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Washington (CNN) -- Parts of Arizona's sweeping immigration law received a surprising amount of support from a short-handed Supreme Court Wednesday.

States throughout the country considering their own tough immigration laws are closely following the proceedings over what has become a thorny issue.

Fed up with illegal immigrants crossing from Mexico -- and what they say is the federal government's inability to stop it -- legislators in Arizona passed a tough immigration law. The federal government sued, saying that Arizona overreached.

"If, in fact, somebody who does not belong in this country is in Arizona, Arizona has no power?" asked Justice Antonin Scalia. "What does sovereignty mean if it does not include the ability to defend your borders?"

Even liberal Justice Sonia Sotomayor told the federal governments' lawyer that key parts of his arguments were "not selling very well."

Federal courts had blocked four elements of the state's Support Our Law Enforcement and Safe Neighborhoods Act, known as SB 1070.

While intense oral arguments took place among the justices, outside there were competing demonstrations on the courthouse plaza, with the law's opponents saying it promotes discrimination and racial profiling. Backers say illegal immigration has created public safety and economic crises.

At issue is whether states have any authority to step in to enforce immigration matters or whether that is the exclusive role of the federal government. In dry legal terms, this constitutional question is known as pre-emption.

Other states watch for outcome

Paul Clement, lawyer for Arizona, told the high court the federal government has long failed to control the problem, and that states have discretion to assist in enforcing immigration laws.

But the Obama administration's solicitor general, Donald Verrilli, strongly countered that assertion, saying immigration matters are under the federal government's exclusive authority and state "interference" would only make matters worse.

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