This entry devotes an existence result for the following semilinear elliptic equation
in the whole space where .
Our aim is to apply the implicit function theorem. It is known in the literature that
Theorem (implicit function theorem). Let be Banach spaces. Let the mapping be continuously Fréchet differentiable.
is a Banach space isomorphism from onto , then there exist neighborhoods of and of and a Frechet differentiable function such that
and if and only if , for all .
Let us now consider
Let us define
It is not hard to see that Fréchet derivative of at with respect to in the direction is given by
Since defines an isomorphism from to , it is clear to see that our PDE is solvable for small enough in the -norm.