I feel like I'm stuck between a rock and a hard place with

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I feel like I'm stuck between a rock and a hard place with my friend.

Background: we work together in the same role. He has depression, and as a result, he hates his job and where we work - he thinks he's shit at the job, that no one likes him there etc. It doesn't help that I'm actually really good at the job and that he sees me getting regular praise from our boss and being asked to take on additional responsibilities.

Where we're working isn't where I want to be either, it's just not going to lead me to where I want my career to go. And honestly, I'm getting sick of working with him. Because of this and because he hates it there, we're both in the process of looking for a new job, and we're both looking for similar jobs.

I've been doing this by sending out my CV (approaching places is the best way to get a job in our field), and I've been getting some encouraging responses. Meanwhile, he's only applying to advertised jobs and getting rejected from every single one of them. Obviously, this doesn't help his mental health and self esteem.

I've offered to help him out by looking at his CV and encouraging him to approach places he wants to work directly, but he just comes up with excuses to not do stuff. On weekends it's "I don't want to think about that right now, my head's not in a good place" and on weekdays it's "I'm tired from work".

Now the problem is that if I find a new job, it's going to affect his self-esteem. I don't want to leave him alone there either, since he hates the place. I could let him know about places that I've approached that are hiring, but if it's a place I'm applying for then he'd be competing with me, and if I'm being honest, I'm much more likely to get the job. This would affect his mental state even more.

>TL;DR: friend I work with wants to find a new job, but isn't putting in effort in applying for new jobs
Any advice?