I have a question in mind but let me generalize it slightly.
Suppose I am looking at some pde like $$-\Delta u + t u = f(u)$$ in $B_1$ (here $u=u(x)$) with $u=0$ on $ \partial B_1$ where $B_1$ is the unit ball centred at the origin in $R^N$. For concreteness lets assume $f(u)=u^p$ where $p>1$. Lets assume I can show for all $t>0$ there is a bounded positive radial solution of the given pde.
My goal is to prove that for $t=1$ the solution is nondegenerate; meaning the kernel of the linearized operator is trivial. Now since this equation is not exactly the one I have in mind
I don't want to prove directly that the solution is nondegenerate since this might not extend to my case.
Using some other tricks I believe I can show that the solution for $t=1$ is nondegerate provided these solutions indexed by $t$ are sufficiently smooth in $t$. I believe the usual method to prove smoothness in $t$ is to use the implicit function theorem (or something close) but of course I can't do that here since I am really trying to prove one of the hypothesis. Any comments would be greatly appreciated.