UX designers will face many challenges in general but they can’t be considered as problems. Each company, working environment, peers, projects, clients, requirements everything will be a challenge and this does not just apply to a designer. Any designation will have challenges in their own realm but the solution lies in the challenge itself.
Typical problems if we have to say a UX designer will face is working in a team where design is not given much importance. When people think the design is all about making fancy things, applying colors, making some sketches then the designer will really get pissed off. Constantly facing such situations where design is underrated then it can be considered as a problem rather than a challenge.
Many designers will face the problem of not liking the work they do, they either stay in the company for the sake of salary and other dependencies they have. In this process, they get tuned to working with compromises and eventually dislike their career. Make a right choice, always work in a place where you have freedom on your work and enough time to keep the brain at peace.