# Smooth dependence of solution to elliptic pde depending on parameter

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.

• 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.
– GabS
Aug 12, 2020 at 15:09
• 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 $t$. Aug 13, 2020 at 1:04