Ngô Quốc Anh

March 1, 2011

The implicit function theorem: A PDE example

Filed under: Giải Tích 3, PDEs — Tags: — Ngô Quốc Anh @ 23:29

This entry devotes an existence result for the following semilinear elliptic equation $-\Delta u + u = u^p+f(x)$

in the whole space $\mathbb R^n$ where $0.

Our aim is to apply the implicit function theorem. It is known in the literature that

Theorem (implicit function theorem). Let $X, Y, Z$ be Banach spaces. Let the mapping $f:X\times Y\to Z$ be continuously Fréchet differentiable.

If $(x_0,y_0)\in X\times Y, \quad F(x_0,y_0) = 0$,

and $y\mapsto DF(x_0,y_0)(0,y)$

is a Banach space isomorphism from $Y$ onto $Z$, then there exist neighborhoods $U$ of $x_0$ and $V$ of $y_0$ and a Frechet differentiable function $g:U\to V$ such that $F(x,g(x)) = 0$

and $F(x,y) = 0$ if and only if $y = g(x)$, for all $(x,y)\in U\times V$.

Let us now consider $X=L^2(\mathbb R^n), \quad Y=H_+^2(\mathbb R^n), \quad Z=L^2(\mathbb R^n)$.

Let us define $F(f,u)=-\Delta u + u - u^p-f(x), \quad f \in X, \quad u \in Y, \quad x \in \mathbb R^n$.

It is not hard to see that Fréchet derivative of $F$ at $(f,u)$ with respect to $u$ in the direction $v$ is given by ${D_u}F(f,u)v = - \Delta v + v - p{u^{p - 1}}v$.

Since $-\Delta +I$ defines an isomorphism from $Y$ to $Z$, it is clear to see that our PDE is solvable for $f$ small enough in the $X$-norm.

1. 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 $F(x_0,y_0)=0$, there exists a mapping $g$ between neighborhoods of $x_0$ and $y_0$ such that $F\big( x,g(x) \big)=0$. In the example above, the point $(x_0, y_0)$ is nothing but $(0, u_0)$ where $0 \in X$ is a function and $u_0$ is a solution to $-\Delta u+u=u^p$. As such, by a small neighborhood of $0$ we mean functions $f$ such that $\|f\|_X \ll 1$. 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

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

3. 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

4. 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

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