I have a generic type question, but I will pose it with a specific example. Suppose $1<p<\frac{N+2}{N-2}$ and $B_1$ is the unit ball in $ R^N$ for $N \ge 3$. Consider the pde $$-\Delta u(x) = t u + u^p$$ in $B_1$ with $u=0$ on $ \partial B_1$ where $ 0<t< \lambda_1$ (the first eigenvalue of $ -\Delta$ in $H_0^1(\Omega)$). Lets assume $ 1< \lambda_1$ and lets index the solutions $u=u_t$. By some variational techniques we prove the existence for all $t$ is the desired range ((so lets assume they are some sort of ground state solutions)). One would expect some smoothness in $t$.
So this is exactly my question.
``Can we typically assume the map $ t \mapsto u_t$ has some smoothness in $t$ for $t$ near $1$?
My end goal is to try and show the solution corresponding to $t=1$ is nondegenerate (in the sense that the linearized kernel is trivial) and hence of course I don't want to try and invoke the implicit function theorem (since if I could ...then i would already have proven my end goal).
With some fairly standard methods I seem to be able to proof the following: if the solution is smooth in $t$ (say existence of one derivative) then in fact the kernel is empty at $ t=1$.
