You are going to want to calculate an integral two different ways

First note that

The right hand side of the above equation is non negative so we get that

Now consider the flux integral out of the disk

This is zero because u=0 on the boundary.

Now if we use the divergence theorem we get

This gives

Combining these two equations gives

So the integrand must be equal to zero so

If it is the first cases we are done if it is the plug that into the pde and conclue.