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.