschmidt, "highlights an all-out effort to increase our worldwide market share." How far that can go is not clear, because as Knight acknowledged with a grin, his record as a business prognosticator does not equal his record as a shoemaker.

"In 1976," he said, "we said that by 1981 we could be at $30 million" in annual sales. "It's gone better than anyone forecast."