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.

Hi,

Why, in the end, f should enough small?

Comment by R.T — February 5, 2014 @ 13:30

Hi there, the IFT basically says that as long as , there exists a mapping between neighborhoods of and such that . In the example above, the point is nothing but where is a function and is a solution to . As such, by a small neighborhood of we mean functions such that . I hope this is clear for you now.

Comment by Ngô Quốc Anh — February 5, 2014 @ 22:51

Why is $-\Delta+1-u^{p-1}$ an isomorphism? Could you elaborate the bijection here?

Comment by Han — November 28, 2021 @ 14:17

[…] Please see this page: The implicit function theorem: A PDE example. […]

Pingback by Implicit function theorem and PDE; do we get uniqueness? - MathHub — April 4, 2016 @ 15:24

Hi, I have a couple of questions.

1. For what values of p can this method be applied to your example?

2. Is there anything special about using R^n as the domain. That is, can we still establish the existence for any subset of R^n?

Comment by Melusi — August 10, 2016 @ 17:43

Hi, what if f is large in the X-norm. How can we say that there is a solution? iteration?

Comment by boytaehun@naver.com — October 20, 2016 @ 13:26

Thanks for your comment, it is likely that such a result cannot hold in view of a perturbation result.

Comment by Ngô Quốc Anh — October 21, 2016 @ 10:12