"To endure this challenge" means to persevere or withstand a difficult situation or problem without giving up or being overwhelmed by it. Full definition
We might even be able to do more than endure these challenging times. (novofoundation.org)
But the fact that he had to endure the challenge left his leadership damaged. (theguardian.com)
«There are 1,001 ways to highlight the gulf in resources between PSG and their domestic rivals, but the comparison with Lille is of particular interest because the 2011 champions were expected to mount the most enduring challenge to [Carlo] Ancelotti's men this season. (tomwfootball.com)