"face various challenges" is grammatically correct and commonly used in written English.
Example: Our company must face various challenges in order to stay competitive in the market.
"China is and will face various challenges and risks during economic expansion.
The sector continues to face various challenges ranging from financial constraints to a shortage of skilled labour.
And those who make it will face various challenges: they will most likely have to walk some distance, the weather could be harsh, the crowds will be huge and accommodations are scarce.
Researchers conducting clinical studies face various challenges.
Current North American electric power systems face various challenges: –Distribution.
Internationalization is a complex process, in which firms face various challenges that may lead to opportunities or threads.
ASUNCIÓN, Paraguay — Horacio Cartes, a Paraguayan tobacco magnate, faced various challenges during his presidential bid.
The vast scale of the trip meant the three individuals faced various challenges in terms of illness, diet and threats from individuals they encountered along the way.
The central character, an Indian woman named Radhika, faces various challenges with the assistance of players, who can help out with donations of virtual goods, for example.
