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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.

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  • $\begingroup$ Did you look at the paper of Zhang Liqun (Communications in PDE 1992) titled "Uniqueness Of Positive Solutions In A Ball". I believe there is a section on non-degereneracy. $\endgroup$
    – GabS
    Commented Aug 12, 2020 at 15:09
  • $\begingroup$ i did not look at that exact paper. My question is not very well posed since I really have a different example in mind. The above example the radial solution is decreasing in $r$ and then I know how to prove non-degeneracy. So even though my end goal is to prove nondegeneracy I am really more asking about this dependence on $t$ and smoothness in $t$. $\endgroup$
    – Math604
    Commented Aug 13, 2020 at 1:04

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