My Blog: Anh-Tourguide: The Implicit Function Theorem: A PDE Example

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 $0<u \in H^1(\mathbb R^n)$ .

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 $(x,y)\in U\times V$ .