Dear Mathoverflowers:

I am interested in radial positive solutions of $-\Delta u(r) = r^\alpha u(r)^p$ in the unit ball in $ R^N$ with $ u=0$ on the boundary.

Here $p>1$ and $ \alpha >0$. (There is a positive solution provided $ p<\frac{N+2+2\alpha}{N-2}$, Ni 82).

I am interested in when I can say the associated linearized operator $L:= -\Delta - p r^\alpha u(r)^{p-1}$ does not have zero as an eigenvalue.

Are there any standard methods for attempting to show this?

thanks in advance.

craig