Here is the question:

"Consider the boundary value problem in lR^2:

Laplacian(u)=(1/r)*d/dr(r*du/dr)=c .for r^2=x^2+y^2<1

du/dr=2 .on r=1

Show there is no solution unless c=4, and find solutions in this case."

I think I am supposed to use the Fredholm alternative:

L(u) = f has a solution if and only if <f,v>=0 for all v in ker(L*) where L* is the adjoint.

My problem is basically finding the boundary conditions for L*. Since <u,v> involves the integral between 0 and 1, and so the 1/r is infinity when r=0.

I get my adjoint problem to be:

v''-d/dr(v/r)=0

subject to

[v(r)u'(r)-u(r)v'(r)+u(r)v(r)/r](1,0)=0

where [f(r)](1,0)=f(1)-f(0)

You can see the third term doesn't really work when I stick in r=0.

Any help would be much appreciated.