Solskjaer watched his side comprehensively beaten at Goodison Park on Sunday, with Richarlison, Gylfi Sigurdsson, Lucas Digne and Theo Walcott all on the scoresheet.
The defeat was Man Utd's sixth in eight matches, as the club continue their fall back to earth after Solskjaer won 10 of his first 11 matches in charge.
After the game, the Norwegian made a startling admission about his players' desire to play for Man Utd.
Solskjaer was asked if he felt his players cared enough to play for the club and replied: "I don't know, I don't know.
"You've got to ask them, I've asked them. You're not going to get the answer from me. But if you want to play at this club it has to mean more.
"I want them to be the hardest working team. That's what we were, under Sir Alex [Ferguson].
"You ran more than anyone every single week, a shift up and down that pitch for your teammates, and we have to stick together as a team.
"Giggsy [Ryan Giggs], Becks [David Beckham], Gary Neville, Dennis Irwin, whoever. [Regardless of] the talent you've got, you ran more than anyone. Every single week you'd shift up and down that pitch for your team-mates.
"We have to stick together as a team. We can't change the whole squad but it's one step at a time. I'm going to be successful here and there are players there that won't be part of that successful team, but there are many of them that do have it."
Solskjaer could only offer his apologies to the traveling United fans before turning his attention to a massive week ahead as the club continue their chase for a top-four place.
"That's the biggest incentive we can have, Old Trafford at home in front of our fantastic fans," Solskjaer added.
"[The fans] were unbelievable again and they never, ever moaned at our players which they could have. They've been supportive and I'm sure they will be on Wednesday night.
"We've got to focus on these two games. We're still in with a chance to get the points we need for the top four so sometimes, it sounds too easy to say but we have to look forward.
"You cannot dwell too much on this."