The Gunners' hopes of ending their silverware drought, which stretches back to 2005, took another blow on Saturday when they were knocked out of the FA Cup by Championship side Blackburn.
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The Arsenal fans voiced their discontent at the end of the match at Emirates Stadium but Wilshere has urged the club to unite and repeat the Blues' unlikely European triumph.
"Chelsea are an inspiration for everyone," Wilshere told reporters. "They weren't having a good season and OK they got a bit lucky in some games.
"But they showed great character and we need to do that. We need to come together as a team. We have good leaders here, a good team atmosphere. That's going to help us.
"Our fans are always great, especially in Europe. They love the European nights here. We need everyone behind us that night. They will be naturally disappointed and we accept that. But we need everyone behind us."
Wilshere admitted that Arsenal would need to quickly bounce back from the disappointment of going out of a second cup competition to lower-league opposition.
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Arsene Wenger's team had previously been knocked out of the Capital One Cup at the quarter-final stage to League Two outfit Bradford.
"It's disappointing to go out of the FA Cup," he rued. "We know what this trophy means in England and we wanted to go all the way.
"Everyone said it was our best chance of a trophy. But obviously it wasn't! We have to regroup now. We have a big game here against Bayern. We need the fans behind us.
"We know what Bayern are capable of. They have a great team. We have to show our character. We've beaten big European teams here before so we need to do it again."