The parents of terminally ill toddler Alfie Evans have asked the Supreme Court to consider an appeal on ending his life support for a second time.
On Monday the Court of Appeal refused to overrule a decision that would allow the 23-month-old from Merseyside to have treatment in Italy.
Protesters stayed away from Alder Hey after a plea by Alfie's dad Tom Evans.
Alfie has been in Alder Hey since December 2016 with a rare undiagnosed degenerative neurological condition.
In recent days, police received reports of intimidation and verbal abuse after hundreds of supporters gathered outside the hospital in recent days.
Mr Evans, 21, and Kate James, 20, from Bootle, Merseyside, have lost legal fights in the High Court and the Court of Appeal, as well as bids for an appeal to be heard rejected by the Supreme Court and European Court of Human Rights.
A Supreme Court spokesperson confirmed that Alfie's parents have filed their latest application to appeal.
"They wish to challenge the decision that Alder Hey Children's NHS Foundation Trust can withdraw artificial ventilation from their child."
Lord Justice Davis said doctors had agreed that there was "no hope" for Alfie and said: "We cannot have a kind of legal 'Groundhog Day' where you come back again and again and again on the same point."
Speaking about the hospital protest, Mr Evans said in a statement posted on Alfie's Army Facebook page: "Deepest apologies from me and Kate James to the parents and staff affected by the protest/demonstration, it wasn't mine Kate's or anyone else's intention to harm or cause conflict or upset.
"We just wanted to take our son to give him the chance he deserves."
The Christian Legal Centre, which is representing Alfie's parents, submitted written submissions by the 16:00 BST deadline.