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.If

,

and

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.