By James Reynolds
BBC correspondent in Gaza City
The Israeli action overnight, which included taking out a power station, bombing a bridge a few miles away, taking up positions about 32 km (20 miles) from here, and carrying out sonic booms, was perhaps designed to show the Palestinians that Israel means business.
Palestinian gunmen have been on patrol overnight
But it is trying to stand back and not engage with Palestinian gunmen.
I've spoken to a lot of Palestinians this morning who say that during the night gunmen were on patrol in various neighbourhoods here in Gaza City and elsewhere in the centre and the south of the Gaza Strip, ready to fight the Israelis if they came in.
I understand they didn't confront the Israeli forces.
Israel kept its distance, perhaps knowing that if it engages with those gunmen, and if many Palestinians are killed, then that would deeply endanger the life of the Israeli hostage.
Despite this there was a fair amount of fear overnight.
I spoke to people who said they were terrified.
There were several sonic booms for example, which came at half hourly intervals, they were thunderingly loud, and scared a lot of people.
So most families simply stayed indoors.
People are now going to work, and although one of the power stations was hit at one or two in the morning by the Israeli army, power has been rerouted.
So electricity has come back to parts of Gaza City, where I'm standing.
Bridges are down but I understand people can use side roads to get from one part of Gaza to the other.
The Israeli ground forces are focusing on one particular area, a southern corner of the Gaza Strip where they're setting up observation posts.
They say this is designed to try to stop the kidnappers moving the hostage around, in particular to stop them smuggling him out of Gaza.
Israeli soldiers are massed on Gaza's borders
One of the issues in Gaza politics over the last few months has been the various different parts of the Palestinian leadership.
You have one democratically elected president, Mahmoud Abbas, who's from the moderate Fatah party, who believes in a two-state solution.
He is being held responsible for everything that happens in the Gaza Strip by Israel.
Indeed Israel has suggested they won't allow him to leave until the soldier is released.
That's one government - then you have a second government, also democratically elected, run by Hamas.
Hamas finds itself being in the position of being an elected government which is meant to be responsible, but having its armed military wing responsible for the fate of the kidnapped soldier.
So you have various different leaderships, various different compromises and conflicts here.