Showing a solution of elliptic PDe is non-degenerate 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
 A: $L$ is invertible if $pr^\alpha u(r)^{p-1}$ is small in a certain sense, cf. Gilbrag-Trudinger. In the general case I don't know anything except to mention the Fredholm or spectral theories. If you need a result for a particular $u$ (and $\alpha$ and $p$) perhaps there are some rigorous numerical methods. I would be interested in other answers.
Update: One potential approach would be to show first that any solution must be radially symmetric and then use ODE methods to the equation obtained.
A: Here I explain where the original problem is coming from,  since it is quite possible I don't need what I am asking for. 
I am interested in obtaining positive solutions of the equation  $ -\Delta u(x) + t H(u) = g(t,x) u^p $ in $B$ with $ u=0$ on the boundary of $B$.  Here $H$ is some second order linear elliptic operator, $t \in R$.   My interest is in obtaining positive solutions $u$ for a various values of $t$ close to zero (I don't care if I can only get solutions for $t$ on one side of zero.   Also $ g(0,x) =|x|^\alpha$.   Also $p$ is in general supercritical.  So what I was attempting to do what perturb off of the radial positive solution using the implicit function theorem,  and this was why I needed the non-degeneracy.    But I would be happy to use another abstract result that would allow me to obtain solutions by perturbing off of the radial solution.   
thanks for all your help. 
